Yeah, so see the point is that you know we are trying to look at families of meromorphic functions we are trying to look at trying to look at normally convergent families of meromorphic functions. The reason is because you want to do topology on the space of meromorphic functions okay and because that is the kind of you know setup that you need to be able to prove the big Picard theorem and the little Picard theorem okay. So now the so you know let me briefly you know remind you if you take inspiration from topology alright what is the topology that you will put on the space of functions normally if you have a topological space and you have you are looking at real valued or complex valued functions then you will restrict yourself to continuous bounded real valued or complex valued functions that is that is a Banach algebra and it is also a topological space it is complete as a metric space the metric is induced by a norm and the norm is essentially convergence in that norm is actually equivalent to uniform convergence okay. So the moral of the story is that if you are going to take inspiration from topology okay then trying to do topology trying to do topology on the space of functions is the same as do studying functions under uniform convergence okay. Now this is topological so that means that you are only worried about continuous functions okay but now suppose you come to complex analysis okay then you are worried about holomorphic functions or analytic functions okay and we are worried about something even worse we are worried about meromorphic functions which are actually which have the additional problem that they can have poles at finitely many at a set of isolated points okay and of course finitely many if your domain is compact is the whole Riemann sphere or extended plane. So if you are looking at say holomorphic functions on a domain okay a domain in the complex plane or a domain in the external complex plane that does not matter suppose you are looking at the analytic functions or holomorphic functions on the domain and you want to do topology on that set of functions okay. Mind you we have already seen that the set of functions it is a ring in fact and then if you look at meromorphic functions it is a field okay it has algebraic structure but we are now worried about the topology okay. So you want to do topology on the set of analytic functions on a domain or you want to do more generally topology on the set of meromorphic functions on a domain which means analytic except for poles. Then you know if you try to draw inspiration from topology you would just say that this is the same as studying them under uniform convergence okay because topologically uniform convergence corresponds to convergence in the space of functions okay but if you do it if you come to the case of analytic functions okay this is not the right thing because you do not get uniform convergence. So I was trying to explain to you last time that for example if you take the geometric series you take the functions that correspond to partial sums of the geometric series okay then they are of course polynomials and they converge absolutely in the unit disc to the sum of the geometric series is which is 1 by 1 minus the variable okay.  Now the point is that this convergence is absolute on the unit disc there is no problem about that but it is not uniform on the whole unit disc okay that was something that I told you I asked you to check it as an exercise I hope you have done it it is very easy to do the convergence is not uniform on the whole unit disc it is only uniform on compact subsets of the unit disc okay. So you do not get convergence but you do not get uniform convergence but you get only normal convergence so the moral of the story is that when you want to do topology on a space of holomorphic functions you should not look at them under uniform convergence you must look at them under normal convergence so that is the first you know that is the first moral first lesson to be learnt that is what you should keep at the back of your mind so that is one thing. So you know more generally if you want to extend this to meromorphic functions things are more complicated because now you have poles okay and the other thing is that of course you know by going from uniform limits to normal limits things are going to be good okay because normal convergence is just uniform convergence on compact sets okay it is weaker than uniform convergence as it is okay but it is good enough for our purposes because as I told you in the last lecture if you have a sequence of holomorphic functions which converge normally to a limit function then that limit function is also holomorphic okay this is something that I explained last time okay essentially it uses you know Cauchy's theorem and Moreira's theorem okay and the fact that analyticities or holomorphicities local property. So therefore there is no harm in relaxing the condition of uniform convergence to the condition of normal convergence which means uniform convergence only on compact sets okay and I also told you philosophically why that is good enough for complex analysis because the moment you say uniform convergence on compact sets you get uniform convergence on close disks because they are also compact and therefore you get uniform convergence on sufficiently small open disks okay that is good enough for the analysis for the differentiation theory and then for the integration theory also it is good because whenever you integrate on a contour the contour is a compact set therefore you will get uniform convergence on the contour okay so that helps in the integration theory. So for all practical purposes uniform convergence on compact sets that is normal convergence is good enough okay so that is what we are going to we have to worry about. Now the other important thing that I want to tell you is that you see at least if you are working with meromorphic functions you know the meromorphic functions have poles okay so at a pole the function is going to behave in a bad way in the sense that the modulus of the function is going to blow up to infinity okay so that is for example that is one of the characterizations of a pole the limit of the function as you approach a pole is going to infinity okay and by that I mean limit goes to the point at infinity okay and of course here you are using the topology on the extended complex plane namely the complex plane with the point at infinity given by the one point compactification which makes it homeomorphic to the three months we have okay. Now there is another pathology and that is the pathology that I was trying to explain towards the end of the last lecture so the pathology was the following you take the exterior of the unit disk mod z greater than 1 that is our variable we are on the z plane the complex plane and you are taking the exterior of the unit disk mod z is greater than 1 and what you are doing is you are looking at these functions powers of z you are looking at 1 which is z power 0 if you want then z and then z squared and z cube and so on okay. Now that is a sequence of functions and the point is that this sequence of functions you can see point wise it will go to infinity because since mod z is greater than 1 mod z to the n to the power of n that is going to go to infinity because it is for a real number greater than 1 you know its higher powers will diverge to infinity so mod z so z power n is so the sequence of functions is going to converge point wise to the function to the constant function infinity namely it is a function that which associates to every point the value infinity okay so you have to worry about this crazy function okay so you have to so this is a pathology that happens that you will have to take care of and the point is that therefore we are forced to introduce a function called infinity okay and this function infinity is what it is just the constant function infinity namely it is the function which maps every value to infinity that is what it is and then if you think of that as a function I mean it is a function of course theoretically if you want it is a function from your domain to the external complex plane because after all in the external complex plane infinity is a valid point okay it is a member of that set so you can really think of the function infinity as the constant function with taking the value infinity provided you extend your values to not just complex values but also the extended plane you include the value at infinity that is one thing so in that sense you can say that this sequence of functions fn of z is equal to z power n that converges to infinity you can say that and when I say that converges to infinity I mean that it converges point wise in the exterior of the unit circle to the function which is infinity okay so you have you have this very nice situation it is a very nice pathology you have these z power n which are all holomorphic functions okay in fact they are entire functions they are just polynomials and they converge to the function infinity in the exterior of the unit disk the convergence is again a normal convergence it is uniform on compactness okay it is still a normal convergence is not just a point wise convergence but it is in fact even a normal convergence in a way I will explain to you so what is the moral of the story the moral of the story is you have a hornest sequence of holomorphic functions you have a hornest sequence of analytic functions which is converging to the function infinity normally that also happens see this is the extreme case that happens and this also has to be taken care of in our arguments okay and mind you if this is happening for holomorphic functions it will happen also for meromorphic functions because you know holomorphic functions are very good melomorphic functions are worse because they have poles so even for a family of holomorphic functions even for a family of analytic functions if you can get normal convergence to the function which is infinity okay you should expect the same thing to happen also for meromorphic functions so what I am trying to tell you is if you sum up all this if you want to study topology on the space of meromorphic functions first of all you must study it with respect to normal convergence okay the second thing is you have to introduce this function keep in mind this function which is the function infinity okay and then you have to do you have to justify this business of trying to make sense of normal convergence okay and so let me begin by trying to you know explain at least in this particular case where this normal convergence comes from okay so let me write the let us do the following thing we will worry about matrix on the plane and matrix on the extended plane which are transported from matrix on the Riemann sphere okay so here is what I am going to do so let me draw a diagram I have this so this is my usual complex plane well x, y and so this is my usual complex plane which is the x, y plane and then I have the well you of course you know we are going to compare everything with the Riemann