Hi, in this lecture we will start a new chapter on non-linear equations. In this we will today discuss a very basic and very interesting method called bisection method. Before going into the method let us define our problem and then we will see some important overviews of the methods that we are going to discuss in this chapter. One of the most frequently occurring problems in scientific work is to find the roots of equations of the form f of x equal to 0, where f is a given function. In our lectures we will take the function f to be defined on a closed interval a b and it is a real valued function. We will also assume that f is a c 1 function defined on a b. What is mean by c 1 function? f should be differentiable at all the points in the interval a b and its derivative should also be a continuous function in the interval a b. This we will always assume throughout this chapter and also we will assume that all the roots of the equation f of x equal to 0 or isolated roots. What is mean by isolated? Suppose you have a function f of x whose graph looks like this say for instance this is the graph of the function f. Then the point of intersection of the graph of the function f with the x axis say it is denoted by r is called a root of the equation f of x equal to 0. When we will say that this root is isolated when we can find a small neighbor root of this root r such that there is no other root of the equation exists in this interval. Then we will say that r is an isolated root. So, we will always assume that our equation has isolated roots. For instance the function whose graph looks like this has two isolated roots r 1 and r 2 and you take another function something like this and then it touches x axis and stays there for some time and then goes. Now you take any point here it is a root of this equation. Now can you find a neighbor root of r in which there is no other root life no because the graph of this function touches the x axis and then stays in an interval and then again it goes up. Therefore you take any small interval however small it is it will always have many other roots of the function. So, this r is not an isolated root right. So, now I hope you understood what is mean by an isolated root and of course, I have already defined what is mean by a root of this equation. Now our interest in this chapter is to device methods that can give us approximation to the root of the equation. What is mean by an approximation to a root? Well to call a point say let us call it as x star if you want to call x star as an approximation to a root r say then we need to check two conditions. One is mod r minus x star should be very small and when you plug in that x star into your function the value that comes out should be very close to 0. How small it should be and how close it should be these are all depends on one's own interest, but generally once you fix what is something that is tolerable to you with respect to that tolerance level you will see whether your x star can be considered as an approximation to your root or not. Generally what we will do is we will take a small number epsilon we will call it as the tolerance parameter. It can be something like 10 to the power of minus 2 or 10 to the power of minus 3 or something like that small number such that we will check whether r minus x star is less than epsilon. Of course, when you are working with numerical methods this kind of conditions are not possible because we do not know what is r that is why we are going for an approximation. Therefore, practically this is not possible for us to check nevertheless this is a criteria just theoretically to accept some number x star as an approximation. Well the next condition is that f of x star should be very close to 0 maybe we can also check f of x star is you can take this as modulus is less than epsilon can also be checked. Now the question is why are we having these two conditions why cannot we just impose this alone. Generally we will be tempted to put only this condition because that itself intuitively tells us that x star is very close to the root, but sometimes what happens is the graph of the function may be like this and then it increases very rapidly in a small neighborhood of the root or now you may be capturing your x star pretty close to r, but its value may be very large this may be something like say 1 or something like that. In that case even though x star is very close to r your function value may be very large and therefore, there is no point in declaring x star as an approximation to r when f of x star is something like order of 1 right. So, that does not seems to be nice therefore, this first condition alone is not enough for us to declare that x star is an approximation. So, we also need x star to be in such a way that the function value is almost 0 therefore, we also need this condition. Now you may ask why cannot we take only this condition and not check this condition well that also has some drawbacks. Let us take another example where the graph of the function goes like this it goes very close to x axis and then goes back and then comes and hits the x axis somewhere here and therefore, by definition the root of this function equation is r which is here. Now if you impose only the second condition that is if you only impose f of x star is approximately equal to 0 or if you are equivalently imposing this condition when you are working on a computer then you may tend to capture a points which is here to be a approximate root for your equation because this distance may be less than epsilon and therefore, you may wrongly capture a point very near to this as a approximation to the root whereas, the actual root is far away from your point x star right. Therefore, to declare that a point x star is an approximation to your root you have to theoretically at least impose these two conditions. In our course whenever we say that a point x star is an approximation to a root of the equation f of x equal to 0 it means mod r minus x star is something very small and f of x star is also very small. In this chapter we are interested in developing iterative methods to capture a isolated root of our equation right. Now what is mean by iterative methods well by this time we know what is mean by a iterative method because we have already introduced some iterative methods in linear systems as well as in the section on computing eigenvalues and eigenvectors right. Therefore, we know what is mean by iterative methods, but still we will repeat it once again the key idea is to start with an initial guess sometimes the initial guess is just one point or sometimes it may be more than one point also. In Jacobi method and Gauss-Seidel method and all we started with only one point, but here in non-linear equations you will see that there are some methods where we will start our initial guess with two points. Therefore, we will say that the initial guess may be with one point or more than one point and once you have this then you have a procedure that is some formula will be given where you will have x n plus 1 given in terms of x n, x n minus 1 and so on up to some x n minus m ok. So, if it is only one point that we take as initial guess then t will only depend on x n if you are taking two points then it will depend on the previous iteration and previous to previous iteration and so on, but we will have a formula different formula leads to different methods right Jacobi method had a formula which was written as b j x n plus c j right that is what we called as x n plus 1 in the case of Jacobi method and similarly in the Gauss-Seidel method we have another formula right that was denoted by b capital G x n plus c g. Similarly, different formulas for t will lead to different methods in this chapter we will try to derive different formulas that can generate sequences that are further seen to be converging to an isolated root of our equation. Well therefore, what is the outcome of an iterative process the outcome of an iterative process in general will be a sequence of numbers and then the question is will this sequence converge and if so what can you tell about the limit of this sequence that is can we say that the limit of this sequence is an isolated root of our equation. These are the questions that we have to answer. So, our task is to develop methods and then answer the convergence questions right. So, that is our interest in this chapter that is we have to devise the procedures iterative procedures and then do the convergence analysis for this procedures. Generally the convergence criteria depends on mainly two facts one is of course, the function that defines our equation f of x equal to 0 and also the convergence criteria depends on the domain in which we are interested in searching for a root and also it is co-domain. Well let us classify the iterative methods that we are going to learn in this chapter the first one is called the bracketing methods. Sometimes they are also called closed domain methods you should not confuse the word closed to domain with closed interval or something it is just a name it is better to use this word bracketing methods. What this bracketing methods does let us see in bracketing methods we will learn two methods one is the bisection method and another one is the regular falsi method. The advantage of the bracketing methods are that they always converge that the sequence generated by these methods will always be a convergent sequence that is the advantage of these methods, but the disadvantage of this methods is that to start this method you have to locate a root of this equation. What is mean by that as an input of