Having introduced abstract simplicial complexes and simplicial maps between them, we assigned a geometric realization, a topology corresponding to a simplicial complex. The next task is to convert the simplicial maps between two simplicial complexes into corresponding continuous functions from the geometric realization. So, today's topic is simplicial maps. Start with two simplicial complexes, K i with V i as vertex sets, a simplicial map between K 1 to, from K 1 to K 2 is nothing but a vector, a vertex map which takes a simplicis of K 1 into a simplicis of K 2, ok. So, once you have such a thing, you want to define a mod phi from mod K 1 to mod K 2, the geometric realization of K 1 and K 2. So, you can call this mod phi also as a geometric realization, it will be a continuous function, but that is the aim, but how to assign, we take the canonical definition namely if phi maps several of the vertices to the same vertex, then at that vertex, you take the sum of all the values of alpha, which are mapped onto that vertex, namely I have defined the mod phi of alpha, alpha is a function remember on the vertex sets. So, mod phi of alpha will be a function on the vertex set of V of K 2, so let V 2 be a vertex of K 2, that means it is inside V 2, mod phi of alpha will be defined as sum of all the alpha V 1 where V 1 is mapped onto V 2, there may be several points which are mapped onto V 2 by this mod phi, so take all of them and evaluate them at alpha and take the sum, ok. So, the right hand side here is a real number, a non-negative real number, this sum may be empty also, there may not be any V 1 mapped onto V 2, this sum may be non-empty, but the sum itself may be 0, for each alpha V 1's may be 0, so that is also possible, the sum total we have to see that it is less than or equal to 1, that is important, it is always non-negative that is fine, if it is bigger than 1 it does not make sense, but this cannot be bigger than 1 because all the time it is sum of, sum of the vertices evaluated at alpha, by alpha, alpha evaluated at only sum of the vertices, even if you evaluate alpha at all the vertices, namely finitely many, the sum total is equal to 1, so this should be always less than or equal to 1, ok. So, as a function it is well defined, it makes sense, one more thing we have to verify, namely when you take mod phi alpha must be a point of mod k2, then its values at all the various points, sum total must be equal to again 1, so if I vary V 2 here, then I am going to take all the points inside V 1, ok, which are mapped on to various points inside V 2, but this sum will be taken over all the V 2's, therefore all the V 1's will be taken care of, therefore this summation will become 1, ok, therefore this summation, this, this left hand side is actually well defined, defines a function from mod k1 to mod k2, ok, this summation is never infinite by the very argument I have given because they are only values of alpha taken at various points, various vertices, more over, one more thing we have to verify, look at all the points wherein mod phi alpha is not zero, that is called support of mod phi alpha, this support of mod phi alpha must be a vertex or must be a simplex of k2, ok, so why that is true, so we need to have support of mod phi alpha as a member of S2, but this is nothing but phi of support of alpha, look at all the points wherein alpha is not zero, only those things will contribute and nothing else, all of this will contribute of course because alpha of V of one of them will be equal to one of alpha phi of that, so all of them will be taken care, so support of mod phi alpha is actually phi of support of alpha, this is a simplex and phi is a simplex map, therefore phi of support of alpha is a simplex, ok, with all these things mod phi is very defined, ok, essentially you have to think of this as summing up the coordinates, so if we are working with product topology automatically this will be a continuous function, right, but we are not working product topology, what we are working with is this weak topology we have to verify that it is continuous, ok, but verifying continuity is easier in the case of weak topology, ok, restricted to any closed simplex F, ok, mod phi is actually a linear map, affine linear map, when I say linear here is affine linear namely mod phi operating upon t times alpha plus 1 minus t times beta, where alpha and beta are elements of this closed simplex, therefore this convex combination makes sense that will be also an element of closed simplex, mod phi of that is t times mod phi of alpha plus 1 minus t times mod phi of beta, can you see why, because mod phi of whatever left hand side operating upon a vertex v, ok, is taken by definition it is this function operating upon, you can see or go back to this definition, operating upon a vertex is the left hand side this alpha operating upon vertices which are mapped onto that, right, so this is operating upon some v2 and then you are taking the sum, but before taking the sum you can pull out this t, t, it is t times alpha v2 plus 1 minus t times beta v2, then you are taking the sum, so t and 1 minus t will come out of respective things leaving mod phi of alpha and mod phi of beta in the integral, so this linearity is very easy to verify, ok, so if you restrict mod phi to closed simplex f, it is continuous because it is affine linear, ok, since this is true for every mod f, ok, where f range is over all the simplex is of k that is even for continuity from mod k1 to mod k2, this is one of the criterion for continuity, that is the definition of the weak topology on mod k1, ok, so we are successful defining mod of phi for each simplexial map phi from k1 to k2, ok, this is in each chunk of simplex on a triangle on a line on tetrahedron so on, it will be linear, only affine linear, ok, there is no origin for all of them, so this is what is going to be what we are going to say, you know, unifying the concept of linear approximation now, so we have generalized just the linear map which is very rigid, ok, so now we can cut down one linear map in a small portion, another linear, another linear and so on, combination of linear, so this is what is going to happen with this one, so in some sense you can think of this mod phi as a linear map, ok, all simplexial maps give back on mod k on the geometric derivation something like a linear map, especially there will be continuous also, another interesting thing here is that under composition this mod is compatible namely first you take phi and then follow it by psi, these two are simplexial maps, you get psi-composified, if you take the mod of that it is modulus of psi-compositive model of phi and modulus of identity is identity that is very clear because look at this definition here, if phi is identity map then alpha v1 will be v1, ok, there is no summation nothing, so phi of alpha will be just alpha, right, similarly if you take another psi on this one, ok, phi mod phi, ok, if you take another component psi of that by variation of definition it is psi of this one you have to take, psi of alpha mod psi of alpha is again you have to take similar definition, so it will be the compositor, so composition is also not difficult to verify, there are finite summation, finite summation you can interchange the summation also if you like there is no need to do that but you can say that the total summation is same thing as once you take this summation and other summation, so composition is also not difficult to verify, ok. Hello sir, like for phi map where simplex is mapped to simplex, is it same happening with mod phi too? Mod phi is called geometric simplex, it is going to be inside geometric simplex, mod f you have to take bracket f closure of f, it will go to phi f closure, it will go inside that that is all, that is what we have to verify, support of phi alpha, this will tell you, this precisely tells you that, no, restricted reach implies it is inside, it is taking inside that one, support does not go out of that, phi of support, this tells you the story, support of mod phi alpha is phi of support of alpha, phi of support of alpha if support of alpha is f and phi of support of alpha is phi f, it is all, that is simplex, what you may have is here you may have an edge, there you may have just a vertex because both the vertices of the edge might have mapped to the same point, here you may have tetrahedron, there you may have a triangle only, if you have an edge here it will not, the image will not be tetrahedron because there are only two elements, no, function, saturated function will take only smaller number of points, if at all, at most that many, these are all finite sets, phi of any of those that will have at most that many elements, but it is simplex in the other simplexial complex, K1 and K2, of course K1 and K2 may be the same also, that is why you can talk about the identity map, identity map is a automatically a simplexial map, now we go ahead what we do with simplexial maps and so on, by a triangulation of a tupholoacal space, we mean a pair, a simplexial complex, K is a simplexial complex and F is a homeomorphism from mod K to X, you start with a tupholoacal space X, it has no structure of a simplexial complex or anything, alright, but K is a simplexial complex and mod K is a tupholoacal space, F to mod K is a, F from mod K to X is a homeomorphism, which is the definition of a triangulation, the word triangulation is borrowed from when you take a surface, you cut it into triangles, but now we can use it for 1 dimension, 0 dimension, n dimension, 50 dimension, all of them, the word is only triangulation, there you do not see triangles, if you go to 3 dimension, there will be triadrons and so on, okay, if X has similar in measure theory, you call the time call volume, 2 dimension volume is area, 1 dimension volume is length, but we do not have many words, so 4 dimension, 5 dimension, we are just taking volume, it is just like that, so if we select a specific triangulation on it, then we call it a simplexial polyhedron, okay, there may be several triangulations for a given tupholoacal space or there may not be any, if there is one, it is a triangulable, if you fix one, then it is called a triangulated manifold or triangulated space or a simplexial polyhedron, sometimes just polyhedron, okay, so do not confuse this polyhedron, it is a convex polyhedron inside RA, convex polyhedron is a special case of a polyhedron, that is all, now we will have some examples here, one by one, okay, recall that the joint of two tupholoacal spaces were defined as a quotient of X cross I cross Y, wherein X cross 0 cross arbitrary Y1 was identified with X cross 0 cross Y2, Y1 Y2 varying over Y, similarly on the other end X cross 1, X1 cross 1 cross Y was identified to X2 cross 1 cross Y, X1 and X2 varying, Y is kept fixed, so this was the definition of what, of the topological joint of X and Y, right, and we had also defined the joint of two simplexial complexes, so these two are not extental, there is a close relation between them, so if K1 and K2 are any two simplexial complexes, then the mod of K1 star K2 is the joint of K1 and K2 is homeomorphic to mod K1 star mod K2, okay, K1 is a polyhedron, K2 is a polyhedron, I mean mod K1, K1 star K2 you do not know that it is a polyhedron, not only it is a polyhedron, it is triangulated by just K1 star K2, mod of K1 star K2 is homeomorphic to this, is the conclusion here, indeed this homeomorphism is such that remember both K1 and K2, mod K1 and K2 are subspaces of this joint and here K1 and K2 are sub complexes of K1 star K2, therefore here also mod K1 and mod K2 will be sub complexes of mod K1 star K2, under this homeomorphism these subspaces are mapped to the corresponding