sphere using the stereographic projection so what I am going to do is let me draw this let me draw this thing here which is a Riemann sphere so this is supposed to be 1 this is minus 1 on the x axis and here is my sphere it is cross section on the plane is the unit circle so it is going to be so this is my Riemann sphere as it is and then of course I have this third axis which I will not call as z I will call it as u because z is supposed to stand for x plus i, y okay z is supposed to be x plus i, y and well and here is an odd pole okay suppose I start with 2 points z1 and z2 on the complex plane okay see what kind of distances can I define on these points what kind of metric can I define on the complex plane there is a usual metric which is d of z1, z2 so I will put d sub e for the Euclidean metric and you know what that is it is simply modulus of z1 minus z2 it is just the distance between these 2 points okay this is the good old metric that we use always in Euclidean space right it is actually the length of this line segment okay joining z1 and z2. Then the other thing that you can do is you can take these take the images of these points on the Riemann sphere because of the stereographic projection and you can measure the distance between those 2 points and call that as the distance between these 2 points so what you can do is so here is the stereographic projection so p2 is this point here on the sphere which is the unique point of intersection of the line joining n and z2 on the sphere okay and similarly p1 is unique point on the Riemann sphere which is the intersection of the surface of the sphere with the line joining n and z1 okay and now you see p1 and p2 lie on the sphere now what I can do is that I can measure the distance between p1 and p2 okay now that distance I am measuring in 3 space okay because now everything once you draw the Riemann sphere you are actually in R3 and your R2 which is the xy plane corresponds to the complex plane alright. So what you could do is you could define this you could define the following new distance also d let me call this as dc for chordal distance so dc is the chordal distance from z1 and z2 it is just length of the chord from p1 to p2 where p1 the stereographic projection of p1 is z1 and the stereographic projection of p2 is z2 okay so this is another distance that I can define it makes see what this distance does is that actually it is the metric in R3 after all the chord joining p1 p2 is exactly the line segment from p1 to p2 in 3 space and I am just taking the length of that line segment so it is actually the metric in R3 it is metric in R3 okay and so it is a metric space you know whenever you have metric on a space and you restrict to a subspace then the subspace also becomes automatically a metric space so this distance will make the Riemann sphere into a metric subspace of R3 okay and what we are doing is that through the stereographic projection you are transporting that metric to the complex plane because after all the stereographic projection is a bijection between the extended complex plane and the Riemann sphere okay. The moment you have a bijection of a set with a metric space you can transport the metric on the metric space to the set so I have just transported the basically what I have done is I have simply transported the metric on R3 restricted to the Riemann sphere I have simply transported it to the plane that is what this DC is so this DC also will D sub C that is also a metric you can check that and that also makes the complex plane into a metric space okay and but the big deal is that you know all these metrics are all equivalent okay namely the topology is that they induce on the complex plane they are all the same that is the whole point okay and that is very important because what it tells you is it tells you the following thing if I want to study convergence of functions you know as long as you are worried about continuous functions I can use any of these metrics and the point is so let me say tell that in advance why I am worried about these extra metrics is because I can also define the distance of a point on the complex plane to the point at infinity okay because that the point at infinity will correspond to a finite point namely the north pole on the Riemann sphere and that distance to that is something that I can measure okay so that is the advantage the advantage is you see I want to be able to measure distance to the point at infinity I cannot do it with the Euclidean metric because first of all the point at infinity is not in my set okay it is an extra point I have added for compactification and once I add this extra point I have to add this extra topology the topology of the one point compactification but then I that is not enough I have to even make it a metric space and where will I get the metric structure the only way is I will have to get this metric structure from the Riemann sphere which is what is homeomorphic to the extended plane okay and therefore I am led to look at the metrics on the Riemann sphere and so this is the Caudal metric okay so D sub C is the Caudal metric and then here is a third metric which is the spherical metric. So D sub S of Z1, Z2 this is the length of the arc of the minor arc from P12, P2 along this length of the minor arc from P1 to P2 along the great circle through P1 and P2. So you see this is a spherical distance what is a spherical distance is a spherical distance actually I am trying to measure distance on the Riemann sphere on the surface of the sphere so it is a curved distance okay and I am trying to measure the shortest curved distance and you know you can imagine this if you basically what one is doing is that one is doing a kind of some kind of Riemann geometry what is happening is that you have a surface okay you imagine some nice smooth surface you have 2 points okay then you can try to connect those 2 points by many arcs on the surface passing on the surface okay and then you can measure the lengths of each of these arcs and you can take the smallest length okay and the arc of smallest length is called a geodesic okay. So now what is happening is that if you take the sphere it is quite easy to see that the geodes if you give me 2 points on the sphere on the surface of the sphere then the geodesic is exactly the following you for those 2 points you get a big circle a great circle of largest radius on the surface of the sphere passing through those 2 points and you take the minor arc okay any 2 points of a circle will split the circle into 2 arcs and you take the minor arc this one of smaller length and take the length of that that is exactly the spherical distance okay and that is what I am denoting as d sub s it is a geodesic for the sphere for any 2 points on the sphere okay. So the great circles of the geodesics okay the minor arcs of the great circles of the geodesics for points on the sphere so what is happening is that now I have all these 3 matrix the beautiful thing is that these 3 matrix give you matrix not only on the complex plane the point is they give you matrix on the extended plane okay see what I have drawn here is for z1 and z2 imagining z1 z2 as points on the complex plane but I can very well make z1 or z2 to be the point at infinity now when I say I make the point z1 or z2 to be the point at infinity I cannot see it on the complex plane okay but I can see its image on the Riemann sphere it is a north 4 so basically what I am doing is I am simply taking 2 points on the Riemann sphere and I am measuring their distance the distance between them either the chordal distance or I am measuring the spherical distance so the moral of the story is that this these distances help you to give a matrix on the extended plane and the topology induced by all these matrix is 1 and the same all these matrix are equivalent okay so this is the fact that you need to check from you know this is very easy in fact to check ideologically let me tell you how to do it how will you check the 2 matrix are equivalent for a topological space it is very simple what you do is that you show that you take a point for each point of the topological space you take small open balls with respect to one metric and show that it contains a small open ball with respect to the other metric and do this for both metrics symmetrically and then you are done okay so you can see pictorially you can see that that is true okay for all the 3 matrix so therefore all these 3 matrix will give you 1 and the same topological space structure on the extended complex plane which is the same as the 1 point compactification and that will be exactly homeomorphic to the Riemann sphere by the stereographic projection okay and the advantage of doing all this why do all this you can ask me why do all this the advantage of doing all this is that now I can say now I can make sense of the following statement a sequence of f functions f n converges normally to infinity to the function infinity it makes sense now because I can say I can say convergence with respect to this metric one of these metrics namely the second and the third one which are also defined for the point at infinity okay and that is the reason why we need to use that okay so let me write this down all the so let me write some somewhere here may be in a may be use a different colour all the metrics below are equivalent on C alright and the last 2 metrics are equivalent on the extended plane the last 2 or the latter 2 so let me make some space let me get rid of this the latter 2 namely these 2 are equivalent on the extended plane C union infinity okay and of course so I rubbed off z equal to x plus I y so let me write it here okay and now here comes the here comes our the advantage of this we can now say that f n is f n of z equal to z power n n greater than or equal to 1 converges normally to infinity on mod z greater than 1 you can say this you can make this statement it makes sense okay and why is that correct you have to do a little bit of you know why I am spending so much time on this is because you see this normal convergence to the function which is infinity is something that is hard to as it is if you do not analyse it is very hard to digest okay so it is very important that you understand what is happening here and you must understand that therefore even if you are looking at normal convergence of you know honest holomorphic analytic functions you can still end up with the function at the function which is infinity okay so you see what is happening see so here is so again let me so again let me draw another diagram so you see here is the here is your complex plane as it is in R2 and this is the origin and of course I have so let me draw the Riemann sphere here okay so this is the situation and you see of course mod z greater than 1 is this is the region of the plane exterior of the unit circle okay so basically so this is mod z greater than 1 it is very hard to draw that and in a 3 dimensional diagram so let me do the following thing let me use another colour you know I am just looking at so it is all these so it is this