this method you have to give an interval say a naught b naught such that there exist an r in a naught b naught such that f of r is equal to 0 that is you have to give an interval in which you have to give an interval in which at least there is one root of your equation this is generally very difficult for you to find you can only find it by trial and error or plot the graph of the function and see where it intersects the x axis and then given interval around that these are something which is not very easy especially you cannot optimize this idea in such a way that a computer can capture the root and give an interval around that. So, this mostly have to be done manually and also it is done mostly by trial and error in that way the bracketing methods are not very commonly used because of this disadvantage and therefore it is always prefer to go for unbracketing methods or open domain methods. In open domain methods we will study three types of methods one is the secant method another one is the well known Newton-Raphson method and fixed point iteration method. Again the advantage of open domain methods is that you do not need to locate the root of the equation in order to start the procedure you can just start with any initial guess chosen arbitrarily just like what we did with the Jacobi method and the Gauss-Seidel method right in those methods we never chosen the initial guess with any property they come as arbitrary vectors similarly in all these three methods we do not need to choose the initial guess with any properties imposed on them you can just start with arbitrary initial guesses, but the disadvantage is that the sequence that they generate may or may not converge that is the problem. Therefore there is always advantage and disadvantage in both these methods one needs to choose the methods according to their needs. Suppose in certain problems you may know an approximate location of your root or in certain problems you want to capture that root with certain properties say for instance you may be given an equation f of x equal to 0 and you want to find the smallest positive root of the equation. Suppose that is given to you may choose the initial guess as 0 and some very large number and then once you choose that then your bracketing methods will surely converge that is what the advantage is they may converge slowly, but certainly they will converge whereas in the open domain method that confidence is not there you may land up getting a diverging or oscillating sequence. So, these are the disadvantages and advantages of these methods while learning we will come to know more about these methods their advantages and disadvantages. So, that we can efficiently use them in our problems as per our need right ok. With this small introduction to a general set up of the iterative methods for non-linear equations. Now, let us go on with some specific methods as I told we will be introducing two methods in the bracketing type of methods and three methods we will discuss in the open domain methods. Let us start our discussion with bi-section methods. Bi-section method is a very interesting and geometrically intuitive method we always prefer to introduce the concepts with something which is geometrically intuitive that is why we took bi-section method as our first method. As I told you have to first locate an interval in which there is at least one root of the equation f of x equal to 0 right. So, your input to this method is of course, the function f that you have to give and then you should also give an interval a naught b naught such that there is at least one root in this interval well I will not always say isolated root because we will always work with those equation which has isolated roots. So, by the word root I always mean isolated root in this chapter. So, how to get this interval is the question suppose the graph of the function is given like this is your y is equal to f of x graph and now I want to find a interval that has this root say r. Now how will you find if your function f is a continuous function again if you recall we have put this condition at the beginning of this lecture itself that we will always assume f to be continuous.  But to define the problem and define the method you just need to have f to be a continuous function assume that f is defined in the interval a naught b naught and how we captured that a naught b naught is that they are found in such a way that f of a naught is less than 0 and f of b naught is greater than 0 or you can also have the other way around that is f of a naught is greater than 0 and f of b naught is less than 0 any of these two conditions should hold. This is equivalent to saying that the product of f of a naught into f of b naught is less than 0. Therefore, we have to search for two real numbers a naught and b naught such that f of a naught into f of b naught is less than 0. Generally, there is no automatic way of finding a naught and b naught especially on a computer one has to do with trial and error either manually or even on a computer you may do it with different trial and error. So, that makes the algorithm much more costly if you go for coding this idea or one has to manually find it and feed it to your code. So, that is the main disadvantage of the bisection method. Also note that these two conditions are the sufficient conditions for the convergence of the bisection method. In fact, these are the sufficient conditions for the regular falsi method also that is to say that under these two conditions these two methods always converge. Well, bisection method is a very nice intuitively clear and also geometrically interesting method and we will discuss this method in our next lecture. Thank you for your attention.


Hello, everyone. My name is James Blair, and this is probably going to be the weirdest thing that you'll see today. I'm going to talk about testing container applications with a new system that I've created called Cogate. Cogate is end-to-end testing for container-based applications, and the idea is that when you open a pull request on GitHub, Cogate will build your images, create an ephemeral Kubernetes environment, and deploy your application, run any tests that you have, merge the changes when you approve them, and after they merge, publish new images. So if you're familiar with Zool, this is very similar to what Zool does, but it is targeted very narrowly to Kubernetes applications, and it does a lot of stuff behind scenes to make this easy for you. So it uses Zool to do speculative image builds, create speculative test environments, project gating, and it has the fabulous auto-hold feature, which if you've been here for the previous talk, then you've learned all about. Auto-holds let you not only log into a Kubernetes cluster to debug your application, but in this case, you can use them as preview environments. So if you want to see what your application looks like, then you can have Cogate deployment and deploy it in one of these ephemeral Kubernetes clusters and then use the auto-hold to preview what it's going to look and function like. So what are speculative builds? These are container builds that depend on unmerged changes in other projects. So let's say that you have container images for your application, and those container images are based on other container images. For instance, you might have a base image that you use for all of your images, and then from that base image you build on top of it. So with Cogate, not only can you, when you open a pull request, can you say build this container image with this pull request, you can actually have that pull request depend on another pull request, possibly even in another repository. And Cogate will not only speculatively build your container image for the pull request that you're opening, but any that it depends on as well. There's really no limit to how deep this wrap-a-hole goes. Speculative test environments. So once we've built these images, Cogate is going to spin up a Kubernetes. It's going to deploy the images that it just built. Again, these changes aren't even merged yet, but it's going to use whatever deployment methodology you tell it to, to deploy and run your app, and then run any tests that it needs to, and then destroy the cluster at the end of this process. When you put all these things together, you get something that we call project gating. And this is when you take a change that is a change to your application, you want to test it, and then automatically have it merge as soon as it passes tests. So basically, you can open up GitHub pull request, review it, say, this looks good to me, and then Cogate is going to deploy your application with it, test it, and if it passes tests, then automatically merge it. The auto-holds feature that I mentioned earlier, like I said, you can use that deployment as an interactive preview environment or use it to diagnose failures. So I'm going to show you a little bit of what Cogate looks like. It's not a full demo, but it's a lightning talk walkthrough so you can get an idea of what this is and how it works. So you can log in with an email address or, of course, log in with GitHub. So it integrates with everything that you've got there. Once you've logged in, this is the list of projects that it has access to. You can fairly easily turn it on and off for any of your GitHub projects. You can also, you'll notice there's a branches column there. You can tell what branches to act on. So it can act on all of the branches or you can tell it to only deal with protected branches. This is sort of depending on your workflow, whether you use the protected branch