spaces, so that will be easy, that will be, that will follow by just looking at the homeomorphism that you are constructing, okay, so what do we do? We construct the map at the mother level, so this is in the left hand side remember what we have is K1 star K2, I am defining map from mod K1 star K2 to mod K1 star K2, so this is a quotient space of mod K1 cross i cross mod K2, so instead of defining it on the joint I am defining a map on the mother itself, then I am verifying that it factors down to the quotient, this is the way continuous functions are defined on quotient spaces, okay, so how do we define this one? An element of this remember can be thought of as a line from a point here and a point there, line segment, right, so a point alpha and a point beta and a point this some value t, so this is a product, right, there is some identifications later on, so I am not going to use identification, I am directly taking the product, assign it to 1 minus t times alpha plus t times beta which is finally supposed to be the line segment, that is why I am defining it like that. Now alpha and beta, what are they, the 1 minus t times alpha plus t times beta should make sense here, right, right, so how does this make sense here? Alpha and beta are functions from vertex set V1, alpha vertex set V1 to i, this vertex set V2 to i, remember K1 star K2 was defined as vertex set of K1 star K2 is a disjoint union of vertices of K1 and K2, V1 and V2 for example, therefore alpha which is function on, function on V1 and beta function of V2 makes sense provided you think of this as 0 on the rest of the chord, that is the meaning of this, on V2 definitely it is 0, this will be 0 on V1, inside V1 whatever definition is there you take that, here definition you take that and now 1 minus t times alpha plus t times beta make sense is whatever, okay. So first of all you have to see that if I put t equal to 0, okay, then beta is not at all coming into picture and this t is 1, this will be alpha. Similarly if t is just 1, then alpha does not come into picture, this will be just beta. Therefore the very definition of this identification makes that mod phi, this phi defines a factors through a continuous function from phi, namely phi twill for mod K1 star K2 to mod K1 K2, okay. So definition is fine, now we have to verify that it is a bijection, continuous the universe is a bijection, okay. The bijection is already taken care by the very fact, by the expression here, what is the meaning of expression on this side, okay. Here you must remember that simplices here are just disjoint union of simplices here and simplices there, right.  So there is a map, let us see why this is, why this is a continuous bijection. Suppose Fj, F1 is K1, inside K1, F2 inside K2 are some faces, then restrict the entire thing to just these faces, okay. The image never goes out of that in the definition.  Okay, now remember this is a full simplex, okay, this and all subsets of this one will be there inside K1 star K2, alright. How many elements are there here? Cardinality of F1 plus cardinality of F2. How many are there here? They are also exactly same and this map is nothing but, you know, this is automatically an isomorphism. In this picture, which is nothing but, you know, it is 1 minus t times alpha plus t times beta dilute all the elements here. So every element here can be broken up into an element here plus element here. The element terms, they may be 0 also, it does not matter. 0 times 1 minus t times 0 plus t times this one, when you take, it is nothing but this, all the functions here, that is what it gives you. So this is a homomorphism, okay. So here you can directly verify that this is actually linear, affine linear isomorphism, okay. From this one, automatically everything follows, okay. Why? Because if you look at the interior of these things, they are not going to, they are not going to intersect each other. So the bijection on the boundary will be bijection on the boundary from boundary to boundary, interior into here. So on the union also it will be a bijection. The second point is easier, namely continuity of the entire thing is verified by continuity restricted into each, each simplex and simplex is just like this. So that is continuous. Therefore it is continuous on K1's mod K1 star mod K2. The converse also for continuity on from this way biject, sorry the inverse, again the same thing on F1 distribution F2, where F1 and F2 we range your all of them. If you verify the continuity, restricted to that, then it will be continuous from K1 star K2 modulus to the K1 star mod K2, the inverse map, okay. So inverse continuity is also verified by this, this homomorphism. Therefore the simplicial joint corresponds to topological joint when you take the geometric realization. So that is the meaning of this. And this feed today is very canonical way we have obtained. You see there is no other structure here, it is just the structure of these intrinsic structure of these two things are used. So this will automatically give you K1 star K2 star K3 all of them. So SOJ2D or K1 star K2 K2 star K1, okay. So this is what you call as canonical bijection, homomorphism. Under a simplicial map, if K1 to K2 there is a simplicial map, K2 to K3, then you go there, correspond a simplicial map, there will be commutative diagram under these homomorphisms. This is the so generic thing but when you specialize, you can take, pay anything but K2 second one will take a single point. Then by definition K star V is nothing but a cone, the cone over K. First take the cone over K and then take the geometric realization, the same thing as take the geometric realization and then take the cone over K, okay. So this one is a cone over a simplicial complex K is in fact the topological cone over mod K, alright. Because it will be K star, mod K star V star with a single point space in the case of topological zone is just a cone, that is what we have seen. In particular K star V mod is contractible, okay. Similarly, if you take X star two points, S naught, S naught has two points, right. You see the suspension we have defined, right. So you can take K star two points here and then take the modulus that will correspond to suspension of K, okay. So this also this part we have already seen, repeated application of S naught star, S naught star, S naught gives you N plus, S naught star, S naught, yeah, S1. So N plus 1 copies will give you S n. This is the old thing. In particular, if you start with some simplicial complex K such that geometric realization of that is a sphere of some dimension, let us say J, then if you take the joint of these two, then take the geometric realization between the same thing as joint of the two spheres, corresponding sphere P1 and P2, which will be equal to S P1 plus P2 plus 1. So this is what we have seen, joint of these two. So this is a special case of this one. So you can use, you can use many special cases to derive various results from this joint. So let us say some more examples, simpler examples. Now the convex hull of our E naught, E1, E n inside r infinity is easily seen to be geometric realization of standard simplest delta n. That is what we have seen actually by definition. We shall now use symbol, now mod delta n for both of them. Whether it is inside r n plus 1, r n plus 2 or r n plus 3 or r infinity, we will use the same symbol mod delta n for geometric realization, the standard n symbol. All spheres and disks are triangular. It means what? There is a polyhedron, there is a simplicial complex, mod of that simplicial complex will be isomorphic to these spaces. So what are these? Suppose n is 0. What is D naught? D naught is a single point, what is S naught? S naught is two points. So there is nothing to prove there. So singleton, double-ton, they are triangular. The simplest polyhedron is mod F for any q simplex F, which is if its dimension is q, it is homeomorphic to delta n mod delta q, that is what we have seen. So we have been telling that this mod F is homeomorphic to the unit disk D q inside r q. So just to recall all these things, I will tell you how we have done it. Namely start with a set of vertices V naught, V 1, V q for F. An element of mod F is described by describing its coordinate functions, values of the coordinate alpha naught, alpha naught, alpha q. It is a function from this set into i such that sum of alpha i is equal to 1, right, where each alpha is nothing but alpha of V i. Thus mod F is subspace of i raised to q plus 1 elements here, consists of elements alpha naught, alpha q with summation of i is equal to 1. You can see that it is the convex of if you take V naught, V 1, V q corresponding to E naught, E 1, E q. Rotate through an angle pi by 4, this will be having a pi, it will make pi by 4 angle with the x axis or x 1 axis, x naught axis whatever. Rotate it so that it will become, it will be contained inside some horizontal plane r q cross t. Next, choose the origin at the barycenter of mod F. What is barycenter of mod F? It is V naught plus V 1 plus V q divided by q plus 1. You think of that as origin. Then you take x going to x by mod x, x by norm x to get the homeomorphism of the boundary phases, union of all the boundary phases to the round sphere s q minus 1. This x by norm x is modulus 1. So, here is a picture. This is a triangle. You can say this one is E naught, E 1, E 2 and the span of that, its triangle is making 45 degrees. You rotate it, bring this point to the x y plane, the 001 x y plane by rotating it rotation because I do not want to change this geometry. Just for a while. So, then this has become horizontal in the horizontal plane. You choose the orthocenter or barycenter as the origin. Then you take x by norm, project it. All of them, this boundary of this triangle goes to the sphere homeomorphically. This is just like a projection map. Once you have a homeomorphism of the boundary to this one, you can use the so-called cone construction to say that the full disk is homeomorphic to the full simplex. So, I will describe that one. So, this is what is going to have triangulation of the pair. This is supposed to be the full disk d cube and the boundary simultaneously of the pair gets triangulated. Triangulation means now what? You have a simplexial complex in the boundary of simplexial complex giving you, when you take mod, respectively it will be d cube and s cube minus 1. So, I will describe this cone construction once for all. Again and again, this will come.  So, this, I want to say that this is going to give you s cube minus 1. So, this part we have seen. Now, you take the cone over this, that is going to be back, the same thing as f is going to give you f back when you take the modulus. If you take the cone over f, see for each subset of f, you have add the extra point. So, that will give you the full set f. The maximum set f, one subset, one point was missing, the cone will come there and it will be the full set. So, this is one way of looking at the boundary of Bf. The cone over that one is homomorphic to gain mod f. Mod level of Bf is homomorphic. So, this is what we have to use on both sides now. So, this is what I am saying. Now, epic to the following fact, simplexial complex f is isomorphic to star of, this is the star, a single point is a cone. This is the join of the single point and Bf. Therefore, mod f is homomorphic to the cone over Bf. I take Bf and take the modulus of that and then take the cone. First you take the modulus or take the first take the cone or then modulus, this gives the same thing, which is in turn homomorphic to the cone, which again is homomorphic to the full disc. This part also we have seen. So, cone over any sphere is homomorphic to the disc. Let us stop here.