exterior of this unit circle on the complex plane which is thought of as a xy plane okay and of course you know the third axis is I am calling it as u now you see so let me change colour again so I have this let me draw this also so that this is u and I have the north pole here okay now you see see this these red lines that I have drawn they are supposed to extend outside the unit circle to the whole exterior of the unit circle and what they are going to give me they are going to give me this domain mod z greater than 1 if you consider it as a domain in the extended plane it is a it is a neighbourhood of infinity okay and what is its image under the stereographic position it is exactly the upper hemisphere okay it is the whole upper hemisphere which is which you can see clearly is a neighbourhood of the north pole okay that is what that is the region you are looking at alright. Now you see take a compact subset of mod z greater than 1 take a compact subset it is a close and bounded subset of mod z greater than 1 okay and so you know if you are looking at a compact subset on the plane okay then any compact subset in mod z greater than 1 on the plane has to be has to lie within a sufficiently well chosen annulus okay it should lie within an annular region consisting of an inner circle and an outer circle centered at the origin sufficiently small inner radius greater than 1 and sufficiently large outer radius greater than of course the inner radius okay so you see if I take some compact set here so here is some compact set K compact set in in in the complex plane then you see this K of course lies inside suitable annulus so it is going to it is going to look like this you know I am going to get something like this I am going to get I am going to get this annulus here so I am going to get this annular region mind you this is the annular region on the complex plane consisting of the region between these two circles and I am also including the boundaries to make it compact okay so it is a close and bounded set it is compact and this is a compact set and the point I want to make is that instead of just considering any compact set K in the complex plane which is lying in this domain mod z greater than 1 it is enough to just consider such annular which lie in the exterior of that circle okay the exterior of the unit circle and you see if you watch carefully how is this annular is going to be given by well this annulus is going to be given by mod z less than small r I mean less than capital R less than or equal to capital R less than or equal to small r which is greater than 1 this is how it is going to be where small r is the radius of the inner circle capital R is the radius of the outer circle okay this is what this annulus is going to be given by and well what is its image going to be on the Riemann sphere under the stereographic projection I am going to get this I am going to get something like this here this is what I will get I will get a curved annular region centered at the north pole alright and now you see now you see that now you can see why the f n converges to the function infinity normally okay because our definition of convergence is in the following sense okay so the definition of convergence will be of course point wise convergence okay but then point wise convergence if you try to write it in the metric it will create a problem when you put the point at infinity okay so I cannot say that for each point z z0 f n of z0 converges to infinity as n tends to infinity I cannot say that I can say that in a topological sense okay but I cannot say that in a metric sense but if I use Euclidean metric but then if I use a spherical metric I can say that okay so the moral of the story is that if you look at the if you look at the distance the spherical distance between a point z in I take the point z in k my compact z okay if I look at the spherical distance between f n of z which is in this case z power n okay and the point at infinity okay and here you see what I mean here is infinity of z see infinity of z I am thinking of the function which is constant function which gives the value infinity to every point so I am what I have written there is actually infinity of z and I am saying the spherical distance between f n of z and infinity of z that can be made uniformly less than epsilon irrespective of z if I choose n sufficiently large so I can make this less than epsilon okay for n greater than or equal to capital N irrespective of so given if I start with an epsilon greater than 0 okay for z in k I can make the spherical distance between f n of z which is z power n and infinity I can make it less than this epsilon for n whenever small n is greater than or equal to large n large enough n such a large enough n exists the point is that this large enough n does not have anything to do with this z it will work for all z in the compact set that is the uniformness that is the uniformness of the convergence on the compact set so n this n is so there exists this n and this is independent of z and you see this is this fact is true this fact is very you see suppose I give you an epsilon okay what is the spherical distance between f n of z and infinity it is actually the spherical distance between z power n and infinity alright and z power n is going to lie where it is going to lie in well mod z power n is going to lie in this annulus okay and you know if you see if I this is the inner radius of this annulus is r power n small r power n the outer radius is capital r power n and you know if I and since r is greater than 1 if I increase n r power n is small r power n itself is going to shoot up okay so moral of the story is that this region if I take its image in the on the Riemann sphere what I am going to get is a sufficiently small annular region surrounding the north pole and clearly the spherical distance can be made less than epsilon for any point in that region okay so that is you know that is pictorial justification for this statement so moral of the story is now you know you are able to justify that this sequence z power n converges normally to the function infinity okay on the exterior of the unit disc in the in the extended plane okay and the point is that you are using spherical metric okay that is the advantage you are using spherical metric because it allows you to give you to measure the point distance even to the point at infinity from a finite point in the complex plane which you cannot do with the Euclide metric and the way all this is done it will also it will also extend the usual definition of normal convergence see suppose you have a sequence of analytic functions which converges to a honest analytic function itself okay a finite valued function a function that does not take values infinity then what is the usual convergence that we talk about the usual convergence that we talk about when you for example when you do first course in complex analysis the usual convergence that you talk about is with respect to the Euclidean metric okay you have to only worry about the Euclidean metric nobody is worried about the point at infinity to begin with okay. Now if you take a usual a honest sequence of holomorphic functions on a domain analytic functions on a domain suppose it is converging normally again to an analytic function on that domain okay then suppose this convergence is in the usual sense I am saying this convergence is also correct with respect to the spherical metric the reason is because the spherical metric when you restrict it to the usual complex plane it is equivalent to the usual Euclidean metric so you do not lose anything. So what I am saying is that this definition of a sequence of functions converging to another function normally on a domain that works irrespective of whether you are using the Euclidean metric or whether you are using the spherical metric but the point is it helps you when infinity values are taken you it helps you because when infinity values are taken you cannot use Euclidean metric you can use only the spherical metric okay. So the normal convergence under the spherical metric is just an extension of the normal convergence under the Euclidean metric they are this as far as a subset of the complex plane is concerned convergence under this spherical metric is the same as convergence under the Euclidean metric normal convergence under the spherical metric is the same as normal convergence under the Euclidean metric because they are equivalent okay. So this is one point that you need to understand. So you know so let me say this so let me write this specifically we need to therefore worry about by that I mean include we need to worry about that is include the function the function infinity that takes the value infinity at every point of your domain then we need to define we may define normal convergence as follows. Let fn be a sequence of holomorphic functions or analytic holomorphic is the same as analytic functions on a domain t in the complex plane we say fn converges converges to f on d normally okay. If the spherical distance between fn of z and f of z goes to 0 normally which means uniformly on compact sets on d where we allow f to be the function infinity this is an exceptional case okay. So you define normal convergence in the following way so what is the normal convergence you have sequence of functions on a domain in the complex plane you say fn converges to f on the domain okay. If the spherical distance okay that converges to 0 okay mind you this spherical distance is a function of z so z is varying on the domain. So I want you to understand this z is varying on the domain this quantity here that is this quantity here is also a function of z it measures for each z it measures the spherical distance between fn of z and f of z okay and what I want is that function of z should go to 0 the constant function 0 uniformly in z on compact subsets of z of the domain. So I want it to go to 0 normally on the domain that is my definition and now the beautiful thing is you know we need this definition because if you take the domain as I told you if you take the domain to be mod z greater than 1 and you take fn of z to be z power n such a definition is necessary. So what it tells you is that now you have to also worry you are not worried only about functions which take complex values you have to also allow functions take the value infinity but then notice if you take the value infinity if you allow functions take the value infinity then you can include meromorphic functions because you can define the value of the meromorphic function at a pole to be infinity and the beautiful thing is the very same definition this very same definition works absolutely well if you change holomorphic by meromorphic. So that is because of a certain symmetry rotational symmetry that is there about for the spherical metric that I will explain in the next lecture but the point is so the important observation is that this same definition works with holomorphic replace by meromorphic and that is all that we need to do all the analysis we want. So I will stop here.