workflow or not. You can set that there. So for this example, what I have here is a very simple application that has two container images and they're in two different repos. The application is a calculator and it's a client server-based calculator because, of course, why wouldn't you use a client server calculator? So there's a calculator server and a calculator client. The client just doesn't arrest call to the server to do arithmetic. So the first thing that we need to do is take a look at our calculator server app and tell Cogate to build an image. This is all it takes to do that. There are some other options if you need them. But if your repo has a Docker file in it, then if you add this Cogate.yaml to it, this is all you need to do to tell Cogate build this repo into an image that is called calculator server. Oh, yeah, I made that bigger so it's easy for you to read. Sorry. So once you add this file and open up a pull request for it, Cogate is going to immediately start building it. So this is something I think it's pretty neat. Basically, any changes that you make to Cogate configuration, they don't have to merge before it starts using them. So you can just propose this change and Cogate will build an image for it. And going back to the Cogate interface, that's what this looks like. Here's the build result. You can see up there that it knows that it built the Docker Hub calculator server image with the latest tag. And if you want, you can click on the View Logs button and see the build blogs. But you know what those look like. So the next thing that we need to do is add our second image. So we added the calculator server image before. Now we're going to switch to the other repo and tell it to build the calculator client image. We also need to tell it to how to actually deploy this application. So I'm going to remember that I have a zoomed in version this time. So this is the Cogate YAML file for the calculator client repo. You can see that we define the image there at the top. So that's just like the calculator server. And then underneath that, we define our deployment. So this is where most of the magic happens. Within that deployment, we tell it what images that we use. So we're going to tell it that we need the calculator client and the calculator server image. And Cogate is going to use that so that whenever you make a change that causes either of these images to be built, it knows that it's going to need to run this deployment. Then we tell it what kind of platform we need. So this can run on a single node Kubernetes cluster. So the platform here is Kubernetes one node. Over time, Cogate will grow any number of different platforms. It's pretty simple right now. We're kind of like in a beta test phase. But we're going to get multi-node Kubernetes systems. We could use EKS, AKS, GKS, all the KSs. And then we tell Cogate how to actually deploy it. So in this case, we're telling it to run kubectl apply. So you can deploy this app by doing kubectl apply deploy.yaml. And then we tell it what. So Cogate expects something to run. That thing needs to not only deploy the application, but also do some kind of evaluation and return success or failure for whether this works or not. And so we, as part of the deployment, we have a job that actually does some arithmetic. And that job, then we just need to wait for that to complete. So there's other methods of application. Instead of doing kubectl apply, you could also use customize. You could create what we call an action container. So that's basically, in this instance, that would be a third image that gets built. And inside of that image is all of your deployment code. So if you wanted to run Helm or something like that, then what you would do is build an image that runs your Helm charts. So with the concept of an action container, it's very powerful. You can basically do anything that way. So this is what our deployment build looks like. It's a little bit different than the image builds. You can see that not only it tells you which image builds were used, and not only can you see the logs, but down here at the bottom of the page, we actually have, and Cogate has an understanding of what resources existed in the cluster during the job. So we've got a little card up there for each of those resources. So it's pretty easy to just click on any of those and get basically the equivalent of a kubectl describe or something like that. So here's what happens if you click on the info icon for the deployment. You get the kubectl describe for that. So you can see all of these Kubernetes resources. You can see all of the logs very easily just from the web interface. Here's our pod that actually ran the arithmetic. So these are the logs from our calculator server doing, adding 1 plus 4. So it's really easy to see the results of the build that way. So let's see. That seems duplicative. Oh, so auto holds. So if you set an auto hold for, that's what this is showing. There's a little hold. So if you flip the hold switch on the build, then you're telling Cogate that you want an auto hold. You want to hold the end of this build. And so once it completes, not only do you have all the things that I showed you before, but there's a button there that says kubconfig. And if you click on that, then it will show you the kubconfig for that cluster. So you can copy and paste that to your local environment and then start running kubectl commands on it and have at it, do whatever you want with the cluster. And then when you're done, you can just hit the delete hold button and it'll release the cluster back from when it came. So that's the quick demo for Cogate. So feel free to take a look at the website. It's got a sort of tutorial walkthrough based on this calculator client server thing. So you can kind of get hands on with it pretty easily. And like I said, this is very new. Consider it in beta. And please let me know if you have any questions or it's not quite working out for you. You need to do something else. I'm very open to feedback. So my email address is up there too. And I've got a little pamphlet here that you can take as well. So thank you very much.


So, welcome to the next lecture. So, in this lecture we shall introduce the class of, we shall introduce the notion of metric spaces. So, metric spaces provides perhaps the most important class of the logical spaces. So, and so after we introduce metric spaces we will also see I mean we have talked about open subsets, closed subsets, continuous maps, we will see we will give descriptions of these in terms of metric, okay. So, let us begin by recalling the definition of a metric space. So, we recall the definition, let x be a set that d from x cross x to the real numbers which are greater than equal to 0 be a function map which satisfies the following three conditions d of x comma y is equal to 0 if and only if x is equal to y d of x comma y is equal to d of y comma x and d of x comma z is less than equal to d of x comma y plus d of y comma z for all x comma y comma z. This is for all x comma y and x. So, this is called the triangle inequality. So, the d is for distance and we know that if we have a triangle. So, this x this is z and this is y then the distance between x and z is less than equal to the distance between x and y and y and z. So, if these three properties are satisfied then so then we say that then the pair x comma d is called the metric space. The map d is called the metric on x or the distance function. So, we just need any function d which satisfies these three conditions. So, given any set x, there could be lots of metrics on x, but the most common example of a metric space we encounter of metric space we encounter is R n with the Euclidean metric. So, that is if we have two vectors in R n. So, x is x 1 to x n and similarly y is y 1 up to y n. So, then t of x comma y is defined to be the square root of the sum of squares of the difference of x and y. So, let us check that this defines a metric. So, let us check that this defines a metric. So, note that the first two conditions are the first two conditions are trivial to check. The only non-trivial condition is the triangle inequality. So, which we will prove now here as a proposition. So, this d of x comma y we shall write this as norm of x minus y, the two norm. So, we will use this notation instead of writing d of x comma y. So, what we need to prove is that with this in this notation that the norm of x minus y is less than equal to or alternatively I mean we can say, we can define let me just. So, if for a vector x in R n we define the two norm of x as summation x i square the square root of this. So, then clearly then I do not need to write this as a definition. So, then clearly d of x comma y is equal to the two norm of x minus y. And check that this defines a metric to check that this defines a metric, we need the following proposition that is precisely the following proposition the norm of x minus y is less than equal to for all. So, let us prove this. So, it is enough to check that and this left as an easy exercise that it is enough to check this for all. So, in order to do this. So, consider the map this inner product defined as. So, we first claim that this absolute value of this quantity is less than equal to. So, let us first prove this claim and to see this. So, let us write let us define w to be equal to y minus t x where t is a real number. And now let us compute So, this inner product of w with w and this is equal to this is bilinear right this is bilinear in both. So, what that means is if we fix x that is fix if we fix x then x comma lambda y 1 plus y 2 is equal to or if