Hi everyone. Happy Friday. I'm Joe for jaspyscasebrakes.com and we're coming at you with 2020 Bowman Sapphire Baseball five box picker team and number 16 Got another round of Sapphire baseball up on the site. Go and get it ladies gentlemen jaspyscasebrakes.com All card ship and every single card is going to be penny sleeved and obviously the top The top prospect names and the parallels obviously will be will be top-loaded as well Now if you bought at least two teams, you've got a chance at the Tigers. So let's see Who's gonna get the Tigers in this quick little five box break? Let's sort by column a Adam got two teams. That's an entry John Samuelson got two teams. That's an entry Chris with four teams That's two entries EA also has four teams. That's two entries Jack has three teams, but the purpose of promo To count for the promo. We still get an entry Joseph entry Joshua B entry Mark the set four teams two entries Michael Coons two teams an entry and TJ two teams an entry All right now. Let's grab those names here and good luck. Let's roll it Let's randomize that list five and a one six times name on top gets the Tigers after six one two three four Five and six and final time after six there you go Chris Garosi. Congrats to you You have the Tigers in this one Good luck Spencer Torkelsons are what we're chasing of course You can see the boxes on the top camera right up there as well Good luck. Now. I don't know if I know Sean Jaspy did a number of these sapphire boxes last night Does anyone remember a Spencer Torkelson? Parallel any color on Spencer Torkelson? So I did a bunch of breaks yesterday. I did not see we've so we've seen a handful of Spencer Torkelson cards Just the regular sapphire cards, but no parallel also no one of ones in general Oh, there was an orange Spiro said oh, that's nice Or a couple I guess I'll have to we'll have to find the one of one then all right and remember all card ship All cards will be sleeved nice peak crow Armstrong For the Mets that'll be for Joseph and your parallels first one is a Yellow angel David Calabresi 38 out of 99 for the halos. That's gonna be for Chris Chris Garosi Yeah, I think the ace of Lacey's will probably be top-loaded When I see the next one, I'll do it JJ Goss. That's a goal to 15 nice nine out of 15 I remember everything will be sleeved though And I think the shipping team will be We'll take care of any of the ones that I miss All right next box Let's roll. Good luck. All right, you're nothing too crazy. They're your parallels. We've got a Jordan and wogu green 41 out of 50 Cubs EA with the cubbies and behind Ethan Hankins is a Ray Ian Seymour 15 out of 99 yellow parallel for TJ and Tampa Bay next box Parallels in the middle boom Let's see what the regular sapphires hold and when you get these folks especially for the Bowman first guys You know, it'd be worth looking up even if it's just a base. It's worth looking up the value Or just holding on to these for a while You know to see, you know when they get called up a lot of value in this sort of stuff. All right We've got Ben Hernandez 14 out of 99 yellow parallel for Kansas City Joshua B second rounder And behind Bryce Bonnan is Jack Blomgren Nice purple three out of ten really nice low number for Adam Ewald and the rocks there you go Adam Bryce Bonnan right there next up. It looks like some orange in there. You see a little orange peeking out there All right, here's a Zach vene for Adam in the Rockies He's pretty solid ninth overall pick in 2020 last year and The orange is Tink hence 25 out of 25 Nice one for Jack and the Cardinals one of his last spot mojo teams 60% of the time last spot mojo will get something 100% of the time There you go Jack. Thank you. And we got his name is Robert Pausen His name is Robert Pausen 75 out of 99 Oakland A's Joshua B Highly tied international prospect next little stack here. Good luck. We got plenty more of this Well, not plenty more of this. I think the brakes that you see on the site Jaspy's case breaks calm. That's it That's all we have So after that very final break of sapphire sells out there will not be any more Next up here. We've got Bryce Jarvis 13 out of 99 Joshua Lucas with the diamond bags and the orange behind Hunter Bishop for the Giants is gonna be Brady McConnell 21 out of 25 orange for the Royals. That's a different Josh Joshua beat With that one and there you go ladies and gentlemen. That was five bucks Pick your team number 16 of 2020 Bowman draft baseball sapphire edition more in the store go and get it while supplies last I'm Joe. I'll see you next time for the next one. Bye. Bye


Welcome back. You're still watching news now on Plus TV Africa. And on the foreign scene, Tunisia opened polling stations on Sunday for run-offs in a parliamentary election that drew only 11% turnout in the first round last month. An outcome critics of the president said undermined his claims of public support for sweeping political change. With political parties boycotting the vote, most candidates are independent and attention is likely to focus on whether there will be higher participation than there was in December. President Kai Said has decreed the new mostly powerless parliament as part of a reconfigured presidential system that he introduced after shutting down the previous parliament in 2021 and assuming broad control over the state. Said's critics accused him of seeking to dismantle the democratic system enacted after Tunisia's 2011 revolution which sparked the Arab Spring and the ridiculed December's ultra-low turnout as evidence that his changes lack popular support. The president says his actions have been legal and necessary to save Tunisia from years of economic stagnation and political crisis and has accused his critics of treason. Urgent action against them.


Stevenson holding on for a 3-2 victory over Newman. I'm Christian Taylor, head coach Tori Amoff after this one. Found a way. How was your team able to pick up this victory tonight? Fairly. We hung on tight at the end. It was chaotic for sure. But honestly, 30 is really what held us in the end of the game. She faced 30 shots, including rebounds, just in the third period. And if you watched even a minute of it, you'd know she stood on her head. So we're going to give her that one for sure. Obviously, we had 22 put up a few goals for us tonight, which was very helpful. So tomorrow, we're looking for more of an entire team effort so that we can be a little bit more comfortable in those last few seconds of the game. Yeah, I mean, how about Senor? I mean, first-ever game out there, making her way over from Mechelon, Wisconsin. How impressed are you with her? Quite a debut. She definitely showed us, the team, and herself, that she can make those big saves. She can come in and clutch moments. And it really was that energy boost that we needed on the bench and those momentum shifts that we needed throughout the game. Of course, you mentioned Carly in that hat trick today. I mean, this was a team that lost a lot of good attackers, like Trinity, like Nikki, but how has she now stepped up into this starting attacker role? Fantastically. And it's what we hope for. It's what she hoped for, it's what our teammates hoped for as well. And I'm really excited that she's seeing that kind of happening now so that she has the confidence moving forward to keep doing that throughout the season. Of course, thank you very much. Congrats on the win. Thank you. I'm off. I'm Christian Taylor on GoMustangSports.com.


Ranger Bill, warrior of the woodland, struggling against extreme odds, traveling dangerous trails, fighting the many enemies of nature. This is the job of the guardian of the forest, Ranger Bill. Pouring rain, freezing cold, blistering heat, snow, floods, bears, rattlesnakes, mountain lions. Yes, all this in exchange for the satisfaction and pride of a job well done. Hi there, boys and girls. Today I'm going to tell you about an adventure that wasn't a bit like anything we'd ever been up against before. Just before I started to talk, you heard a certain noise, and I'm pretty sure you recognized it as the crack and whine of a rifle. But this was a very special rifle, one with great power. This kind of rifle is called an express rifle, probably because it shoots bullets so very fast. Well anyway, all of us, but especially Henry, will never forget that particular rifle. We'll find out why in the story I call The Secret of Seal Island. Bill looks sad. Bill is sad, Ray. Well, I don't like the idea of having to type out these long reports. I'm an outdoor man anyway. Besides, I'm not much of a typist. And yet, all of that wouldn't matter if it wasn't that these are such unhappy reports. Yeah, same old thing. Yeah, same old thing. That poacher. We catch him someday. But when? Half my men are on the trail. You've been out, I've been out, but we never even see him. He hard to find that for sure. I've got plaster casts of his tracks, photographs of his tracks. I know those boot marks perfectly. But I've never seen his face. He's a bad man. Shoot more game? Yeah. Now he's unbelievable. My men found three more grizzly carcasses today. Three in one day. How you know for sure, this man killed all this game? His tracks. Plus the fact that he's an excellent shot. Game poacher, no good. He's been shooting stuff out of season, anything he wants, whenever he wants it. Take hide and sell, huh? Oh, sure. He's a fur hunter, all right. You know it's a good market for bears, kids, even though the animals are shot illegally. Poacher, very bad man. Kill game out of season. Kill any game he wants. Same as stealing. Yeah. What gets me is that we don't even know who this man is. Indy never named for a man like that.  What's that? You know, Luke Karo. Wolverine. Oh, yeah, sure. Wolverine, only animal in the world who killed just for sake of killing. Very fierce, smart. Nobody ever see, hard to catch.  That certainly is a good description of our poacher. Well, at least we have a name for him now, Grey Wolf. We'll call him the Wolverine. Not far to the top of the rimrock. Keep heads down. Yeah, I know. And when we get to the edge, we can see the whole valley spread out before us like a map. See, Bill, see little pile of rocks there at edge. We go there, look through binocular, and not have to stick our heads up over top of skyline. Good. You see anything, Bill? No, not yet. I'll quarter the whole valley. You think the Wolverine is up here? Tracks go in, not come out. Well, I certainly hope... Grey Wolf. You see? Yeah. Here, take the binoculars. They're over there on the far edge by that chimney-butt. Big man. Big black beard. Let him all right. But how can we be sure it's the Wolverine? Here, you take a look. You see pack of furs on ground beside him. Maybe two. Maybe three bear hides. Oh, yeah. Nobody else have hides this time of year. Of course. Well, now that we've seen him, how do we get him? We go back down, back into valley, and circle way, way around, long way. Walk quiet. Come up behind Wolverine. Catch him. But it'll take us most of the day, Grey. Yeah, he'd be there. He'd wait there. Watch for a game. We catch him. Come on. When we saw him, he was just on the other side of this little chimney rock. And here's what we'll do.  Right. That way we'll come on in from two sides at once. You ready? I'm ready. All right. When I count three. One. Two. Nobody here, Bill. So I see.  Well, it's a note. For us. He left it here for us. Well, what note, say, Bill? Dear Ranger Bill, you're supposed to be a woodsman, but I guess you forgot that if you're facing into the sun, then the sun glints on the glasses in your binoculars. I saw you before you saw me. Too bad, all that long walk for nothing. You'll never catch me, signed... Wolverine. The Wolverine. Old summertime. Why on earth do you play that tune? All you do is complain about the heat. It kind of takes my mind off of my suffering. Hey, just fling a few more of them ice cubes into this here. Lemonade here, will you? Okay. Here you are. Hey, thank you. And then made up a tune all about her. Mr. Bobble, I feel when she's as hot as this. Okay. Let's hear it. Does hot weather affect you that way? Terrible. Please, give me some more of them ice cubes. Not many left in the bucket. Here you are. Hey, thank you. Now I've done rid another tune, and only this one is about how good I feel when she's nice and cold. Go ahead. Is that part of the title? About the ice cubes, I mean? Not sure. It just aren't anymore. They're all gone. None of the refridge, either. You used them all up. I never can get enough ice cubes. I wish I was sitting right on the iceberg. Guess what I wish. What a thing to say. Why, none of us will ever see an iceberg.  