Hello friends, I am Sanjay Gupta. In this video I am going to demonstrate you the difference between static binding and dynamic binding in C++. Before starting, you can note my information. You can follow my YouTube channel that is Sanjay Gupta underscore tax school. You can download my Android app that is Techimus which is available on Google Play. Now coming on to the topic. For demonstration purpose I have implemented a code for you. Here header file IO stream is included. Then STD namespace is used. Then first class is A which is having an attribute X. Inside public section I have implemented first function that is void read. It will display a message enter a value on console and the entered value will be stored inside X. Then show method is implemented which will display value of X on console. Then A class is narrated into class B. So class A is base class and class B is treated as derived class here. And after that main function is implemented. In class B you can see an attribute Y is declared. Then I have implemented read function which will display message enter a value on console. The entered value will be stored inside Y and that value is received through scene. Then show method is defined which will display a message on console and it will display value of Y on console. So this is the termination of class B. So class A members are available in class B and in main function pointer of class A is created that is P1 and you can see P1 equals to new B object is created. So P1 is of base class pointer. So the comments is saying pointer is of class A that is P1 and object of class B is created. So object is of class B is available. So here object is belonging to class B and pointer belongs to class A and then read and show functions are called through P1. So now we have to understand the binding here. Here P1 is calling read and show functions. So P1 belongs to A class. So in case of compilation time read and show functions of base class will be linked with this P1 because P1 belongs to A class. When we execute this program then read and show function of base class will be executed because right now virtual keyword is not used. So I am first executing this code so that you can understand the output. I am entering a value that is 3 and it is printing X equals to 3. You can see that X equals to and value of X is displayed through show function of class A. It means this read and show functions are executed for base class. So this is static binding. Static binding means at the compilation time read and show functions are linked for base class and those base class functions are executed at runtime. So static binding is worked at runtime so that is why it is known as static binding concept. Now what is the meaning of dynamic binding? Dynamic binding says that if at the time of compilation some functions are linked with the function calling statement and if at execution time those functions will be changed then that process is known as dynamic binding. So now you can see I have created object of class B which is pointed by pointer of class A but I want class B function should be executed. So for that purpose you have to use a spatial keyword that is virtual keyword. So here you can see I have created these two functions as virtual function now. So class A that is base class its functions are now virtual functions. So virtual function has a property that if compilation time some set of functions are linked with a pointer then at execution time virtual keyword checks which class address is available in the pointer. So here you can see pointer is receiving address of B class that is derived class. So virtual function transfers control from base class to the respective derived class. In our example the derived class is B. So read and show functions of B class will be executed now. You can see it by execution. It is asking for a value. I am entering 4. You can see here this time it is showing y equals to 4. Previously it was showing x equals to 4. So this time derived class read and show functions are executed because base class functions are now virtual functions. So this concept is known as dynamic binding. At compilation time functions of base class are linked but at execution time functions of derived classes are executed and this all is done with the help of virtual function or virtual keyword. So friends if you are not using virtual keyword then the pointer type is used preferably but if you are using virtual keyword or virtual function then the address which is stored inside the function is used preferably. So this is the difference between static binding and dynamic binding. I am summarizing it now. Static in case of static binding the compilation time function call and execution time function execution both are same but in case of dynamic binding the compilation time function call and execution time function execution both are different. In case of dynamic binding linking is done with base class but execution is done with derived class functions. So this is all about static and dynamic binding in C++. I hope you have understood the concept. If you want to watch more programming related videos you can follow my YouTube channel that is sanjaygupta underscore tech school. Thank you for watching this video.


It's not so, because there are supposed to be two windows and we didn't get two windows. So if this full screen is here, we're good. That is good, but we have a title bar. It's not going away. Have you seen this before? Okay, now it's, oh, I see. Mark, I guess we'll have the title mark. That's not too much. The last thing is this is going to give you a beep at 30 minutes. Let's just let you know it's time for that. At 30? Oh. Got it. All right, so let's get started. So, hello OpenStack Summit. I mean, Open Infra Summit. It's nice to be back at the summit after such a long break. We and the SAF team are very attached to OpenStack, so it's nice to meet you all in person again. So my marketing manager had a long history of saying that you have to introduce yourself, otherwise it's malpractice. Those are things I worked on in the past. So I've had the privilege of spending basically my entire career in open source, aside for brief stint in academia. I'm the product management director for the SAF platform at IBM and at Red Hat, and previously I was the Ubuntu server PM at Canonical, and if you round out the decade, I was the dreaded systems management czar at SUSE. That was my actual title. I was the maintainer of MAN for a very long time, and I wrote a book by O'Reilly about AWS, which is why the picture looks funny like that. And I think that's enough about me. So, security. This is a pretty advanced session. We have the obligatory introduction notes, but this is SAF Day. It's the afternoon. If you don't know what SAF is by this point, see me afterwards for Cliff Notes. Similar thing for Rook. You should have absorbed something about Rook, but here the focus is open stack, so maybe you didn't. Rook is a Kubernetes operator for SAF, and essentially it manages SAF clusters running on a Kubernetes fabric. That's the way you should think of it. It's interesting because it automates some of the day-to-day management or encapsulates multiple options in ways that are not possible outside of a containerized and virtualized environment. So, that's another interesting component that we use all the time. So, now let's go to the actual security. Let's dive right in. Now, what is security? Security hardens a specific point of the infrastructure. That is what security best practices do. Cherry-picking practices without the model of the threat and of the attacker that you're worrying about is not a viable strategy. It's like saying that everything is a priority, which is the same thing as saying that nothing is. So, you have to decide instead of having... That's not the meaning to management, but instead of saying management like things that result in everything is a priority, you have to have an actual priority, something that is actually meaningful. Otherwise, you will wind up in scenarios like cover your computer in concrete and bury it at the bottom of the ocean because you have to stop every possible imaginable thing, which is not very useful, clearly. So, are you facing script kiddies at one end of the spectrum? Or GRU, the Russian Secret Service, at the other? These are very different threat models. Some attackers want to steal your data. Others don't care about your data. They just want to encrypt it and hold you for ransom, cryptolock you for ransom, as they say now. Others may be just satisfied with complete disruption, bringing you down. It doesn't matter where the data is or if you have it or not. So, for these reasons, you have to decide who is in your threat model and who isn't. Typically, the worst-case scenario is the privileged insider. Somebody that is inside your security perimeter and has root credentials. If you have that in your security profile, then you're going to have some very interesting challenges to deal with. Now, once you have this, you have the basic framework to make choices. And so, let's start from the network. In SEF, the network is segmented in multiple logical networks. The public security zone is an entirely untrusted area of the cloud. It could be the internet as a whole or just networks external to your cluster that you have no authority over. Data transmissions crossing the public security zone should make use of encryption. Note that the public zone, as I just defined, it does not include the storage cluster front-end, the SEF public underscore network. That's an unfortunate naming, but that is a different thing. That it defines the storage front-end and properly belongs in what is called the storage access zone. So anyway, the SEF client zone next refers to network accessing SEF clients. These could be the object gateway, for example, the SEF file system or block storage. SEF clients are not always excluded from the public security zone. For instance, it is possible to expose the object gateways S3 or Swift APIs in the public security zone. That's exceedingly common. Next is the