we fix x then it is linear in y right. And similarly if we fix y then linear in x right. So, using the bilinearity we get that this is equal to minus 2 t 2 x right. And it is also symmetric bilinear and plus symmetric right because clearly for symmetry x comma y is clearly equal to y comma x. So, using the fact that this bilinear and symmetric we get this about this is precisely equal to t square minus 2 t. So, for a fixed x and y we view this as a quadratic in t right. There is a degree to polynomial in t and as this and as it only takes positive value greater values greater than equal to 0 it follows that the discriminant is less than equal to 0 it follows that the discriminant is less than equal to 0 right. But what is the discriminant? So, the discriminant is 4 is less than equal to 4 into which proves our claim right this is the claim which we want to So, using this claim we will now prove the proposition. So, note that. So, we have x minus y o square is equal to this is equal to and this is less than equal to yeah. So, over here we have used the claim. So, oh sorry we have not used the claim yet this is less we are just taking absolute value this is less than equal to now here we have used the claim. So, this implies it and note that the norm of y is same as norm of minus y. So, therefore, we can just replace y by minus y and we get. So, thus so this shows that this R n comma d is a metric space. So, this proves triangle inequality holds and therefore, this is a metric space. So, this is the main example of metric space. So, now given a metric space how can we use the metric to define a topology on it. So, let us see that. So, given a metric space x we can use the metric to define a topology for that for x in x and an epsilon positive define the open ball. Of radius epsilon around x as follows ok. So, b epsilon x is defined to be those y in x says that the distance of x from y is strictly less than epsilon ok. And we consider the collection there is something which we are familiar with b contained in the power sort of x defined as. So, b is the collection of all these open balls around various points of x x is in x and epsilon is positive. So, it is easily check that. So, it is easily check b satisfies the two conditions for being a basis. So, thus b defines a topology tau on x which has b as basis right. And so, therefore, in other words. So, thus a subset u contained in x is open in this topology if and only if it satisfies the problem following property for every u in x or sorry for every x in u there exists an epsilon positive such that this open ball of radius epsilon around x is completely contained inside. So, this is very similar to the examples that we saw in the beginning of the course that is R n. So, if we have a topological space x over here. So, a set is open if and only if given any point x we can find a ball of radius epsilon around x which is completely contained inside. So, the epsilon of course depends on x. So, it is trivial to check and this is left as an exercise that the topology defined on R n using the Euclidean metric. Is the standard topology and this check is left as an exercise. So, now we want to understand what it means to be in the closure in terms in a metric space. So, in terms of the metric. So, definition let x be a metric space let x n for n greater than equal to 1 be a collection of points of x of points of x. So, we say x n converges to x if for every epsilon positive there is an n sufficiently large such that for all let us say n naught and this n naught depends on epsilon depends on epsilon. So, is that for all n greater than equal to n naught the x n's are contained in the epsilon ball around x. So, this x n. So, I should write x n converges to we say x n converges to x right and it is often written as x n converges to x ok. So, roughly if we have x x is here and we have sequence of x n's. So, we want these this entire the sequence to get closer and closer to x as our n tends to infinity right. So, or precisely what we want is here no matter which no matter how small we take epsilon when we take the epsilon ball around x after finitely many n's all the x n should be in that epsilon ball ok. So, let us prove this lemma let x be a matrix space and let a contain in x be a subset then x is in a closure if and only if there is a sequence x n converging to x and x n belongs to a. So, x is in the closure of a if and only if there is a sequence in a which converges to x right. So, let us take a simple example. So, if we take let us say this region. So, then given any point on the boundary given any point of the boundary we can find a sequence inside this red region this region x square plus y square strictly less than 1 right which converges to this point of the boundary. But if we take some point outside this boundary then we will not be able to find a sequence. So, then we can find a small neighborhood around this point outside which does not beat the set a. So, therefore, we want no matter which sequence we take inside a yeah it will not satisfy the definition of converging to x if we take a point outside this circle x square plus y square is equal to 1. So, that is a that is a small example of what is happening. So, let us see a proof. So, let us assume that suppose x belongs to a closure right. So, let us say x is over here right. So, then by the definition of closure every open subset u containing x meets right. So, take u to be b 1 upon n x. So, we take u n to be this collection right and we choose. So, each u n it meets a because x is in the closure right. So, choose any x n in b 1 upon n this epsilon this 1 upon n neighborhood of x. So, we are taking this we are taking smaller and smaller neighborhoods right. And inside each neighborhood we are choosing each of these neighborhoods it meets this red region and we choose one point in the intersection a yeah. So, then we claim that x is equal to x n converge to x right. So, to prove this what we need to show. So, we need to show that given any epsilon positive there exist some n naught very large. So, that such that for n greater than equal to n naught we have x n belongs to e epsilon x right. So, let us show this. So, note that first choose. So, we can choose n naught sufficiently large so that 1 upon n naught is strictly less than epsilon right. So, then we claim that n naught we have b 1 upon n x is obviously contained in b 1 upon n naught x which is contained in b epsilon x right. And moreover this these balls these are nested and so on right. So, from this and x n is here x n plus 1 is here and so on right. So, all this is contained in b. So, therefore, this shows that. So, this implies for all n greater than equal to n naught x n belongs to b epsilon x which implies that x n converge to x ok. So, conversely suppose so in the first part of the proof we showed that if x is in a closure then we can find a sequence of x n's in a. So, that x n's converge to x yeah. So, and conversely so suppose x n's is a sequence in a such that x n's converge to x then we need to show that x is in a closure right. So, once again we will use the definition. So, let u containing x be an open subset right. So, by the definition of the topology on x which is given by the metric right. So, then there exists an epsilon positive such that b epsilon x is completely contained inside u. So, u is some set open set which contains and let us say x is here. So, we can find an epsilon. So, z b epsilon x is completely contained inside u right. And so as x n converge to x. So, this implies that for all n greater than equal to n naught there exists n naught such that for all n greater than equal to n naught x n belong to b epsilon x right. So, this implies that x n belongs to b epsilon x intersected a which is containing u intersection right. So, thus u intersection a is non empty right. So, thus x belongs to a closure. So, we have given a nice. So, in case of metric spaces we have given a nice and intuitive criterion for what it means for a point to be in the closure of a set right. So, right. So, once again we return to our earlier example. So, the closure of this set we are taking this open region a is equal to x square plus y square strictly less than 1 right. So, the closure of this is exactly the set a closure is equal to x square plus y square is less than equal to 1 ok. So, we will end this lecture here.


The BSU men's basketball team has seemed to turn a corner on their season. After a nine game losing streak, the Beavers have gone three and three over their last six games. Now the focus is on how to turn these improved performances into a winning streak. Over the last three weekends, the Beavers have won on Friday, followed by a blowout loss on Saturday of at least 20 points. The team realizes that it's a troubling trend that has to stop soon in order to rack up some wins and move up the standings. We see that we play a lot better on Friday nights. We've been working on it trying to get that sweet mentality when we go into Saturday nights because if you can get a sweep it's a big momentum in the next week as well. The challenge for us is we just got to play better basketball consistently and you can play good basketball occasionally but we're trying to play consistently better and I think we still have the right mindset and we're still working at it and hopefully we'll continue to grow and get better. The team starts a four game homestand tomorrow against Concordia St. Paul. The two teams are split in the all time series at 16 games apiece. If you enjoyed this segment of Lakeland News, please consider making a tax deductible contribution to Lakeland PBS.