A hundred in the shade and no ice cubes. Gray Wolf, do you get seasick? Well, that Bill, do you feel okay? Sure. Henry, how about you? Do you get seasick? I don't even know. I've never been on the sea to find out. Same here, but why the question? It's plain, and it's a mighty, mighty strange story, too. It's in three parts, really. Part one is, do you remember how we hunted for months and months for that game poacher we called the Wolverine? We never caught him. Sure. We haven't had any poaching for a long time now, so we just figured that you and Gray Wolf at least scared him out of the country. Exactly. You've happened to hit the nail right on the head. The Wolverine is out of this territory. Oh, good. But he's still up to his old tricks. Poaching. And that's part two. I have here on my desk a long-detailed report from Washington and from the Alaskan Game and Fisheries Commission. It appears that somebody has developed a new and fiendish method of shooting the fur seals up in those arctic waters. That's against international law. And who, evidently, has a sloop or small vessel with a supercharged engine, has been shooting the rare fur seal, probably from a long distance away, from his boat, no doubt, so as not to startle the herd, and then skinning them and, we suppose, selling the highest.  Why don't they catch him? They've never even seen him. They do have the plaster casts of his boot tracks and micro-photographs of the bullets he's used. Washington compared the Alaskan report with mine. I get it. Yeah, the plaster casts of the footprints and the micro-photographs of the bullets they match. Mm-hmm. Beyond a shadow of a down. It's our old enemy, the same man doing his poaching now in Alaskan waters. Wolverine. I think so all the time. But what this got to do with us, Bill? Well, this is part three. And it sure sounds strange on a blazing hot summer day like this.  So Washington wants us to go along on the expedition that's being fitted out to capture. We can identify him positively. The arctic. Mm-hmm. And since it's summer and you have no school... Hey, you're going to take me along. Yeah. Stompy can hold down the fort here. Since we are going right up into the arctic ocean. Enjoy this hot weather while you can. We're going to where it's mighty, mighty cool. It's getting colder by the minute. Another way to get them, colder it'll get. Cotton storm weather? Are we in the first seal waters? Two, three more days, boy. No time sailing vessel like this one don't go as fast as in modern ships. I like this kind better though. But how do we know when we're in the first seal waters? Still worrying, huh? Well, first we sight a seal or maybe a couple just little brown heads bobbing in the waves. Then we'll keep sighting more and more until finally we get to the islands. The islands? Yeah, seal islands we call them. Just rocks sticking up out of these here icy seas. The seals like them. They gather there by the hundreds. Thousands if it happens to be a big enough island. Mostly they're harmless critters and easy to shoot. That's why it's again international law to shoot them. A man shooting at them from a long ways off can just shoot about as many as he likes without the seals catching up on as to what's going on. That makes it cruel. Oh, look ahead, boy. Kind of nor by nor east. I could see something captain storm weather. Is it an island, a seal island covered with snow? Nope, that ain't no island, boy. That there is an iceberg. And a big one too. That old bird there means we're getting up into the northern seas. Up into the land ice and snow. Sure is big. Yeah, plenty big. They come bigger than that sometimes. You know much about them? Iceberg? Well, we learned in school that the part you see above the waterline, I mean, that's only a tiny part of the iceberg. Right. One eighth of the iceberg sticks out of the water. The other eight, seven eighths is under the water. Oh, that's why they're so dangerous, isn't it? Well, your ship could hit the underwater part of an iceberg, even though you thought you were far enough away to pass by it. Yeah, we'll just circle around this and give it a wide berth. Captain, what happens to icebergs? Oh, when they get down into warmer waters, they just mellow away, boy, just mellow away. If they get into regular steamer lanes, the government sends out warnings. What's that? Iceberg, you watch. Wow. What? That whole iceberg split practically right in half. There are two icebergs there now. All right. That's called calving. It's a pretty sight to see, but we're lucky we weren't any closer. Captain Stormwell, over there on the right. Captain Stormwell, I see it. I see it playing his day. Yeah, that's so do I, boy. But it ain't over to the right. It's over to the stabbered. But I will say you got mighty sharp eyes for a landlubber. You spotted it first, and that means we're getting there. That little head of bobbing along the water. That's a fur seal. Islands all around us now. Yeah. We pass quite a few of those bare rocks you call islands. We haven't seen any seals. Just a few of them in the water. There'll be plenty as soon as we get further up. We may not have to go as far as you think. See that little island off there to Starboard? Captain Stormwell there? Let me see. Let me focus my spyglass here. Those tiny dots. Are they seals? Yeah. And what's more? And what's more, they ain't moving. They ain't moving a mite. And that ain't good. Your hands! Get busy on them davits! Stand by the lower of the stabber along boats. We're going over there! There they are. All dead. All shot from a great way off, looks like. And the pearls have been taken off slick as a whistle. What do you got to say, Bill? Three nations have signed the solemn pact to prevent this vicious, wanton illegal poaching. I know. Who'd done it? Like a tracks and snow. I know those tracks anywhere. To Indian footprints, same as photograph. Those footprints belong only one man. But who? A man we call Wolverine. Yeah. Shooting and skinning for seals is against the law. Sometimes certain restricted permits are issued, but we know that this fellow has not. What are we going to do now, Bill? Henry, we follow this man all the way to the onyx. We know now what he's up here for. It may be a long and a difficult pursuit, but we'll catch him. No sign of him, Captain? There is a sign. We've been tacking back and forth in these waters for ten days now. Yeah, I know. We're about as far north as we can go. Pretty soon we're going to have to turn around and then...  Bend down those hatches, men! Make everything ship shape. We're in for a blow. Hang on to that railing, Bill. She's getting rough. I'll say it is. Captain, do you think... Jordan, sail there! This looks like a humdinger. That would better. Hey, uh... Captain Schwann with it. Look! I'm looking. Right where you're looking. I see the same thing you see. A slope, riding with bare sticks. A rig like that couldn't carry a square inch of canvas in this wind. A long leanship, painted black. She's trying to use her engines. See that propeller wash under her stern? She's turning, turning away. Certain. She's seen us and she's trying to get away. Well, that must mean... Of course that's what it means. That their exhilarated swoop is the ship we're after. Otherwise she wouldn't be trying to buck the stern and get away from us. That's the Wolverine ship, for certain. But what can we do, Captain? We can catch her. That's what... But she has a powerful motor. This is just a sailing vessel. All the better. These waves make a motor useless. See how the propeller lifts right out of the water? Bill, you'll see all the sailing vessel can save the day. This wind is their enemy. But our best friend... Throw on more canvas there! We're chasing the Wolverine! What are we going to do? What matter, Henry? Look at this fog. They can't see ten feet ahead! I'm afraid the Wolverine ship will escape. Don't forget. They can't see either. They have to stop and go slow the same as us. I suppose. Bill says we've been chasing that black swoop for thirty six hours now. Day and night, but... of course the sun really never sets. ever sat there you might be long what's been our turn on watch for a long time it seems yeah hard to watch for nothing to see but fuck why I hope that other boat doesn't get away everybody hopes same thing Henry yeah I know but hey Grey Wolf do you notice anything unusual no something some seems different some oh they will find out what it is the fog it's left and see it's getting clear by the minute we keep sharp look out now maybe you see black sloop right if you take the leeward side I'll take the start right I'd like the captain told us to fog must have gone blow away quick only these these ways were so high what's that silly gray wolf the sloop the black sloop right here boy only about a quarter mile away both of us had to just ride a for the wind during that fog now that she's lifted go on behold we're ending up closer to the black sloop than ever before hard of helm and give chase right and looks by the time we catch up to him we'll be right in the lee of that monstrous iceberg up ahead and that'll be a help on a condo it'll break the force of the wind yep they don't never come much bigger than that one boy that's because we're so far north it ain't hard had time to break up none I sure is red biggest I ever see to come to think of it now what do you suppose is the matter with them fellas that's loop huh what do you mean they must be plumb crazy they're near enough to that bird they ought to be veering but instead they're steering straight for it maybe a runner broke oh she'd yaw if that happened nope they're steering her straight toward that I wonder they're up to some sort of trick I think sartan but what it is I can't figure not for the life of me my stars are going to make a landing right on that iceberg that's right but what for they can't escape away from us that way besides nobody can live on an iceberg not for long anyway little boat right up by iceberg now she shallow draft well look at that somebody's getting out somebody's being made to get out Henry see those rifles those men in the boat are pointing at the man who just jumped out of that iceberg looking just about see still pretty far away this is incredible watch the sloop captain storm with her yeah she's doing just about what you and me had figured she do bill uh-huh there she goes back and away and turning and head not to open sea they left that man alone on the iceberg right and if you take a good close look at him with a spyglass I'll guarantee it'll be the Wolverine why why they've marooned him Henry those men and his crew have figured he was too dangerous to be with so they put him off in that iceberg marooned him what about the sloop bill now let it go can arrange to get it later right now we're going to follow our original plan we're going to go and capture the Wolverine and not stay still he'd take off back through those ice hills we'll find him once we set foot on the iceberg we'd better hurry we are hurrying gray wolf sure would better make more hurry man not alone on iceberg what but an indian sees more than we can Henry never see before but see now you look over biggest ice hill great white bear i see it a polar bear and a giant one that poacher has done been marooned on an iceberg with a polar bear no sign of the Wolverine or the polar bear and either one of them can be very far away this is a big berg that we could walk clear around it in 10 15 minutes i'm sure glad we got these shoes on with a spike gotta have him boy now then let's spread out keep about 40 50 feet apart gray wolf you go over to port there right and i'll take the next position then you bill right and you boy take the other end okay now then we'll keep in sight watch where you put your feet get a good grip before you take the step and go slow and above all don't do no hollering but that man knows we're here sure he does sunny icebergs is funny a loud sudden noise might start one to kraken we don't want this thing to bust apart while we're on it oh well you don't have to whisper and if and you do call call easy like we understand let's take our positions and go forward now slowly sooner or later we'll see the poacher what we do then bill i'm counting on him giving up surrendering when he sees that we mean business he lifts his rifle to fire at us you shoot shoot right out of his hand gray wolf i will captain say big noise might split iceberg that's a chance we'll have to take and the polar bear now he's probably