storage access zone. That is an internal network providing SEF clients with access to the storage cluster itself. And finally, the cluster zone refers to the most internal network, and that provides storage nodes with connectivity for replication, heartbeat, backfill, recovery tasks. This zone includes the SEF cluster's back-end network called the cluster underscore network there in SEF. And operators often run clear text traffic in the cluster zone because they rely on the physical separation of this network. Or on the logical separation, usually this is on a VLAN at the very least. This, again, going back to previous comments, would not be a valid choice, for example, if privileged insider is in your threat model because the privileged insider very likely would have the keys to the cabinet, can go in and tap this network. But if your threat model doesn't include that, then you can have the performance of a cluster that doesn't encrypt its internal communication if you're not worried about people putting a network tap inside your rack. So, again, everything is on the threat model first. These four zones are separately mapped or combined depending on the use case and the threat model in use. Now, components spanning the boundary of two security zones, which are demons in SEF, two security zones with different trust or authentication requirements must be carefully configured. There are natural, maybe not weak points, but there are natural points to attack when you're trying to escalate privileges. You've got access to one network, you're trying to hop into a more privileged one. And so the crossover points should always be configured to meet the requirements of the zone with the highest security requirements or the higher level of trust, if you want to think about it, in those terms. In many cases, the security controls should be a primary concern due to the likelihood of these components being probed and the possibility that you just fat finger some configuration thing and you create a passage between two security zones by misconfiguring the way things are set up. Operators should consider exceeding zone requirements at integration points, which usually it's an empty thing to say because, again, you have to be able to use the technology that you're deploying. But in storage, it's actually often a possibility. Storage is a little bit simpler in this regard than compute. And so if your storage use case gives you the possibility of ratching up the configuration security of the integration points by all means, that's a good hardening practice. For example, here, let's say the cluster zone can be isolated from other zones easily because there is no reason for it to connect to other zones. Conversely, an object gateway in the client security zone instead needs access to a lot of things. The monitors, ports 6789, all the OSDs on ports 6800 to 7300 to access the actual data storage will likely expose the S3 API to the public security zone, ports 80 and 443. So some things you can't close, but there are plenty of things that you can. So those are the low-hanging fruit. Now, normally I'm joined by Sage McTuggart, who leads our security team, but they could not be here today. The thing about product security that is interesting in CEF is that as you heard yesterday, we have transferred CEF internally between the IBM and Red Hat boundaries. We moved the CEF team from Red Hat to IBM. So now we are looking at IBM product security not Red Hat, which is slightly different. If possible, even more paranoid is my impression. So from the point of view of the customer, that is good. You may get more patches, but it doesn't matter because the deployment model is containerized. So when you get one container, how many security fixes have been addressed in the container is really not that relevant. So potentially, if it doesn't slow us down and fix things that in the past we would have considered relevant because they were not accessible in a storage system like vulnerability in the... Usually the canonical example is saying that there is a vulnerability in the printer driver in Grell and how does that matter when you're running storage on top of that? IBM tends to be very strict about what is in the images. So we've become stricter about purging the images with dependencies that we're not using. But it's fairly easy to just uninstall these things that are in the stock rel image that we start from. And we're probably going to be shipping more fixes for things that are rated medium or lower in the CVEs. I don't think that it changes things much, but it's better or same, right? So that's probably the most visible change. Let me see. The new team is called Product Security. But because we're still getting revved up in terms of what processes we're adding as part of IBM compared to Red Hat, we're trying not to say things that then don't happen later. So bear with us as we figure it out. The general line is that it's going to be same or better in terms of security. Generally speaking, the standard for Red Hat is that vulnerabilities are addressed if they're exploitable. Otherwise, Red Hat tries to minimize the number of fixes that ship to reduce the noise in terms of what the operators have to manage in terms of the constant stream of updates. As I already mentioned, IBM takes a different take, and it tends to be if they exist, patch it. That is probably the most visible difference. We are getting more disciplined in terms of doing penetration testing and regular scans. I don't think I can tell you how often just yet, but it's more often than in the past, which is nice. And of course, nothing changes in terms of our relationship with upstream. Everything that we fix goes to upstream first, any CVE that we address is available in the community project, just as is in the products. So let's move on to encryption. Server aside, operators overwhelmingly choose to encrypt data at rest using the Linux, using the Lax mechanism. All data and metadata of a self-storage cluster can be secured using a variety of DM crypt configurations and nearly all of the existing Red Hat customers choose to. I stopped tracking how popular was at rest encryption in 2015, because at that point it was already overwhelmingly over 50%. It's a given that at rest the data is encrypted. The OSD is encrypted, period. We're at the point where actually replacing encrypted drives is a single operation in CFADM, and it's a very simple workflow in the dashboard, so that all the configuration mechanisms that need to take place when you are taking a dead drive out and putting a new one in are all handled for you. We also have a security best practice of locating the monitors on separate hosts from the storage demons. This is because the at rest encryption keys are kept in the monitors, so that way by having anti-affinity, if somebody steals one machine of your cluster, they either have the keys or they have the data, but they don't have both. They need to take two machines or the right two machines to be in proper. This obviously is not possible at the edge where you overlay everything in three nodes, but for normal self-clusters with seven nodes, 11 nodes or hundreds of nodes, this very much makes sense. And there is one other thing that I get asked about all the time, which is, okay, well, no, actually, I get asked only by the Washington DC crowd, but since this is the OpenStack crowd, you're on the same technical level. What happens with the keys that are being kept by the monitor? How are those secured? Those are at rest on the file system of the monitor, so the file system of the monitor should be also de-encrypted. The Linux boot partition should be de-encrypted. And then it's in the hands of, however, you secure the boot up of your machines, right? You're in your data center, you've already solved that problem of how to authenticate boot up. It applies there as well. Encryption-wise, the object store gateway has additional capabilities, including encryption at ingestion time, so you can have RGW encrypted data as it comes in, rather than LUX encrypting it as it gets stored on the file system. The difference there is a little bit academic. The part that is interesting is that you can think of it as when you're encrypting the data at rest, the keys are managed by the cluster operator and they're centrally managed and they are per machine keys. When you're having encryption done at RGW, the keys are per user keys and they're managed by the user. So that's the biggest, most relevant difference. Key rotation with tools like Vault is supported in RGW. There is support for Amazon AWS, SSSE, KMS, and there are more things coming. Further, you can use a Department of Defense certified cryptography under FIPS 142 and this year we hope FIPS 143 is supplied by RHEL in the version that you're deploying. So if you use RHEL, you know that there is a mode you install RHEL in that's called FIPS mode. It only gives you Department of Defense cryptography in that case and SEF very strictly only uses crypto from the operating system. So at that point, SEF is also using certified crypto. Encryption in transit now. For encryption in transit, a network communication can be secured by turning on the SEF protocol encryption in the Messenger v2.1 protocol or later, a protocol that was introduced with Nautilus. Now, here it's different than at rest. Network communication in clear text may be fine. The extreme case was the one that I was giving you. The internal network that the class and turn nodes are using to replicate data is very often physically secured with its own NICs. So if your threat model doesn't include someone that has the keys to the cabinet, there is no need to purchase additional CPUs and RAM to just encrypt traffic between the nodes themselves. The most common scenario where that kind of communication gets encrypted is corporate policy. We have some customers. I believe one is in France. I think it's due to a French regulator and a couple of others mostly in Europe that are adopting policies that say we encrypt everything no matter what. So from the point of view of the operations team, if