The entire globe, even as it trembles in passion with the birth of new nations and shrinks in the hand of a dispassionate science, is today the site of a momentous conflict. As each side attempts to prove to the world the superiority of its position, the conflict is fought with the words of diplomats, with gestures of friendship and help to uncommitted countries, even with cultural demonstrations. It is fought indeed on every level of man's experience, for the stakes are high. As one of the adversaries in the conflict, we see a challenge as great as any in our historic past, a challenge not we hope to be met and joined in battle, but to be faced and fought in the hearts and hopes of men. It is the challenge of ideas. I am Edward R. Murrow. For a little while, I would like to review with you the great conflict of our times, one which demands and must get the attention and the involvement of each one of us. This conflict is the reason why most of you are in uniform, and some of you will be going overseas, servicemen, families of servicemen and civilians working for the Department of Defense. Because of this conflict, many of you are already abroad, more than a million Americans in 91 foreign countries. First and foremost, of course, Americans are overseas to help preserve the peace. By maintaining an alert readiness to meet the threat of aggression, America's armed forces in concert with those of its allies are serving as a deterrent to war. Some of you may wonder how directly involved in the contest you are or will be. Perhaps your active duty is behind you or soon will be, and with it, you may believe your involvement in the contest ends. But the truth is that involvement in this conflict is far-ranging indeed. It touches not only members of the armed forces and their families, but everyone who supports the mission of the armed forces, every defense worker, every taxpayer who puts the bill, who in fact is not involved. And the conflict itself, how can it be defined? Well, let's look at it this way. The Communist bloc would like to see the entire world under Communist domination. Over the years, as the strength and determination of the free world has gradually convinced Communist leaders that aggressive war would be a reckless and costly gamble, they have begun to talk more and more about their ability to win from us in the arena of ideas. This of course is fine with us, for we are a people with a traditionally great faith in our ideas, the ideas that have moved mountains and created wealth and shaped us as free men. And we are confident that history can do no other than award us the victory in any contest in which ideas are the weapon. But I would like to say this, confidence by itself without effort does not win contests. Victory in this conflict depends on much, much more than confidence.  It is total competition with an antagonist who is putting into it everything within his capability. It is not a conflict between peoples, but between basic values and systems of government. Between the principles of life each believes in. When we talk of these conflicting values, we are obliged to speak of that special quality which we call for want of a better phrase, individual liberty. And when we talk of this liberty in America, we talk with many voices, for we are a diverse nation. And there are perhaps as many concepts of what America is as there are people among us. But although we look at America through different eyes, there are many insights into the essence of our moving and growing nation that we share. I'm John Wayne. Is there any better or equal hope in the world, Lincoln asked, than the ultimate justice of the people? We Americans believe there is not. The stonework of our national life is made of this belief. We believe in many things, but this belief that man is a responsible being bears out our own unique stamp as a nation. As a people we are active and often noisy. We are industrious, often times to the bafflement of ourselves and our friends. We relax as hard as we worked. We are proud. We are sentimental. Beauty is of national concern to us, for some of us its pursuit is a deadly serious past time. The rest of us simply enjoy the results. Some of our national spirit shows up in the monuments we erect, the large ones and the small ones. It rings through the music which animates us on occasion some difference of feeling. But other times we are united. It rings in many more, but above everything else we are free. We believe in the ultimate justice which we as free men can create. Our heritage of freedom is our most priceless possession. And before us have died to keep it alive, and many in our times have done the same. But although men have died to preserve it, may die again. The glory of this heritage is the vision of life that it has given us, a life of dignity and nobility of spirit, such as men have never before been enriched by in the long march through the history they have recorded. There is a phrase which recurs in our national documents, the literature of our story as a people which points to the source of our belief in individual freedom, and defines perhaps better than any other the nub of the conflict between the opposing systems of values which sweep the world today. That phrase is, this nation under God, that phrase and others like it, set the standard of our concept of man. As a creature of God, man is a being with dignity and conscience, with the ability to determine right from wrong and the obligation to act on that right. From this belief in man as a responsible being, flow the beliefs in his other qualifications which we accept, usually even without bothering to think about them, as parts of that vague condition which we call the American way. His ability, indeed his right to explore the truth in all things, his ability to govern himself, and his ability to handle his own economic problems. The opposing point of view in this conflict rests on a fundamentally different vision of man. As a creature not of God, but of the state, in this system the value of individual man diminishes sharply and the state is all important.  What we oppose fundamentally is the aggressive nature of the communist state, its unceasing effort to expand wherever it can, to grow bigger, to take over, to supplant. This deadly impulse toward aggression we oppose as a continual threat to peace. These are the contrasting points of view between the major antagonists in the conflict which has become known over the last decade as the Cold War. For an appraisal of this continuing and protracted conflict, we can go to a reporter who has watched the growing conflict with the perception of a trained military observer. I am Hanson Baldwin. I have here the policy statements of each of our services on the contest we're engaged in with the communists. What they boil down to is this, is a contest for the minds and hearts of people around the world, all people. The areas in white on this map represent the part of the world composed of ourselves and those who share our beliefs. The communists seek constantly to win an advantage in the minds of the people here and to separate us. The black portion of the map is the communist dominated part of the world controlled by the Soviet Union and the Chinese Communist regime. And of tremendous importance is that part of the world which is attached to neither camp. The uncommitted nations of the world represented in gray on the map. For the most part, these nations are absorbed with their own problems, either they are newly independent after years of domination by a foreign power or they are still seeking independence. A strong tide of nationalism sweeps over this part of the world and with it a mistrust of anything they feel is associated with the old ways of colonialism. They are not committed to either side in the conflict. Our own policy toward them has been one strictly of watchful, non-intrusive friendship giving help when help is asked for but otherwise keeping hands off their internal affairs. The communists have been more direct. Their local party members have not hesitated to exploit to their own advantage the passions that smolder in these new states and to identify themselves in the minds of the people there whenever possible with anti-colonialism and us with its opposite. It is part of their worldwide technique. How do our military services adapt themselves to this conflict in which perhaps no gun may be fired? By recognizing that the activities and performances of all their members in every foreign country in which they are stationed will either add to or detract from the goodwill our nation needs in this contest for the respect of the peoples of the world. The president of the United States has called developments of good relationships between United States citizens abroad and members of their host countries basic to the full attainment of our foreign policy objectives and the services have responded by promoting such good relationships as part of their overall policy. The communists initiated the Cold War for the same reason they would have started a hot war had that been feasible. A would now should they ever decide that the free world was weak enough for them to risk it whatever the degree of tension they create it is a substitute for hot war another means of