died in the water swim away but now what if he hasn't that's another chance we have to take everybody ready and i'd like this over in plenty of bad man poacher no good never we should see him pretty soon and i'd like this iceberg said foward show up for the bear him or the bear both he must be laying low some of his probably watching us can't say i realized this expedition no how i thought surely we'd see him before this almost across the iceberg must be hiding behind an ice hill waiting for us and then i wish he'd stay closer wow it sure is cold he's lonely well says i should keep even closer if i see us getting away let's see i'm getting farther away there he is where over there to the right over behind that ice hammock over by henry henry watch out that's him all right black beard and all now that he sees us he's standing up and have rifle bill i wait for your orders you say fire are ready good i think the bear the polar bear he's right behind henry and coming fast can the boy shoot yeah sure but look look his rifle's jam must be froze tight ray wolf you're the best shot get that bear i can't shoot from here henry between us and bear if i shoot now i hit henry henry run run to the side henry henry slip and fall bear coming fast shoot gray wolf henry still in line of bullet let's stop the bear he's down for good gray wolf how can i ever thank you for i not shoot bear i'm not dare shoot huh get henry other man shoot bear save henry's life other man why why then it must have been she's gone she's cracking that shot done at this here iceberg is cracking in two what do we do now hang on and pray kept the storm with her all shook up but still ship shape henry take a gander over yonder this old iceberg done split right spang in two and over yonder's the other half they're setting on it is henry and that poacher oh no like to me lord make everything all right like to me like wolverine binding up henry's leg taking good care of him it's a miracle the dingy still there let's get her into and roll over there fast as we can roll come on so when i saw that polar bear coming for me oh i was scared when i found my rifle was jammed then i fell and sprained my ankle well then i was really scared i tell you bill i never prayed so hard in all my life and your prayers were answered henry and you wolverine i don't know your real name how was it you decided to fire at the bear instead of us you saved henry's life i done plenty of bad things in my life i ain't no ornament to society but i can't stand still and see a young boy clawed up by a bear well we're grateful you know that plenty grateful he fixed my leg and everything i ain't looking for no thanks no favors i done wrong in a north i'm ready for my punishment but i done found out one thing found it out in spite of myself you might say well what the like you say i had the choice of shooting you or the bear when i was standing there in them few seconds something come over me like a voice saying to do the good thing shoot the bear now can you understand a thing like that yes yes i can and on the long voyage home i think we can explain a few things to you so that you will understand too i'm ready to listen henry what on earth are you doing i can get some ice to take back to stumpy he always needs ice for his lemonade well see you next week for more adventure with


what was the central issue with the mixed flow reactor in non-isothermal asking question is very bad think you know starting straight way is best I think ya Anupriya what was the problem why you are so much worried about this fellow mixed flow reactor so what happens feedback is always happy what happens what is the issue there are multiple study states ya I mean why should I worry if there are multiple study states because you will be either getting exceeding temperature or you will be getting lower conversion ya particularly unfortunately when you have calculated from energy balance a temperature which is in that middle point okay and also you have conditions corresponding to a line which goes through that three points and unfortunately what you have is only middle point without knowing so then you may be thinking that that is the temperature may be around 70 degrees 80 degrees but suddenly it may shoot up to 150 or 200 degrees where everything will be burnt okay so that is the reason why immediately after design is done for mixed flow reactor non-isothermal design then you have to check really whether these study states you know the particularly temperature whether it is correct temperature or not if it is not correct temperature and definitely when you are well you have to operate only at that point that middle point because if you go above some explosion may be occurring if you come below there may not be reaction at all so you have to operate only at that point what you have to do now you have to go for control and have an excellent control and if it is going away see earlier on its own if one temperature increase is there one degree centigrade so then it was coming back on its own you know under study state conditions and even if there is one point less and one degree less less than the operating temperature may be 199 again it will go back to that particular temperature but this unsteady state temperature is dangerous if it moves up then it goes to the next study state may be 200 kilometers 200 degrees away so then you will have a problem similarly when you come down then it may go to that that is called extinguishing point that means the entire reaction may get extinguished that means particularly in the case of combustion that is called extinguishing like you know when I told you when you put too much gas why it is switching off on its own was that the amount of gas is more and the heat that is available in the flame is not sufficient so that means all the gas should be heated to that temperature but when you suddenly increase the flow rate you have more amount of cold fluid cold feed where there is no there is no the temperature cannot sustain on its own right that is why it will get extinguished switched off so that is why if you have I think very low means anyway it may not catch fire so those are the two extremes so I also gave you some practical examples what you see everyday you know match stick so that will make some interest in you I mean the daily things what we see also we can see in the actual reactors and all that what we have discussed is only for simply irreversible reaction you are going to R so what kind of these graphs you get if I have for example reversible exothermic reaction of this type what kind of QG curve you get by the by you can just think you are not at all expanding your brain so I think you are interested in submitting assignments in the Moodle but all that is there but I think no mood in the class absolutely you do not have any mood in the class I think you are only working on Moodle Moodle Moodle can you expect I think the S shaped curve can justify now I cannot keep on telling the same thing grinding grinding finally you know like shrinking core model the particle may disappear you may disappear out sometime because of my grinding okay but I think nothing is happening no reaction absolute calm before storm okay I think absolutely no idea no okay can you justify why I think S shaped S shaped curve you get for A going to R I explained also this what is S shaped but I am asking what is the reason physics I say all the time I think now chemistry is about to be over I am talking about physics physics physics you are talking about chemistry chemistry chemistry okay your chemistry with other teachers and all that when the increases rate increases so the initially it increases the QG increases but at very high temperature initial increase means you should increase rate but see we have S shaped curve what is the justification for S shaped curve unfortunately if you apply to some college where you want to join for you know Phd and all that then if this kind of question is asked what do you do I say they won't ask you to derive they will simply ask you explain how I am sure he is telling very we will give you a chance initially temperature is low initially temperature is low concentration is high but rate of reaction is still low concentration and temperature shoots up so the heat generated you already gone to shooting up temperature I am talking about S shaped where it goes like this after the initial part initial part what happens temperature is low even though concentration is high rate of reaction is low heat generation is low temperature increase is low how many lows okay and then after sometime the concentration even though it is low temperature will pick up so now when it is started raising very steeply then you have the right combination of temperature as well as concentration even though concentration is low temperature is picking up but finally temperature is very high temperature is not picking up concentration that is why it has to get flatted again remember I say how many years I have to tell you same thing you know but I think all of you have a wonderful technique that the moment you cross the door I think you will forget everything whatever is discussed is it chapam or you know some kind of curse that no one should remember after class is over anyway it is unfair to ask you because I think my time is going okay so for A going to R this kind of QG will be like this the down trend is not there for irreversible reaction irreversible reaction why should here be that down trend because of the back quote reaction okay because of the back quote reaction so now you know it reaches a maximum that means rate is reaching a maximum so that means heat release will be the maximum then rate decreases you have also seen you know that our graphs okay so that is why you get this kind of shape for A going to R you know irreversible reactions okay I mean even if it is second order A plus B going to R plus S also shape is same good and here temperature concentration of R is more concentration and temperature both are high concentration of R is more what is the relationship between concentration and R concentration and temperature both are high how concentration is more first of all it is minus R A okay how concentration is more irreversible reaction I am talking about irreversible reaction that is R CR is more so after reaching the maximum point what is the reversible but what rate minus R A okay the difference between forward reaction and my back quote reaction is minus R A okay minus R A equal to K 1 C A minus K 2 CR what is K 2 CR is the back quote reaction what is K 1 CR is the forward reaction so what is overall reaction again we are going to L K G yeah so R that R only we are talking it reaches maximum even though his doubt is that you know concentration of CR increasing where it should fall it will naturally fall but temperature is increasing heat release only decreasing temperature is increasing right temperature is increasing okay but only heat release will be less it reaches maximum and then from there it comes down right yeah so that is what so now we are going to draw QR lines here by the way where I have to operate on this line here okay right so when I start for example you know I may have a situation where I may have okay this this or maybe I think I have drawn it to okay this which one you choose because all of them T 0 is same this is T 0 is same so depending on the slope UA and also you have sigma of F i that that equation is same so because this is QR line this is Q G line this is QR line QR QR QR but different okay so normally we will try to choose this why because conversion will be maximum corresponding to this Q G QR okay so but you know why I am telling this one is that here it is invetable for you to have controller yeah that will give multiple study states but if I go here safe no problem but still the concentration is I mean the conversion is very very low so that is the reason if you have situation you try to go here but this is again very very low the other one is not I think so here somewhere otherwise what you can do is you can now draw a line like this so that means you are deliberately changing T T 0 right so and also the slope means you know F A sigma of F i C P i plus UA so all those variables you have to try to manipulate and T 0 also you have to push it right trying to push T 0 you know that is why so many beautiful things are there in reactor design so much brain is required for doing you know reactor design right so way I told this one is that when T 0 is moved we do not know whether it will be economical or not