what is happening between the SEF nodes in one cluster is regarded as network communication, the operations team may decide that it's actually better to buy a few more CPUs and a little bit of RAM rather than slugging it out with their security team for an exception there. So most of the network encryption that we see for the SEF protocol is coming from that kind of compliance reason. But there are plenty of cases where it's actually necessary for actual practical security reasons. CFFS talks to the entire set of OSDs. RBD talks to the entire set of OSDs. If the clients are transiting any network that is less than 100% locked out, it makes sense to encrypt that. In the OpenStack-specific scenario, this has been addressed for years in a slightly different way, which is that in the NOVA VM, you deploy DMCrypt to encrypt the file system of the VM internally. And then all the communication from the virtual disk of the NOVA VM to SEF is in the clear in terms of SEF protocol, but the payload is DMCryptedData. So you have perhaps not ideal because you're using at rest encryption for in-flight data. So maybe some cryptanalyst may wrap my knuckles, but in general, that is how OpenStack has done it for all this time. And it's worked just perfectly fine. However, if you want, you can replace DMCrypted at rest in the VM and use Messenger v2 encryption instead. It's a little bit of... It's logical that you use DMCrypt in the VM because you want to encrypt the VM at rest anyway. So you kind of get two birds with one stone there, or I think with one scone is the new way to say it. By encrypting both the VM at rest and the network protocol. But that is very OpenStack specific. How much does this encryption cost is the other question. And so you have to size the cluster. If you're going to encrypt the in-flight protocol, you have to size RAM and especially CPU to account for these overheads. In most cases, the performance impact is not that significant. You have to account for it in the cluster architecture, but in terms of the overall user performance, you are going to see the same performance because there is no significant enough latency that is not overshadowed by the latency of network communication. So we usually don't see other slowness. You just need a little bit more hardware when you design the cluster. Things vary between large block sizes and small block sizes, but that's a general heuristic. All right, looking at more specific protocols, DS3 service is usually secured between RGW and DS3 client with TLS on port 443, obviously. Also, it is possible to serve plain HTTP on port 80 for some reason you want to do that. I guess for some services that may be desirable, maybe you don't want to encrypt images or something like that. The interesting bit is that TLS termination at HA proxy is a special case. So TLS would end at HA proxy, then HA proxy needs to talk to RGW. That hop is in the clear. You have to account for that in your security model. The other thing that's generally applicable is that standard practices like maintaining the firewall of all your nodes are then the firewall around the cluster, but individually firewalling the nodes so that they don't expose things that are... that is not necessary to expose is an obvious best practice and that should be absolutely followed. Rook specific. This is a little bit of Kubernetes bombardment, so I apologize if you're not too familiar with that, but it's only one slide. Rook can use CRD's custom resource definitions to encode many settings like configuring trust certificates for the Rados gateway web server. Rook supports at rest data encryption as we discussed earlier with in-flight self-protocol encryption management being added in 1.9. It's essentially a management tool, so you see there usually a little bit of latency between self-versions and support for it in Rook. The Kubernetes user permission system applies to PVs, Kubernetes persistent volumes. So permissions, quotas, and all the other accoutrements that come with Kubernetes storage management are all there. Rook doesn't need to do anything for that. Rook also support key management systems and CSI interface, allowing individual volumes to be encrypted with their own key. This kind of helps a little bit with managing key rotation, revocation, and limiting the scope of each key. Control plane. So how do you manage this? This is very standard. As popularized by Ansible, SSH is used by SAP ADM, and so SAP ADM, SAP Ansible, and other deployment tools or day one tools tend to be SSH-centric. So that's something everybody tends to be very familiar with. These provide paths for install and upgrade management as part of host management. And then on the management dashboard, there is a thing that is interesting in terms of being another decision. You have to decide where you're placing the dashboard in terms of what network. You can place the dashboard in a fairly secluded network inside of SAP where only the SAP administrator can reach it. You can put it in a more public network where it's more accessible within your corporate infrastructure. It's a little bit more accessible. That is something that different users choose to do differently. It's one of the decisions you must make. Ultimately, it needs to be reachable by the operator's workstation to be useful, right? But where is the operator workstation is variable depending on who the customer is. The manager is the demon that provides a significant number of, or mostly all, of the functionality on the dashboard. So that needs to be reachable to have the dashboard accessible. All right, more on identity and access. Cephex is the internal identity system that Cephe uses. Its use of shared secret keys protects clusters from man in the middle attacks by default. But good practices still apply here. So good practice is to grant the key ring, read and write permissions only for the current user and the route with client admin users restricted to route only. And that limits what user can pretend to be a route, essentially. Talking about RGW. RGW supports the key and secret model of AWS that most of us are already familiar with. So nothing too surprising there. It also supports the security model of OpenStack Swift as it needs to be a drop-in replacement for Swift. Of course, the administrator's key and secret need to be treated with the appropriate respect, and that's no big surprise. We use full route privilege keys or administrative keys if you want to call them that way, sparingly to reduce risk profile. RGW's user data is stored in self-pools. So that should be secured as we discussed previously with data at rest, and you have to consider this user data, so you should be aware of it. You can integrate identity providers from your organization to provide identities to RGW so that you don't need to create a whole set of users just for RGW, you can obviously import. You can obviously import the users from your corporate identity vault. With identity and access, we support LDAP and Active Directory users. We recommend secure LDAP. And obviously in the OpenStack context, we support Keystone for OpenStack Clouds. In auditing, a lot of operations are audited. There shouldn't be anything touchy there, but it is considered the best practice to regularly purge your logs so that you don't leave too much data behind. You should aggregate your logs in a central location like ArcSysLog or Splunk or whatever it is and purge them locally out of an exceeding level of paranoia. Data retention is an interesting one. Once data is deleted from a self-cluster, you generally cannot do anything with it. However, there are exceptions. RBD is a trash bin where dynamic use of spare capacity can be used to retain deleted images for a certain number of days or until the space is reclaimed. RGW has versioned buckets, so you retain data as part of previous versions of objects that you have deleted. If you don't want to retain data inadvertently, you have to configure these things correctly so that they don't retain data for you and you don't want to keep it around. Additionally, like in almost any storage medium, individual data blocks used to constitute an object are still there on the disk platters and you certainly cannot overwrite a self-cluster by writing a lot of data to it. It's not going to work. At least it's not going to work reliably, which is what you want to do. So the sanitization or the secure deletion practice is use hard drive encryption, like we discussed before, for other reasons. When you want to sanitize the drive, you throw away the key. The fact that now there are drives that do this in hardware make things even easier. You don't even need to manage the keys yourself. Hardening options are very vendor-dependent. These are Red Hat options and obviously consequently also the IBM options. Others may vary. Now, we're not going to go into discussion of GCC and kernel security options, but if you're lucky enough that you don't know what these things are, you can learn more by looking at documentation from the Ubuntu kernel team. They have a very nice table of all of these. So once you know what they are, then you can decide whether you need them or not. It doesn't really, for most people, well, maybe this is open stack after all, but for most people, we don't go out compiling new binaries. We don't have the time to do that. But you should be aware of what the security level is.  Now, that is it. There are some more reading things for you available, covering the things that we haven't covered. Kubernetes secrets are always a hot topic, especially recrelated. So hacking Kubernetes by O'Reilly and a tutorial by Rani Osnot at Acquia Security Cover, that in detail. Our product, Red Hat, Chef Storage and IBM Storage. Chef have a dedicated hardening guide that expands what I was describing in long form. Kubernetes documentation is a very nice treatment of encrypting data at rest and again, those mysterious options from GCC for the hardening of binaries are also documented. These are all the people that contributed to the presentation so far and that's it for today. Are there any questions?