achieving their end and their end of course is world conquest short of war if possible. World conquest by actions short of war actions that are political economic and ideological how do the communists apply these three methods first we will consider the political method and the techniques the communists use when they attempt to acquire their objectives in this way we would do well to consider the success the communists have had by using the political method alone as you will see from the black area on this map they have expanded between 1940 and 1954 took over 15 countries totaling 714 million people since 1954 there have been major internal disruptions in nine additional countries outside the iron curtain now how do they go about applying this political method the communist coup in Czechoslovakia in 1948 provides us with a blueprint of the ideal formula for political conquest which includes taking over a country without outside intervention destruction of its parliamentary form of government murder of its democratic leaders when necessary rig election with only the communist candidates on the ballot once the penetration is complete all this is backed by the naked force of military occupation by such devices the Soviet Union turned its neighboring states into satellite political and economic puppets in China political action was pruder but no less effective here it was political action by Congress and so overwhelming as this political conquest ban that the communists have been able to propel red china to a position of major influence in the eastern world the political force method includes the suppression of freedom in a country already under communist control without outside interference inside the iron curtain there were revolution without help from the outside in East Germany in 1953 Poland in 1956 Hungary in the same year and Tibet in 1959 all of them brutally suppressed and finally there is the international Communist Party dedicated to the subversion and overthrow of every still free government where the political method may not prevail there is always the economic method I'm Frank McGee let's consider for a moment this matter of the communist use of economic methods to achieve conquest and how it's done the present masters of the Soviet Union have said to us peaceful coexistence until we bury you they propose to officiate that our interment by catching up with us industrially and then surpassing us they're working hard at it today the 200 million inhabitants of the Soviet state bend their backs and their talents to the Kremlin's will and Sputnik and Sputnik's successors have proved conclusively enough the USSR's proficiency in the area of technical ability throughout the vast stretches of this empire you find new industrial cities plants assembly lines the most modern equipment Soviet products of every description awaiting delivery and reaching industrially backward areas of the non-communist world teams of Russian technicians are available to operate this machine this is economic penetration paving the way for political takeover the hand of Soviet friendship which usually precedes an attempt at economic penetration has been reaching into every part of the world which shows the slightest inclination to receive it recently we've seen it extend into our own hemisphere into turbulent Cuba where it was embraced as a triumph of Soviet policy a few statistics will demonstrate how serious this threat is becoming to the free world of the entire annual Soviet economic capability their gross national product 70% or it may run as high as 80% is devoted to the area of heavy industry the US and free world share of gross national product devoted to the same area is certainly not much more than 20% in terms of volume applied to this area of gross national product the Soviet Union is beginning to surge ahead finally the communist attempt ideological penetration perhaps their strongest efforts go into this kind of penetration into the free world before we talk about ideological penetration itself let's take a look at the reasons why this form of penetration is so important to the communists for one thing communist aggression has alarmed many free nations and produced their united will to resist these nations have implemented their will to resist all other forms of Soviet penetration through a number of trade agreements the United States and other industrially strong nations of the free world are actively promoting better living conditions all over the world this has the effect not only of helping these countries to improve their industrial and economic position but also of providing a defense against communist entry which flourishes and depressed economic area the military potential of the free world has been revived by such mutual defense packs as a North Atlantic Treaty Organization in Europe the Southeast Asia Treaty Organization to counter aggression in this part of the world the ANZUS Treaty and the Rio Treaty to provide hemisphere defense all over the world our allies are now setting up their own defense bases with the technical know-how to operate them this kind of military help by us is not by any means a one-sided operation the nations involved contribute as heavily as they can no less valuable are the morale and friendly relations being generated between servicemen of all the participating countries in joint training program demonstrating their respective skills to each other they cost the language barrier to comradeship this banding together by nations with a common purpose has produced strength not only in the military and economic field but in the realm of moral and spiritual values as well at a meeting of the NATO Atlantic Congress in 1959 a statement was adopted which called attention to the moral unity binding the member nations this document recognizes respect for human dignity as the inalienable basis of civilization among the fundamental rights of man specifically named are respect for the sanctity of human life the right to freedom of speech of conscience and religion of opinion and believe the right of every man to work and to receive his just reward the right of the family to stability and the right of parents over their children and their education in all these ways economically militarily morally the world outside the communist orbit slowly draws itself together the practical communists appraise this consolidation of free world defenses it confronts them with a serious deterrence to their world ambitions hence they must destroy allied unity or at least weaken it how well here we return to the subject of ideological penetration for if all other forms of penetration don't work if the united strength of the non-communist world makes political penetration unfeasible and economic penetration unsuccessful if cultural penetration does not do the job the communists can always hope to succeed by resorting to ideological warfare against their strongest opponent meaning us American go home Wall Street warmongers down with American imperialism peace yes peace their propaganda screen on communist terms naturally inside the free world of communist and their sympathizers seek to foment suspicion and distrust of our motive even attempt to influence our allies to eliminate our overseas bases which are protecting them as well as us their mediums of mass communication are tireless in vilifying us as a nation and as individuals such as depicting you as a brute and a ravager is there a chance that the communist will succeed is there a possibility that our allies will accept this malicious perversion of us and everything we stand for in large part that is going to depend on how thoroughly we understand this conflict and how dedicated we are to victory understanding is a continuing process and it embraces many different aspects of our day to day living it is at root a function which involves the flow of information I am Lowell Thomas we live amid the greatest abundance of information any people has ever had but survey after survey reveals an astonishingly low informational level among us to be sure the right to remain uninformed is one of the privileges of a democracy but in a contest such as the one we're engaged in it is one of the surest ways of losing our freedom being informed does not mean only keeping up to date on current events important though that is it means also being aware of developments within our own nation and the forces that move through it its weaknesses as well as its strengths we all too often dishonor our freedom and demean it but that's not the full story if it were our system would have collapsed long before this the full story is in the progress we consistently make through law and through education the way free men must the communists would like to have you as well as people in every country in the world believe otherwise and once they have succeeded in letting the idea take root that our deficiencies are our chief characteristics once they have managed to erode our faith in ourselves at least a little that much of the battle is lost a snake of a whisper is abroad that Americans have lost touch with their past is it true it can't be if we are to survive and when if we understand the conflict then we understand the need to keep America strong in our greatest challenge by keeping