that means before entering the reactor you have to increase the temperature of the feet so you have to use a different heat exchanger so some other heating system so all these things overall picture is all these things you have to take into picture and then find out whether simply at a lower temperature T 0 is it economical operating here or drawing something like this may be economical because that slope is very very steep there okay slope is I mean so slope is very very steep when I want to change the slope what are the conditions I have feed rate sigma f i c p i is nothing but feed rate only f not so that means f not should be very large when you have f not very large what will happen to the overall reactor size volume should increase because residence time will be decreasing because again f not is nothing but volumetric flow rate into C A not no so that is why so many things so many connections particularly non isothermal design when you are doing all these problems will come those are the issues you have to appreciate atleast you should know what are the problems if you are able to find out what are the problems and solution you can find out later that is why all the time I have been telling what are the issues what are the problems do you know the problem in the design of mixed flow reactor or plug flow reactor plug flow reactor this problem will not come very happy why swami I told you I told in the class not only you everyone yesterday I told why this multiple study states will not exist in a p of r do you remember what is the reason there is not any back mixing any accumulation of heat in plug flow will be simply washed out okay there is no communication backward this that increase in temperature heat accumulation and all that will be simply washed away if there is a temperature rise it will just move away but the problem there is another problem is parametric density right that means at a particular point particularly that is very dangerous with temperature T not you know feed temperature when feed temperature is increased slowly at one point it shoots up you cannot control anything in the reactor this information I am not able to give you more except words only so I think by less and amundsum there is a paper by less and amundsum bi I think by less and amundsum under amundsum if you search lot of this information you will get so there actually they have varied various T nots feed temperatures and then shown at what temperature at what feed temperature it shoots up like you have shown that figure no Ram Krishna that kind of figures you can see okay and another my question the plug flow yesterday I think it skipped from my mind to ask you if I have a recycle reactor study states see extending brain you should question me yes because there is feedback okay see how much beautiful information is there in your brain if you want to bring it out but Swami started laughing you know subject is laughable or okay good fine so this is one thing this is the same again to find out you know study states and all that writing energy balance material balance and then drawing the line or solving both the equations and then trying to find out okay good that is the one so I may also have a reaction something like this so that is why last year I have also not done I think after seeing you I thought I want to do more and more okay because you are silently absorbing I do not know how much you are absorbing because this A going to R R going to S so very nice funny graphs you will get okay QG and temperature right if both are exothermic reactions exo exo Swami guess they are irreversible reactions right they are irreversible are excellent otherwise like like this very bad snake like this it goes okay this is like snake you know if you put that okay so now in this case how many study states you have this also lowest one 1, 2, 3, 4 okay yeah I think this also 5 5 right and as usual this will be unsteady state intermediate this will be unsteady state okay anyway I think that is only just for information and I can also give you if this is exothermic and this is endothermic just for information only what you are different shapes you know how many beautiful things are there in reaction engineering I just want to show it to you okay this is r, r going to s but this is exo and this is endo very good beautiful so it goes like this also good so now next one you have qg and t but here I have a going to r r going to s but the first one is endo and this is exo yes that is why I have drawn this also it goes below this if you like snakes I think you will get lot of snakes in this multiple steady states okay good yeah there is another thing also which we have to just discuss about ignition and yeah extension extension and ignition points okay so those points we will discuss for a going to r normal okay so for normal x shaped curve qg versus t and of course also q r is there both are there let me draw like this curve okay so I may have a line like this that we know already or I may also have which is just touching here this is 1 this is 0.1 this is 0.2 and this one we know your experts in this now this is 3 this is 4 this is 5 okay now next logically it may be like this so this is sorry sorry so this point is put as 4 that I will this is 5 this is 6 this is 7 yeah so this is 8 then this of course we know this is 9 so meaning of this is if you keep t not changing what are the various stages you know the entire curve moves okay how the reactor move how the conditions inside the reactor moves that means we are starting here this one we know 0.1 and this is 2 3 4 5 6 7 8 9 okay good so now if I plot them again this is just yeah t versus t not for example this one is t not 1 t not 2 t not 3 t not 4 t not 5 like that t not 1 2 t not 5 so t not is the variable here that means I am just trying to find out what will be t t is in the reactor when I vary t not okay so you will get yeah so I mean when you travel here this point will be there this is 1 then it is moving slowly slowly increasing this is also temperature is increasing here we have another temperature so all that I can show simply as like this okay this is 0.1 this may be 0.2 0.3 and then it goes like this goes like this okay so this this point is 0.4 here you will have somewhere 0.5 this will be 0.6 this is too much this I turned it too much yeah so here this is 0.6 yeah this will be 0.6 then of course here you will have 0.7 0. slightly away 0.8 then here 0.9 okay so this point also slightly I have to draw okay so this is 0.1 0.2 this is third point so like that so what will happen is you know it is a kind of hysteresis when you are moving from 1 to 2 right so 1 to 2 1 is here and then suddenly it may go to 6 without travelling all that distance okay so that means like this right and then that means this is the one so this point is called ignition point that means suddenly I have a temperature here and next temperature is very high it is not next small temperature next small temperature that means it is not going through all you know that increment temperatures the way we are imagining right so suddenly when I have this two point this temperature for this is this okay so it may touch instead of going to 3 4 and 5 when I am increasing temperature t0 right so and from this temperature it may go like this good so that is one way so this this is called this corresponding state is called ignition that means suddenly it is catching fire see what will happen if I come down so that means from this side t0 5 to t0 5 to t0 5 sorry t0 1 to t0 5 so now when I am coming down here like this this is fine okay and at this point you can see I am coming here from here to 4 so that is what it straight falls it is not curve I mean it is not slightly slanted may be I think here this goes may be this also I have to remove so this is the point when I am coming from this side and this is the point when I am going this side so this point what do you think this is ignition point see here you are coming from this side and then you have 9 anyway high temperature then I have 8 and suddenly from here to 4 so the reaction can extinguished can get extinguished so this is extension point extension on this on this line so that is why normally it is shown that that is the kind of hysteresis what we have so it goes like this and when you are coming like this correct no when you are going you are going like this when you are increasing temperature feed but when you are decreasing feed then you are coming like this so you have this one overall so this is only just for your information and of course probably I will not give on this one the reason is it takes time you have to plot that and then carefully look at those points anyway if you correctly calculate and then draw it definitely you will get all those points so that is why it may take time so only may be 1 problem or 5 problems I have to give you either do 1 problem or 5 problems so this is what I just want to tell you about this some more things about multiple steady state mixture flow reactor so as I told you there is a lot of information available actually there is a book called control of chemical reactor by J M Douglas remember correctly J M Douglas no not Douglas anyway I think sorry I have forgotten now so that is also another book only on control of reactors in fact you know in M Tech we had earlier some kind of history M Tech we had earlier 6 streams 6 streams were there we had biochemical engineering environmental engineering polymer engineering chemical reaction engineering transport phenomena and control 6 covered so I think at that time again may be 30 for the M Tech input so in every branch every stream only 5 5 5 5 but some streams I think we used to get only 2 like that so at that time in chemical reaction engineering stream all these courses were there we had a separate reactor theory course chemical and catalytic reaction engineering course and then non catalytic reaction engineering course separately and then stability of chemical reactors that was another course and one more word I think only these 5 courses and the remaining things anyway you have to take transport phenomena and reactor theory was common to everyone but those who were there in the reaction engineering stream they have to take the remaining 4 extra that is chemical and catalytic reaction engineering non catalytic reaction engineering stability of reactors and one more course I told I think all these courses good so at that time we used to teach all that now we have also forgotten because I think streams are removed and now I think our brain also is becoming blend it is not sharp because when you teach also we also remember in fact the most beneficiary of teaching is the teacher not the student unfortunately okay good so I think with this I have covered all these I think anything else is only one or two things now just left out that is our normal multiple reactors how do you get the maximum this we postponed at that time we discussed if you have activation energy for one reactor is more than the activation energy of the other reactor how do you do how do you operate the reactor so that information I told you that when you are discussing non isothermal reactors we will just discuss that quickly I will go through that is multiple reactions temperature effects okay yeah so if I have A going to and A going to S this is desired our symbol so here this is one and two reactions I have a case where E1 greater than E2 and other case is E1 less than E2 how do you operate this how do you operate this you have the rule already with you yeah when E1 is greater than E2 use high temperature in the so that this will be forming more and more and that is what you wanted you know that is the desired product so that is what so here I have high T high temperature and here low T good very simple so this only just remind you all this again so now I have here this is desired product and I have one and two reaction again you have case here E1 greater than E2 and E1 less than E2 what is the no condition if I use high temperature okay so this will be yeah so this will be forming more when you go to temperature scheme after some time you have to go to low temperature but straight away if I have this one more definitely I have to use low temperatures so that is why here simply here also high T and also low T but when I ask the temperature scheme then you have to say that okay sorry initially use high temperature and then later you go to low temperatures okay good okay so that is the one and yeah when I have this one A going to R R going to all three I have R, S and T this