One is good, ladies and gentlemen. Welcome back to another video. Let's get some basics out of the way before we start this video. Now, I will be going over my updated shoe collection, obviously, but I'll also be doing a lot of fashion tips. This is going to include some do's and don'ts for wearing shoes with stuff. Also, big disclaimer. If you clicked on this video thinking I was going to have like a giant row of Air Jordans, I don't do that anymore. And that's kind of part of my whole fashion tip thing here. I ended up actually selling a lot of my old Jordans. So I know you guys are going to be sad. And I did mention this before, but I sold my space jams. I sold my own C12s. I sold my CP312s. Let's hop straight into the shoes part of this, right? We're going to start with the basics. These are the shoes everyone should have.  I will see people with the coolest shoe collection. They'll have every color in the book other than basic white and basic black. And I have no idea how they make outfits. On the screen right now, actually, I'm going to show you guys a few recommendations of some cheaper shoes that you can get.    Also, apologies if my lighting gets kind of wonky. It's a weird day outside.  You know what I'm about to pull out, right? My OG classics, the triple black Janaskis, these things have actually been worn on my feet since eighth grade. Not this exact pair, but these shoes, triple black Janaskis. Eighth grade till now, I'm going to be a senior in college. This specific shoe has stayed in style for me for nine years. Now that's not my only pair of all-black shoes though. Recently, I finally got a different all-black shoe. And I actually love these. These things are so dope. These are Air Max 270s. One of my favorite shoes right now. Also have this exact pair, but with the salmon camouflage background. I'll try and pull the picture right now. Love those two for like a tiny pop of color. But I don't want to get too crazy with it. I just wore these to see the point the whole day. Comfy as hell. But I don't really want them in any other color.  As I mentioned before, all white.  So they've got the like red, white and blue thing going on. I love straight up converse. The bad thing about these is I'm not a huge fan of how these look if you're not wearing jeans. So like right now it's summer.  Which is why I am seriously running out of daylight here. So I do apologize, but that is why I have these. These are actually in the kicks shopping vlog, which I just posted. That's what I got these for. These vapor max fly nets are so comfy and they look super good with like low cut socks and shorts. So that's what these are for. I don't recommend this purchase up because these boys are expensive. So unless you have like solid disposable income, this is probably not the move. All right. That's it for the basics. Now we're going to move into the curveballs. You ready for this one? You ready for this curveball? This is I am in my shoe shop. What is this? My sneaker updated sneaker collection.   This will be the style of dress shoe. Right now it's the wingtipped, which is like a little narrow whatever. This is called monk strap. These things are fire. If you're trying to impress your lady friend and you're going to prom or you're going to something really nice. Hey, don't take it from me boys. You'll see one day. All right. Now we get into the part where I have style advice and I have two of my favorite pairs of shoes right now. Okay. Here's the thing. You have two options when it comes to shoes. Do I turn on the lights behind me? Is it literally that? Lighting YouTube videos is not fun. I'm so serious. You have two options when it comes to your outfit. Your shirt makes the outfit or your shoes make the outfit not full.  I like want to puke in my mouth. Now this is totally my personal opinion. If you wear that and you love it, that's what's important because the most important thing about fashion is the confidence that you have when you leave your house. You want to look good because you want to look good. So when you're walking around, you feel good. So if you want to wear that, go ahead my friends. But in my personal style opinion, no. Now also, as you guys know, I've been doing this channel for like four years now. Guys always ask about my clothes back in high school. I would always get a question about where I got a sweatshirt, where I got a hoodie. Well, I have an awesome surprise for you. Now that surprise is that today's video is sponsored by Poshmark. Poshmark is basically an app where you can just clean out your closet. It's awesome. You can get really nice stuff like Nike Adidas and even designer brands for like 70% off. Well, I have a ton of stuff that you guys have always asked about that I've completely grown out of. To show you a few things in my closet right now, for one is this blue Nike golf polo. Got this from my dad. It did not fit him and I never got around to returning it. So it's now up for grabs. Two, I have this. Okay, I don't know soccer. Someone gave this to me as a gift. It's Nike. It's super nice. I'm just not a big fan. Primitive hoodie. You guys know I'm sad to see it go, but I just can't do the color yellow. And this last one, I thought this would be awesome because you can't buy the Maroon MMG OG merch anymore. But I have extras like personally, so I thought I'd throw one of these up. I thought that'd be awesome. So if you're interested, I'll also let you guys see all the other stuff that I put in my closet. I haven't exactly decided yet, but you'll definitely see it on your screen right now. For the most part, it's going to be all the Nike stuff under retail that I grew out of. So if you click the link in the description, that's going to take you to Poshmark. From there, you can see my closet, which is at Matthew Mahur, same as my Instagram. I'm going to keep updating this thing as I'm cleaning out my closet and stuff. But yeah, again, guys, make sure you click that link in the top of the description to start your own closet on Poshmark. And I hope you enjoy the rest of the video. That brings me to one of my favorite shoes right now. I literally don't know what this shoe is called. I saw it on Nike.com and it's so awesome. It's a Nike Air of some sort because it's got the bubble, but they have so much pop that's so cool. But if I'm going to wear this shoe out, if I'm going to wear this shoe anywhere, I can't wear like a matching t-shirt to this. You cannot wear any of the colors on this shoe other than white or black. If you are wearing the shoe out, nice jeans that you cuff at the bottom.  I see so many people get a shoe like this and then they want to like, Oh dude, I need coral pink on my shirt. Like I need a coral pink shirt to match with this. Please don't. Please don't. But that felt like a rant. It's not. I just want you guys to look good for the ladies. I'm trying to help you out. This shoe also a super recent purchase of mine. I would wear it very similarly to these, except these you wear more like converses and these you wear like Air Max's. So this I would wear with light jeans. I would wear a white t-shirt, light like light wash jeans, maybe with rips in them and then these.  Both of these are awesome shoes though. I love these. All right, I have two pairs left. I know I'm going to get roasted for this pair, but custom converse. I put all white everything pink interior, which is like got my sock residue on it. That's not cool. I gave it the black and white logo and then you could choose the patch and the patch color and I chose this little pink rose. I just think it's so cool. I don't get a lot of use out of these. I just thought I'd show you guys them because they're custom and they're pretty cool. All right, and finally the last shoe. I know like I said guys, I really, I really cleaned out my shoe closet and I don't have like crazy Jordans anymore. I do have more shoes than this. They're just back home, but I didn't think they were important enough to even bring. This is my all-time favorite shoe. These are my babies. I love these so much. Now I would never, ever, ever, ever, ever recommend you guys ever purchase a shoe like this. I am embarrassed at how expensive these shoes are. They're like Gucci slippers. I guess you could call it. They call it like a moccasin, but it's the most giant massive flex in the whole universe. So get flexed on. All right boys, recap of what we learned. If you're going to wear crazy, awesome, cool shoes, your favorite pair with lots of colors on it, try and keep the rest of your outfit downtoned. Always have a player of black shoes and always have a pair of white shoes.  You can have a clean out for every single day for cheap. You know, you could do like this. This t-shirt's like $10, I think. This decent pair of jeans and white commerce or white shoes. That's a great looking outfit. I would feel great going out in that. And you can do that for four or five days a week. And then on that one day a week, when you're really trying to kill it, you got something important at school, you got an event to go to, you got something you really want to do. That's when you bring out your good outfit. I know this is kind of a random video, but I like talking about this stuff. And I'm not posting madden. So I thought you guys would enjoy. Hope you enjoyed guys. That's all for now. Make sure you click that link in the top of the description to start your own closet on Poshmark. Thanks for watching as always. And I'll see you guys later. Peace out.


I'm not a big fan of Mondays and of course it's a Monday, but this is a celebration Monday And of course, I have to it's a tough job that we do most people are home right now But we have to come to work. That's good. But of course, we are expecting Sam so Jay on Zoom right now We haven't been able to get to him yet, but we'll continue to show as it comes on I've got Assas with me on the show. Yeah, good morning. Welcome on show bro. Thank you very much. And Basically Let's look at the results from the Premiership matches played from the weekend. Yeah, and There were not so many surprises, but there were so so many Way wins. Yes, there was there was one surprise the everton show see results everton coming out victorious Gritzen Park, but on the whole there was let's start with that. Something is wrong with Chelsea Definitely, they have been on a downturn in form since the rumble did match. Definitely since the Defeats are Burnabout coming out of champions the Chelsea have not been the same. They had a match against my United During the week. They drew 1-1 Everybody gave them the win. Exactly. But then this match against everton, Frank Lampard had his side well drilled and Chelsea Didn't have any answers to that performance, basically Okay, let's look at them all the matches. So what are the surprises do we have? No, I think Liverpool obviously the one I think we have the slides Yeah, for the results of the matches here. Okay, so let's have that now, please If your results, yes Liverpool won against Newcastle They needed that for them to keep the title risk of in Man City also turning out a big performance at Leeds United It's a surprise to some people that actually Liverpool could actually get a win because yes Because Newcastle are one near this park. Just one since Eddie how's coming Newcastle have been a different team And that much was a quite a bit of and might get a go from Naby Kato really well worked go But yeah, they were to dig in to get a result and they put impression man's in the title is Tottenham Hotspur against Lister City Tottenham got a 3-1 win and