alive the standards and the traditions which have been her glory what are the bulwarks of our past upon which our society has been built there are many and we have discussed some of them our spiritual heritage our belief in individual liberty but there is yet another as important as any of these I am Helen Hayes many of us who are living now will never see the end of the conflict which dominates our lives leaders change and leadership passes from one generation to the next but for generations or certainly for many years the challenge which confronts us will continue so it becomes not just a hope but an urgent condition for a peaceful future that those who are catapulted into the problems of maturity with every passing year will understand the challenge and devote themselves to it even as you and I must interest cannot lag efforts cannot cease it will require the sternest kind of dedication bread as it were into the American character the training grounds in which this discipline will be planted and perfected are the institutions which have shaped our society particularly the home which has always been in our culture the nucleus of society itself the home is the wellspring of the strongest qualities of citizenship and it must remain so if we ought to preserve the toughness of moral fiber which is our strongest heritage a toughness of moral fiber a belief in the right as God gives us to see the right as Lincoln put it in words which are carved into our national soul these are the sinewy threads of our past keeping America strong this is our challenge keeping the vision of liberty bright keeping the threads of our past the strands of moral toughness with which our history is bound moral toughness the quality which made men work instead of week on the hostile shores of New England which bled mortally yet never died on a hundred fields and oceans stretching out from valley forward which moved to strong men west bringing with them first only their pride then their railroads which built cities and governance and created wealth and above all made freedom the birthright of every man that quality must still live within us or without it we are surely lost and it will live so long as we keep faith in our future and faith in our past this is the challenge as compelling as severe as crucial as Americans in any age have ever faced


Hi guys, so welcome to part two of this left hand rhythm pattern series over a melody in the right hand. So we've already constructed a melody in the right hand. So we're going to take that same melody and build forward. We've looked at the pulse, we've looked at the toggling pulse, we've looked at the quicker pulse or the disco pulse, we've looked at We Will Rock You, we've looked at the Thresio, which is a very nice Latin pattern. The left hand approach is to look at certain beats, very few beats, like maybe one beat or just two beats over the entire bar and just hit them with a lot of power and a lot of energy, right? And I call them as just stabs or choppy stabs. So you go, maybe I just want to hit the two of the bar, one E and a two E and a three E and a four E and maybe the two and the four, one E and a two E and a three and a four E and a four E. Right, I hit that at the two, so it's a little tricky because normally you'd want your left hand to begin at the one but this adds a lot of impact for your song. So you do a stab at two, then you can also do a stab before the two and before the four, that's the end. So it'll be one E and a two E and a three and a four E and a one E and a two E and a four E and a four E. go 1e and a 2 okay you can do that stab those stab points you can also do 1e and a 2e and a 3e or something like that or 1e and a 2e and a 3e and a 4e and a 1e and a 2 right so specific beats you could target in your left hand so this will really enhance your melody and also train your independence on the piano okay another style or another rhythm pattern which will really work is what I call as melodic missing beats so your melody will not have all the beats of the 16th note phrase isn't it there will be gaps and that's what makes a good melody there needs to be some breathing space as well so if there was no gaps it would be rather annoying I guess it would be this sounds like a machine gun shooting so there are gaps in this melody where are the gaps for instance in beat number one itself 1e and a so where the e and the a are gaps 1e and a 2e and a so 2 and the and are in beat 2 those were gaps so you fill in those gaps with your left hand the other hand and it sounds really interesting it sounds very percussive almost like a tabla player or a jembe or a darbuka player of sorts so you could work on maybe something like this a good way to start off would be just hit hit the left hand not on the piano maybe on the body it already creates a groove of its own so you I'm right now hitting all the missing beats you don't have to you can hit a few right but you get the idea I hope you don't need your left hand should never collide with the right hand let the right hand play where it's composed on and the left hand fills up all the missing spaces so let's see how that sounds so it's sort of like a no-brainer for the left hand the pattern of the left hand is already there it's inbuilt it's just what the right hand is not playing so it's easy to build and it's very customized and it will always sound unique because every melody tends to also be unique so the missing beats will also be unique right moving on so let's now look at an extension of the Latin pattern which we learned in part one again if you haven't watched part one please head over and watch it and if you haven't subscribed to our channel it'll be awesome if you can subscribe and stuff so the Latin pattern moving forward from the thresio could be looked at as the clave and we call a clave well there are many claves let me just show you one of them I'll show you the song clave which is one e and a two e and a three e and a four e and a one e and a two e and a three e and a four so that's the hit point it's sort of like a thresio and then a sense of relaxation ta ta ta ta ta ta ta ta ta ta ta one e and a two e and a three and a four e and a four so this is going to be a little tricky to incorporate both so I would suggest first get the left hand this called as the song clave or the three two song clave because it's three beats a group of three beats and then a phrase of two beats we look at it that way one e and a two e and a three and a four e and a one e and a two e and a three right one e and a two e and a three e and a four e and a one e and a two e and a okay and to add more interesting challenges you could also not play the left hand together with root and octave well start with root and octave together and then try to dissipate them play the root ta ta ta ta ta ta ta ta ta ta ta ta ta taa right let's see how that sounds now so that's your song clave another very very popular latin rhythm for you to practice on and it's very scalable you can use these rhythms for or these rhythmic concepts for I'm hoping a lot more melodies to come so that's about the clave and the next rhythm pattern which you could try out in your left hand will be pretty much what I call as following the drummer and essentially you're following the kick drum and the snare drum of the drum player and you try to beatbox it or try to vocalize it in some crude way so that it triggers your left hand to play it. So for example, doom, ta, doop, doop, ta, doop, ta, doop, ta, doop, ta, doop, ta, which is one and two and three and four, right? But when I say doop, I kind of mean kick drum and ta, I mean snare drum. So if you create a drum groove like doop, ta, doop, doop, ta, doop, ta, first say it with the pulse, doop, ta, doop, doop, ta, doop, ta, doop, ta, and now try to play it with your piano left hand. Doop, ta, doop, doop, ta, kick, snare kick, kick, snare kick. So this is an eighth note groove and then the back to the melody. This feels really cool once you get it. It's sort of like there's two instruments happening with one guy or one player. There's like a drum kit and a tune right there. Doop, ta, doop, doop, ta, doop, ta, doop, ta, doop, ta, doop, ta, doop. So right now I'm beatboxing it. Also try singing. Come. Ta, doop, ta, doop, ta, doop, ta, doop, ta, doop, ta, doop, ta, doop, ta, doop, ta, sometimes actually playing the piano may be a bit easier than singing but I'd encourage you to do both because you need to do both. And this feels really cool, doesn't it? The drums there and a melody instrument here. That's what the piano is all about. That's the fun of playing the piano. So if I now try and attempt the melody over the doop, ta, doop, doop, ta drum groove. Singing really helps, moving the body really helps, connecting with what you're playing, enjoying what you're playing really helps. And have fun. Don't worry about the mistakes. The mistakes are just going to get better and eventually it'll be perfect. So you go. That's about following a drummer. You could follow any drum groove, find a few on the internet or jam with a drummer friend and develop what the drummer is playing on your left hand on the piano. Moving forward. We have this idea of playing dotted 8th note pulses, right, a very consistent set of a stream of dotted 8th