is the desired product yeah this one I will just nicely leave it for you this is one two three this is the desired product I will give you some kind of equation 1 by T opt equal to optimum temperature here you cannot use only one temperature somehow optimum temperature sorry you can use one temperature but that has to depend on E1 and E2 and E3 if this one is less and these two are more okay activation energies or this may be low this may be high this may be low all combinations are there okay but even then with all these combinations you will get one nice equation you please derive this equation I do not want to derive that it is very simple only thing is you have to write equation for maximum what is that it is rate there maximum yield not conversion not rate both do not have any meaning here it is yield you write an equation for yield and differentiate with respect to temperature then you will get this T opt so that equation is R by E3 minus E1 ln of E3 minus E1 E1 K30 this 30 is frequency factor for the third reaction okay good so divided by E1 minus E2 K20 so that is the equation R is R is not product there oh sorry yeah logical logical doubt that R is you know gas constant okay good so that is the one and the last one which I want to tell you is our famous reaction what is that denby reaction okay good so now when I go to denby reaction it may be you can write here A plus B going to okay R going to S this is going to T this is going to U okay so now we call this one as 1 2 3 4 okay I mean various combinations are possible the first combination is E1 greater than E2 and E3 greater than E3 greater than E4 what is the temperature scheme Pooja this is the desired product oh desired product very good question this one is desired very good question not R here we have SC is the one I will ask you here I will tell here yes and I will ask in the examination R okay in between yes so that means uniformly we have high temperature that is right okay good so then you have E1 less than E2 reverse of this and E3 less than E4 this is low temperature right so now I have here another case E1 less than E2 and E3 greater than E4 initially low temperature and then slowly increase the temperature I am happy I say I think at least I think you are experts of this temperature schemes okay activation energies somehow I think you like activation energy all of you are able to answer this okay this is increasing temperature how do I write okay correct no arrow good so then the other one is of course reverse of this is E1 greater than E2 and E3 less than E4 this is decreasing temperature maybe I think here I have to put increasing temperature is increasing temperature okay good actually those are good symbols of when you are doing PHD work or MS work it is better always if you have 4 5 variables okay and the dependent variable you have to see how it is getting increasing or decreasing so write like this okay if delta P as an example because we easily understand delta P so delta P is increasing with velocity okay so write delta P increasing with velocity delta P increasing u right so like that with viscosity what happens with density density of fluid diameter of the pipe what happens that will give you very clear picture for your either equation development or correlation development if you write like this all the parameters because all of you have to do some project definitely okay good so with this I think very happy non exothermal I have covered this here much more than earlier that is why always Rahul is scolding me sir you are partial to this you are all very very lenient to this batch okay plus you are also telling them more okay maybe I think this is my last thing so that is why I have to tell more because otherwise I do not know where I have to tell later okay good so this is the one but I tell you everything will be happy in the class but the moment you cross the door nothing will be happy for you next class is much worse next class you come all looking this is fresh CD okay with I think you can put any memory there but what is the use fresh CD nothing is there inside so that is what is happening every time in every class so now I think this is closed now and now we have what is left is residence time distributions right without writing there I will introduce you for residence time distribution then we will write in the next class residence time distribution we discussed about ideal reactors we also discussed about ideal reactors residence time distribution okay merit what is the residence time distribution for plug 0 why it is 0 because it does not have any accumulation what do you mean by accumulation what is the question of accumulation we are talking about study state reactors use correct words another thing always you will lose marks the moment you use wrong words without knowing or without thinking in the interviews if you do not know you keep quiet that is much better but if you are trying to say something which you have not understood or you do not have any meaning for that and simply say some word people will definitely catch you not only merit everyone so that is why where is accumulation in PFR we are talking that you tell residence time distribution equal to 0 because every particle will spend exactly same time what kind of stupid question you are asking because where is the distribution there because each and every particle will exactly spend 10 seconds means 10 seconds if it is 11 seconds or 9 seconds only distribution comes okay Swami just behind him what is the residence time distribution for mixed flow infinity infinity is a point it is not distribution you see again catching points see if you say only distribution means it will start from somewhere and end somewhere I said 0 0 to infinity ideal reactor we are talking about only ideal reactor okay if you ideal okay ya it is 0 to infinity right now I think who I think we will go to puja residence time distribution for batch reactor what is that same as PFR I have to go to PFR and search it why 0 here it is by definition but there by force because you are not allowing anything to go out anything to enter in batch reactor you are closing all the doors like you know batch system whether you like the class or not 50 minutes you cannot go out so all of you have exactly the same residence time here what is residence time distribution 50 minutes for this class residence time distribution 0 mentally I am not talking physically you are there mentally you may not listen to me even one word also okay throughout the class you may be sitting here you may be in your own matrix correct no I think how can I check unfortunately I am not new to check and then cross out of the matrix in batch reactor also it is 0 but batch reactor normally we do not take into picture when you are talking about RTD particularly when it is flow only because when it is flow it has got its own choice to go out or coming okay for the particles whereas in batch system where is the choice you are closing all the doors that is why all residence time distribution is focused only on flow systems okay and why it is 0 for mixing flow sorry for plug flow why it is 0 to infinity for mixing flow why because you just gave the value but now I am asking why by definition of we define our PFR such that there is no distribution of particles and each and every particle has to spend exactly same time what is the best example please remember that I think other than conveyor belt any reactor only conveyor belt will take all the particles exactly because again once they enter they cannot walk they will be there that particle it does not have legs no so that is why I think it will be taken and then at the end whatever you fix 10 minutes 20 minutes like that it will come out example of one conveyor belt reactor coal combustion mainly it is coal combustion okay because coal has always plug flow coal particle if I have uniform size of the particle and at the end of 10 minutes each and every particle would have burnt same exact same way right but in reality that is not there that reality we will see in the next semester okay good so this is the one now this ideal resolution distribution will be spoiled by non-ideality parameter what is the statement ideal distribution that is 0 distribution for PFR 0 to infinite distribution for for mixed flow CSTR will be spoiled by non-ideal parameters like axial mixing dead zones yeah re-circulations bypassing these are most of the time these are only 4 5 okay so that is why we have to now find out the residence term distribution for normal reactor real reactor not ideal reactor and then to find out which non-ideality would have created this kind of distribution it is like many people are going to hospital nowadays everyone is sick right yeah so they go there you go there and then they give you a test maybe blood test because they cannot find out what is wrong with your body so by taking blood test they will know okay this parameter is not well so probably this is the cause that is what exactly what we are going to also do right otherwise you have not come to that stage maybe you will come very quickly you know this diabetes stage because now it seems in India diabetes comes around 2025 earlier it was 60 okay because the kind of food what you take is wonderful food so that is the reason why I think diabetes is pre-pwn okay I think really I think in Andhra it seems the diabetic capital of the world because those are the people they take this much rice and that rice has only sugar nothing else okay yeah so that is the reason why Andhra I am not joking correct Andhra is the capital of the world for diabetes 3 times of 4th time also okay 4th time they take snacks again made of rice only okay that is why that is very bad that is why North India it is slightly less because wheat is slightly better than rice as far as diabetes is concerned they will also get a little bit later and also their problem is early morning breakfast is 100 sweets correct no for example Bengal 99 99 then how many breakfast no sweets lunch lunch time everyday then what do you take for breakfast paratha route everyday okay but I think which state I think Rajasthan all sweets only morning or Gujarat we are great in that somewhere some non-ideality will be there always so that is why particularly when sugar patients go there they will give glucose into your mouth early what is that empty stomach you have to go and then they take the blood test and afterwards they will pour you 5 ml or 10 ml into your mouth and then after 1 hour or half an hour may be half an hour and also may be 1 hour 2 hours 1.5 hours 2 hours 2.5 hours they take the samples what they are doing they are exactly doing tracer test that is what we also inject tracer and at the outlet we try to we try to collect samples then depending on that tracer concentration sugar concentration glucose concentration then they will find out for healthy body there is a curve ideal reactor correct no for healthy body how the sugar will be absorbed by the body there is a curve so now for non-healthy body it can go anywhere for healthy body there is only one curve for non-healthy body there are thousands of curves you may have some kind of problem I may have some kind of problem so that is why they try to check and then try to find out whether they have healthy body or wrong body so if they have that one sugar then they will correct you and all that that is different that is what exactly we are going to find out here because we know the ideal reactor you know response the graphs but only thing is now we have to develop some equations also for experimental reactors and then compare always with that curve what is the experimental curve so that is what is the overall picture and unfortunately I have to take I think you know more time I have to spend maybe next class and this residence time distribution normally we talk only for one phase what we have talked till now is only for homogeneous reactors two phases like coal combustion I have coal particles solid and also air entering for combustion now I have to find out how coal is entering and leaving and how air is entering and leaving same thing for example catalytic reactions catalyst is in the packet bed and air is the reactants are entering getting converted and products are coming out now find out what is happening to catalyst but fortunately here catalyst is batch it is not moving so RTD for catalyst will not come correct but if I take moving bed I can use moving bed catalytic reactor so the particles also are moving may be co-current going up or may be counter current solids coming gas going up now I have to find out gas is moving in plug flow particles are coming down in plug flow or gas is moving in mixing flow only two flows ideally or solids are coming in mixing flow you have to find out a little bit extension slurry reactors so when you have slurry reactor you have three phases what are the phases gas liquid and solid solid is the I mean if you take example of hydrogenation of vegetable oils what is the catalyst what is the gas hydrogen and what is the liquid unsaturated oil now you have many possibilities I can take for example oil as batch solids also as batch and gas only bubbly so when these two are batch I do not have to worry much about okay but I have to worry about bubbles whether they are going in plug flow or whether they are going in mixing flow all the bubbles are spending exactly same time or some bubbles are coming very quickly some bubbles are coming very late then you have zero to infinite residence time distribution like a story I am trying to tell you so now what will happen if I have liquid also moving liquid also continuously coming continuously going out now that will be in mixing flow or whether it is really in mixing flow or not we have to check so that is why for multiple reactions for every phase you have to check whether it is in mixing flow so these are the techniques we are trying to develop in the next few classes