They've not been winning in the past few and against the Lister City and luckily for us We got a go from Kelechi Yeah, yeah, you know Yeah, but that much was crucial for Tottenham's top four ways and their big players turn up Harry Kane We've all gone human son of the other two. So it was a big performance I'm going into the North London Derby coming in this weekend. So yes Tottenham needed that been definitely and until Anthony Conta at the end of the match was excited It was delighted by the results West Ham again at it against Arsenal That was one I was thinking would be It would be a banana peel for Arsenal Yeah, they had to dig in. It was a tough match at the end of the match. The scenes was crazy Also, yeah coming up victorious against West Ham. West Ham, I think I have bigger fish to fight now the Europa League They have a match against Frankfurt to focus on this week during the week So I think the West Ham were not fully concentrated on that match But also yeah big performance from them and they're delighted to win that match Everybody thought the Manchester City would actually rest a few players because they have a match Next week, but they brought out all their big guns against Leeds United I wondered why Leeds were four-nil they win I think Manchester now from now to the end of the season They have every match for them is a final because they're being kept on their toes by Liverpool And yes, they did the one against Leeds few folders are in that outstanding form to assist Gabriels scoring goals. So yes, they have a big match this week Champereague, so yeah, they have to focus on that match. And I think that this week defines Manchester City season, definitely Well, okay. We're talking about everything against Chelsea already. Now. We're looking at Manchester United flop and Manchester and I said fanfare for the record, but they're going up against Brentford who can be But I'm not feel for any big team. We've seen Brentford beat Chelsea quite possibly and Yes, and since the signing of Princeton Ericsson, Brentford have been in superb form and Manchester United We know have not been in the best form this year. So and That match can be any scoreline. There's nothing that's really difficult to predict But now we will see this they're playing this evening. So we'll see how that turns out But I'm expecting Brentford to put in a shift at Old Trafford Mmm, okay Let's go to the French lochampioner Leon brought up a 1,000 Lochampioner victories with a 3-0 victory at High Fly in Marseille at the Stade Veldrum after the quiet opening 45 minutes Castello Lekeba scrambled home time On the opener for Peter Botts side 10 minutes into the second half Moussa Dembele double Leon's lead when he headed in his 17th league goal of the season and the cross from the right flank brought the decisive third Karl Toko Ekambi Finishing to make it four goals in his last three outings in the league one Second place to Marseille have opened the door for other challenges following two defeats in the last four Now let's look at the French when I was growing up. I didn't know I didn't hear into the Pargui so German PSG Yeah, Leon and some other teams but not PSG But Leon is still trying to get the a Marseille. I've had a fantastic run so far and Leon were able to break a record with a win over Marseille at home There are 1,000 to win in their history and I think that's nice, but Leon are they coming back? As you said That result was a surprise because Marseille are high flying this season in the second league But Leon with the 1,000 free no win. It was quite comfortable seeing the highlights there And then yeah, it's nice to see other French teams apart from PSG playing good football Yes, but definitely growing up PSG wasn't the name the house on him That is now the likes of Leon and Marseille with once dominating French football And I see them coming up against each other and playing good football Leon would be delighted to get that result because they need to qualify for Europe If not the Champions League then definitely the Europa League having lost to Estamina for the finals So it's a big result for them and yes a piece of history for the club and that they can move on from now This is the one I want to talk about and it affects all of us Every one of us in the world today, not only Nigeria in the world today Now Naomi Osaka failed to survive her second round match at the Madrid Open As she was bitten by home player Sarah Surab's Atoma Now Osaka produced plenty of unforced errors and also appeared to be suffering with a cough injury And the stubborn sorry best thermal took advantage to win 6-3-6-1 in just under an hour and a half Before I go to the technicality of tennis Let's first of all go to the fact that Naomi Osaka consistently complains about mental issues And you see when you work on TV or you work on radio or you are in the public glare People don't care How you feel people don't care what you're going through all the ones is they want to have a good time They want to enjoy what you're doing and I think it's tough on us sometimes I think when you're at the top of a game because you know Osaka has been world number one. She's she's won four grand slams So people assume that she has the confidence in her abilities, but she has personal life People don't realize that when you see a celebrity on on TV or sports personality You think they have the confidence to get through every day life for the human being just like you and I And yes, she's going through a tough time this period and she lost this match And she's not really the best player on clay her grand slams have been in eggs. I was going to go You know Naomi people are looking at the fact that the headlines that we made is Maybe it's still the mental issues. Maybe it's still she's still going through some issues and all that But the truth is from time. She's not really been the best on clay. Yeah, some people are just some players are just They struggle at certain quotes. So Naomi hasn't she's not a good clay court player. Clay course player. Yes Hard court player. She wins the u.s. Open the Australian Open. She struggles in this this this surface So yes, it wasn't so much of a surprise for her to lose this match But yes, definitely being one of the top players in the world when she does lose a match people will come up with rumors And so this meant this meant to help issues playing up But yes, she is human being just like me and I so she has she's prone to making errors as we saw in the highlights Now people suspected That she would not win this particular competition Because um, first of all, she's not the best on clay. Yeah, and then secondly people are saying that um There are some sectors of haters who actually say They don't want Naomi to win anything anymore Wow I think really because some people said it was rude for what to have left that competition and said Mental issues, but I think mental issues is a big reason to leave a competition really is a big reason I think she's entitled to have her own opinions and she's in control of It's all about life of her life basically and this match she she she lost this match because she's not good on clay as we discussed And she also she's playing A crowd favorite to span expand. Yeah So who is good uncle? He's good uncle. So you can't really fault her determination or her efforts It just she met she came up against a better opponent on the day. Oh, so sad Now Giannis acting to combo Was at his freakish best As the Milwaukee box with the Boston Celtics in their eastern conference semifinal series opener The greek dropped 24 points 13 rebounds and 12 assists Triple doubled to propel the defending mba champions To a 101 80 29 fig victory The highlights of giannis standout performance came midway to the fourth when he threw himself an alley hoop Of the backboard Boston stars jay sin tatton and jay lade brown And the order tough showing night combining for just one 10 to 31 From the floor game two isn't boston on tuesday nights Okay, now um Milwaukee box giannis and tato kumbua Can they make it happen again this season? Well, let's start with the eastern conference first Yes, this this win was a big win because they won this match at the home of the Boston Celtics at the garden Yeah, so they have the home advantage going into the rest of the the series But as you saw giannis was in imperious form 24 points 13 rebounds 12 assists He led this team from start to finish But special mention for grew holiday as well 25 points, which was um the game high But giannis isn't in form and when giannis is in form the box always have a chance Exactly the one this last time he did everything right the assists the three pointers the he did everything right even defensively He was on top of his game. Yeah, so I think but I think boston Boston Celtics will have something to say because they have too many good players for this this type of fizzle out the likes of Marco smart jason terryton will come back in and we'll see a different game next game Hopefully now katie taylor For those of you who don't know her she's the light-weight world champion in boxing and She won with a splits with a split decision victory over aramanda serrano On her historic night in madison square garden new york In front of a rashes sellout crowd now the irish woman who stretched out on beaten professional record to 21 Zero and serrano were the first woman to headline a fight at the world's most famous arena And put on a short stopping performance equal to the occasion Now taylor kept the fight on the outside to begin with before deciding to stand and trade in the fourth and fifth And she almost paid dearly For the decision as serrano connected promoter match room boxing said the fight attracted more media applications Now britain antony joshua's heavyweight fight at wembley against olblad mara flisco And against alexander usick the tottenham hot sports stadium now katie Somebody has to stop katie really In the women's boxing somebody has to stop her because she just doesn't stop winning now She's so good and people know her so much that she beats the media publicity of Vlad mara flisco alexander usick antony joshua i think she's that good She's an incredible athlete. She's probably the best. She's the best female boxer out there Like definitely lightweight division and yes this i think this fight was on the Boxing promoter johannis quoted the biggest fights female fighting history Yeah, i'm having rumors of both fighters got around one million dollars in prize money But katie taylor definitely the irish produced stunning fighters. Yes, they do katie taylor so they have a history of producing this fight I don't I don't really see anybody that can compete with katie at the moment. So it's kind of unfair How opponent this match was a tough fight about katie's two over came in the split decision. So I don't really see anybody that can keep up with katie at the moment. No Okay, now Let me leave you with irish boxer katie Now she's so good She has gotten more media publicity for our fight at the madison square garden than any other male Boxing fighter. What's that? Thank you very much. Thank you very much. La walla while he's behind the cameras You can't see him, but he can see you. Thank you very much dibola. They blew the arsenal fan Don't take him seriously really toning. Thank you very much. Downstairs. We've got d1. Dio Thank you very much for manning the controls My name is wily scott. You're not seeing time tomorrow. Like I always advise you at the end of every show If not for anything At least for your heart Do some sports