notes. So a dotted 8th note is where you divide the beat into four equal units, but you don't hit it at 1e and a 2e and that will still be the pulse. You go 1e and a 2e and a 3e and a 4e and a 1e and a 2e and a 3e and so you get very different accents or very different accented points by just giving yourself this dotted flavor or dividing the beat which was already divided into four into three accented unit divisions. So that's 1, 2, 3, 1, 2, 3, 1, 2, 3, even though the pulse is 1, 2, 3, 1, 2, 3, 1, 2, 3, 1, 2, 3, 1, 2, 3, 1, 2, 3, 1, 2, 3, 1, 2, 3, right. So it's a little tricky to practice so I'd suggest you slow it down and write it down and then feel the pulse and then execute. So it'll be 1, 2e and a 3e and a 4e and a 1e and a 2e and a 3e. Takes a while to resolve, right, and slowly. Here and there you may make a mistake and resolve it, but it's okay. You could continue from then on and do the same 3, 3, 3, 3 divisions, right. For example, summer of 69, right, you could do something like this or just dotted eighth notes. Okay, let me try that with the melody. Very nice practice to kind of understand every subdivision of the 16th note and really isolate those subdivisions without anything before or after because you're dividing it fairly regularly, right, you're dividing it by 3. So that is about dotted eighth notes. Try practicing it. It's a very good rhythmic tool to build your rhythmic chops and your independence. And last but not least, I want to talk about the same hand, the left hand, but with some finger independence which will make your sound a lot more grand and epic. So with finger independence in the left hand, it's essentially hold the root, what you're playing with pretty much the pinky. And then try to do a rhythm pattern with just the thumb or just the index finger if you're playing the fifth. So with just the thumb, see what's happening there. My pinky is lingering on and my thumb is playing at certain beats, maybe off beats, maybe the clave, maybe the thresio, maybe any, we will rock you, maybe anything we've done in the past or anything you come up with, but hold the pinky whilst doing something with the thumb. So or maybe just use that as a ghost, like what drummers do, you know, think of it as a snare ghost, let's see how that works. Currently doing dotted, now the clave, something like this or melodic missing beats. I'll really add another layer of your piano, right? You have the pinky doing its thing, you have the other finger doing its thing, you sort of isolated the fingers of the left hand itself. So all through we've been doing right hand versus left hand or right hand melody, left hand rhythm. Now within the left hand itself, there's a little bit of stuff going on, right guys? So we've done five left hand rhythm patterns in part one. I hope you've also watched that. If you have, thanks a ton for sticking this far and watching all these rhythm exercises, and we've done five more in part two. I really hope that all these rhythm patterns help you with your playing, not only to play this melody, I've used this melody to kind of give you a template on which to build your piano playing on. You could take a popular song, you could take your own song and try to spice things up or try to bring in these left hand patterns over the course. Okay. And the thing about rhythm is once you've got it, you've sort of got it. You're not going to leave that rhythm when you work hard and get the pulse, when you work hard and get the thresio or whatever it may be. It's pretty much with you for life. So when it comes to piano independence or independence of the hands, the way I look at it is it's more independence of tasks. The melody is there and the baseline is there or the groove is here and the chords are here. So that's another perspective, perhaps to look at your independence, because it's not just how do you develop hand independence? No, it's about what you do there versus what you're going to do here. And it'll really enhance the fun you're going to have on this amazing instrument. Right, guys? Again, this is Jason here from the Nathaniel School of Music. If you found the lesson useful, please subscribe to our channel. Share the video with all your musician friends. Keep rocking and I will see you in the next one. Cheers.


Hello everyone, it's Missy! Welcome back to my channel! Today's tutorial is from a picture I posted on Instagram. Make sure to like and subscribe and click that notification bell so you never miss any future videos, and let's get started! For this hairstyle, I've parted my hair on the right side, as I usually do. I always like to spray my part with some hairspray to try and tame down any flyways.  This will make it easier to create some volume at the crown of my head. So I'm just going to slide a couple bobby pins horizontally over the section and then gently pull on small pieces of hair at the crown to add some volume and also ensure that it's symmetrical.  I'm going to spray this with hair spray and then smooth it down with my fingers. And I'm just going to twist the section together in an upwards direction and then lay it across the bobby pins from the crown section. I'm going to pin the section with some bobby pins and I'm just sliding them into the hair so that they don't show. And then I'm going to go back over the section and gently pull on the edges so the twist is a lot fuller. And now I'm going to pick up a similar section on the left side of my head right in front of the crown section. And I'm going to repeat the same steps twisting the section upwards and draping it underneath the previously pinned section. I'm just going to secure it in place with some bobby pins and then go back over the section pulling on the edges so the twist is a lot fuller. And to keep the style clean I'm also going to spray it with some medium hold hair spray and then just smooth down any loose pieces as I go. So now I'm going to take another section on the right side of my head right in front of the previous section and I'm going to repeat the same steps again twisting the section upwards then drape it underneath the previously pinned sections and then secure it in place with some bobby pins. You just want to make sure to use as many bobby pins as you need for each of these twists. This will just ensure that it stays in really well. I like to slide them into the twist so they're only gripping a few pieces of hair but they'll stay hidden really well this way. And then of course once it's pinned I'm going to go back over the twist and pull on the edges to add some volume and make it fuller and it will also give the overall hairstyle a looser and softer look. And then I'm just going to take the final section of hair from the left side of my head right in front of the previous sections, twist the hair together again and I'm also going to pull on the edges to add some volume to the twist like I did before and then just pin it in place with some bobby pins. Now I'm going to divide out a section of hair at the center back of my head right underneath the twists. I'm going to divide this into three pieces and braid this into a fringe braid. So I'm going to start by crossing the left strand over the middle and then cross the right strand over the middle. Then I'm going to cross the left strand over the middle, pick up a new section of hair right next to the braid on the left side and cross it over the middle strand as well. Then I'm going to cross the right strand over the middle, pick up a new section of hair on the right side of the braid and cross it over the middle strand bringing it into the braid. And then I'm going to cross the left strand over the middle, pick up a new section of hair on the left side and cross it over the middle strand as well. And then I'm going to cross the right strand over the middle, pick up a new section on the right side of the braid and cross it over bringing it into the braid. I like to bring in fairly large sections. This will give the braid a looser, softer effect. You can bring in smaller pieces if you'd like. It would make the braid a little bit more detailed. Either way it will look beautiful. It's totally up to you. So then once all of the hair is brought into the braid, I'm just going to do two more stitches of the braid, crossing the left over the middle and then the right strand over the middle. And then to finish the braid, I'm going to take the left side strand and I'm going to wrap it around the bottom of the braid two times. And then I'm going to grab a clear elastic band and tie it around the bottom of the braid, making sure to include the ends from this wrapped piece. And then the clear elastic band will hold it in place, wrapped around the bottom of the braid. And then to finish the style, I'm just going to go back over the braid and gently pull on the edges, as I always do, so that the braid is a lot fuller and bigger. This will help the braid show up a little bit better. And it will also give the overall hairstyle a looser and softer look. And then I'm just going to spray this style with a medium-hold hairspray and smooth down any flyaways. Thank you so much for watching. I hope you enjoyed today's video. Make sure to like and subscribe and click that notification bell so you never miss any future videos. And I will see you next time.