Good afternoon, I'm Michael Collins here Minister. I'm going to be moderating today's session so you're very, very welcome indeed as I say you're very welcome to the Institute very welcome to this IEA webinar. And we're delighted to be joined by Minister Angel Logar who's the Minister of Foreign Affairs of Slovenia, who will speak to us about the priorities of the Slovenian EU presidency, which began of course on the 1st of July. And this is the second occasion since joining the EU in 2004. The Slovenia has held the rotating presidency of the Union, the early occasion of course being in 2008 and we look forward very much to hearing the Minister's presentation about Slovenia's presidency focus on strengthening the EU's resilience and strategic autonomy, facilitating the conference on the future of Europe, addressing rule of law issues within the Union and securing the EU's external borders, as well as Slovenia's plans to reinforce the EU's capacity to better respond to crisis, and the special emphasis Slovenia will place on EU relations with the Western Balkans. So the Minister, Minister Logar will speak to us for about 20 minutes, and then we will go to Q&A with your audience, and you'll be able to join the discussion using the Q&A function on Zoom, which I think you're all very familiar with at this stage. Please be free to send your questions in as they occur to you throughout the session, and we will come to as many as we can in the time available to us. A reminder that today's presentation at the IAA, both the presentation and the Q&A are both on the record, and please feel free to join the discussion on Twitter using the handle at IEEA. We're also live streaming this afternoon's discussion, so very welcome to all of you joining us via YouTube. Let me now just formally welcome the Minister and say the Minister has served as Minister of Foreign Affairs of Slovenia since 2020. Before this, he was a member of the Slovenian National Assembly, representing the Slovenian Democratic Party, where he headed a commission of inquiry into abuses in the Slovenian banking system. He was also chairman of the Parliamentary Public Oversight Committee, as well as a member of the Finance and Monetary Affairs, Foreign Policy and EU Affairs committees. He's also served as Minister of Planet Potentially for OECD relations in the Ministry of Foreign Affairs from 2013 to 2014, as director of the Slovenian Government Communications Office, and he was the spokesperson of Slovenia's EU Council presidency in 2008. So Minister, you're very, very welcome to Dublin. I know it's been a very, very busy day for you, having met, of course, our Minister, and also I know opening the new embassy. So congratulations on all that front. The floor is now yours. Thank you very much, Mr. Koinks. Ladies and gentlemen, at the outset, I would like to thank the Institute for hosting me today. I'll bite in a virtual form and form from the other side of the town. It is certainly good to be back in Dublin, even with the pub still in restricted mode. I'm grateful for the opportunity to be able to present the priorities of our presidency. You might be aware, this is already the second Slovenian presidency of the Council, but this one is taking place under very different circumstances than in 2008. In a sense, that one was the last residency that had avoided any major crisis soon after the EU had to deal with the financial and debt crisis, and then with the migrant crisis. Then, of course, came to Brexit, but without exaggeration, I can say that COVID has put the EU and its member state to the most difficult test so far. It has put our health system under tremendous pressure, and it's still affecting our social and economic life. In the short run, it imposed certain limits on our way of life, but in the long term, it would also impact the way we organize ourselves nationally on the EU level and in the global community level. Just like almost every region and country across the world, neither the EU nor the member state were ready for COVID-19 pandemic. During the first days of the outbreak, actually, those were the days when I sworn in as a foreign minister, the coordination and cooperation between the member state was often initially very difficult. We will all remember media reports about long queues of lorries at the border checks. Meanwhile, on a global level, trade flows of critical products and materials were hampered by the export restriction and prioritization of prioritization of domestic production. This made us search for emergency solution to contain the risk of shortage of medicines and medical devices. However, as the event continued to unfold, the EU's response improved in efficiency. Now with the introduction of the digital COVID certificate, I hope the travel will be back this summer after all, at least to some extent, though the last reports on the Delta variant are not very optimistic in that respect. Experience from the previous economic and financial crisis allowed the EU to be better prepared for the economic fallout. The temporary recovery instrument under the name Next Generation EU and worth of 750 billion euros was set up based on green and digital transition. The fact that the EU managed to struck a deal on a recovery fund in a record time demonstrates the capacity of the European Union to rise up to the challenges. If we know such a major crisis, I am pretty sure that EU member state will not be able to agree on such a package. So this is just a snapshot of the situation in which the Slovenian presidency is taking the helm of the Council of the EU for the next six months. The Slovenian priorities are natural and logical reflection of this event and the challenges we are facing. So let me briefly mention them. The first priority of our presidency is resilience. The main is to use the experience gained during the pandemic to run the ad hoc reaction during the crisis into structured and long-term preparedness and resilience to the pandemics, but then again also to other type of crisis. The focus will be on building the European Health Union and starting their work on the proposal for a new European authority named HERA, Health European Response Agency. Our aim is to strengthen the research and development and to scale up the production capacity for medicines and medical devices in the EU. As recent event in the US and also here in Ireland have shown, we can very quickly find ourselves in a situation where a large scale cyber attack could have similar fatal consequences as the spread of the virus. It can harm our economy, critical infrastructure, democracy, and when taking place in the health sector, it can even cause loss of life. The number of cyber incidents in Europe almost doubled in 2020 and they also grew in scale of consequences. One of our key priorities will therefore be to promote a more coordinated response to potential large-scale cyber attack in Europe. One of the most important lessons learned is that the EU should better guarantee the security of supply of critical products. Better understanding of strengths and dependencies of our society and economy will help us design the right measures to reach higher or simply the right level of strategic autonomy while maintaining our trade relations open. We especially believe in enhancing relation with like-minded partners in this regard. I'm cautiously optimistic that during the Slovenian presidency, the European economy will start to pick up after the crisis. Recent data certainly support this view, including those for Slovenia. One of our presidency's first important task is to ensure that national recovery and resilience plans are confirmed as soon as possible. Europe's recovery after COVID-19 needs to be green and digital. It will be our task to start to lead a coherent debate on the climate and energy package fit for 55. That was presented by European Commission recently, I mean yesterday. However, EU's climate goals are not sufficient without cooperation of the largest emitter. So COP26 climate conference in Glasgow will be a turning point in this regard. When it comes to the digital transition, I wish to underline three interrelated areas that deserve particular attention. Regulation of digital services and markets, artificial intelligence and data economy. It is important that we strike the right balance between the competitiveness of our industry and ensuring that this important segment of the economy is regulated in accordance with our basic values. I know how sensitive these issues may be here in Ireland. But these are the topic to be addressed also by conference on the future of Europe, which will be in full swing during our semester as two of plenary session and four of the public panel will take place. In my view, the conference should be about how to make Europe more efficient and effective. I look forward to seeing all citizens stakeholders and think tanks joining the debate on Europe's future priorities and putting forward their proposals. We cannot ignore the need to have an active and visible role of the Union in the international community. Slovenia will devote particular attention to our renewed transatlantic commitment and the situation in our immediate neighborhood here. I would like to highlight the EU Western Balkan Informer Summit, which will be hosted by Slovenia this autumn. The EU's enlargement of course is about implementing reforms and fulfilling political and technical criteria. However, it is also about the strategic role of the EU in its immediate neighborhood. If the EU does not continue with the enlargement process, other global players will take over the leading role in its backcourt. Let me touch at the end also on the subject that is of great importance to Ireland and the whole European Union, the UK relation. They will remain our major occupation for years to come. No one is more aware of this fact than you and Ireland. Presidency, we will spare no effort to ensure full implementation of the trade and cooperation agreement, as well as withdrawal agreement, including in the areas of citizens right, the protocol on Ireland and Northern Ireland, fisheries and level playing field. Pacta Sun Cervanda, and we shouldn't expect nothing less from our British partners. Since 2016 and throughout the negotiation with the UK, we offered full solidarity with Ireland. This process and stability on the Irish island needed to be protected and protocol on Ireland and Northern Ireland is a solid means to this end. Reluctance to fully implement the protocol, not only contribute to instability, but also put at risk the integrity of the single market and the health of our fellow citizens. In this slide, Slovenia closely follows the situation and is ready to work in the best interest of the EU. Our joint efforts will be important to successfully tackle many of the challenges ahead of us, while we prepare ourselves for the end of the pandemic, transforming its impact in the long term in the balance of our economies and societies, building for a new Europe that lies ahead. I will stop here at this point, so we can give to participant an opportunity to comment and as well to ask questions that I shall later on reply. Thank you very much for your attention.