The Jack Benny program, presented by Lucky Strike. Feeling low. Feeling tense. These eight words. For level best. That's just how you'll feel when you light up a lucky. Because lucky's fine tobacco picks you up when you're low, calms you down when you're tense. Put you on the right level to feel and do your level best. It's important to know that fine tobacco can be used for you. And L-S-M-F-T. L-S-M-F-T. Lucky strike means fine tobacco. Smooth, mild, thoroughly enjoyable tobacco. So next time you buy cigarettes, remember Lucky's fine tobacco picks you up when you're low, calms you down when you're tense. Put you on the lucky level where you feel your best and do your best. Yes. Smoke a lucky to feel your level best. Program starring Jack Benny with Mary Livingston, Phil Harris, Rochester, Dennis Day, and yours truly, Don Wilson. Ladies and gentlemen, let's go back about an hour. Jack is in his dressing room getting ready for the broadcast. Rochester, did you run over and get me a sandwich? Yeah, boss, here it is. Good. Oh, for heaven's sake. Sardines. Rochester, I can't understand you. What's the matter, boss? You never get a sandwich. What's the matter, boss? You never get anything right. When I send you for a chicken sandwich, you bring me ham. When I ask you for a ham sandwich, you bring me cheese. When I ask you for cheese, you bring me egg. When I ask for egg, you bring me turkey. And today, you bring me a sardine sandwich. Now, what did I send you for? A sardine sandwich. Oh, yes, I forgot. But look, Rochester, I figured you'd make a mistake. I asked you for a sardine sandwich because I wanted corned beef. I knew that, but they were out of corned beef, so I brought you sardines. All right, I'll eat the sardine sandwich. Did you get me a bottle of Coca-Cola? Yes, boss, here it is. Wait a minute. This is root beer. That's funny. I asked for seven up. Well, at least you got the sandwich on rye. How did that happen? Oh, we never did have trouble with the bread. Well, I better eat it in a hurry. Oh, darn it. And I pick up the sandwich, the sardine slipped out, went into my sleeve. Help me get them out. Mmm, what a mess. Well, I'll eat after the broadcast. Now, help me finish dressing. Okay. Which toupee do you want, boss? The one with the part in the middle or the part on the side? Didn't you bring the black one with the widow's peak? Where's that toupee anymore? Why not? It turned gray while you were dickering with CBS. Oh, stop. Dusty, maybe, but gray, no. Now, Rochester. Who is it? It's me, Jack. Come on in, Mary. Jack, did you take my fountain pen? Yes, yes. Here you are, Mary. You know, I use it. I just put down a wonderful joke I want you to do on the program and surprise the cast. Here, read it. Oh, for heaven's sake, Jack. Another joke about my sister, Babe. Well, it's funny. Go ahead and read it. I want to hear out-of-sound on the program. All right. Say, Jack, my sister, Babe, lost her job in that restaurant. But, Mary, she was with that restaurant five years. How come they let her go? They've got a cat to catch her mice now. Oh, poor Babe. And after she let her fingernails grow. Jack, we can't say a thing like that about Babe. Why not? That's a funny joke. Joke nothing. It really happened to her.  I don't want to hurt her feelings. I'll think of something else. Say, Jack, I meant to ask you. Are you really going on the Ford Theatre program March 4th? Yes, Mary. I certainly am. Well, I hope you're not serious about doing the horn blows at midnight. Uh-oh, Miss Livingston, you shouldn't have said that. Rochester, what's wrong with mentioning the horn blows at midnight? It doesn't bother me, but the red, red turn white. Don't be funny. Now, look, Mary, I'm still going to do that play on the Ford Theatre. Fletcher Markle, the director, gave me a contract, and that's that.  Well, wait a minute. You're not going to do the program on that polo shirt. Why, what's wrong with a polo shirt? Clark Gable wears one. So what? Jane Russell wears sweaters, but I'm a suit gal myself.  Look, Jackson, what'd you stop us for? We were rehearsing a number. I know, Phil, but I just want to compliment you on trying to improve the orchestra. I noticed you got a harp this week. Yeah, but we can't use it anymore, Jackson. That was a mistake. Why? What happened? The rehearsal, Frankie woke up, saw the harp, and thought he was dead. No. Then he tried to fly over the piano and almost broke his neck. Well, it serves him right for sleeping during rehearsal. Jack, let's run through our parts, so the others get here. Okay. Here, Phil, take a script. Look, Jackson, I don't need no script. I'm loaded with jokes. Phil, take a script. I thought of a gag that's a dilly. Get this. On my way down to studio, I stopped off at a bar. Phil, take a script. And while I was there, I met a fellow who came from Venice, Italy. Phil. So I bought him a drink. He bought me a drink, then I bought him a drink, then he bought me a drink. Phil. We kept buying each other drinks for a couple of hours, and when I walked out, he was under the table. Phil, Phil, take a script. Well, don't you get it, Jackson? The guy who was under the table was from Venice, a Venetian. So what? I'm the only guy who ever drank a Venetian blind. If you lived in Beverly Hills, they'd blow up your house. Phil, take a script. Yes, Mary. And don't get too close to Phil. He might exhale and disintegrate your nylons. Now, look, now let's cut out all his foolishness and get down to a... Jack. Oh, hello, Don. Hello. The quartet is here. You better rehearse them first. Oh, the sportsman? All right. Say, did you tell the boys that instead of a popular song, I wanted something a little different this week, you know, something classical? You know, we have music lovers listening in now. Oh, yes, Jack. And they have a wonderful number prepared. It's Ponkele's Dance of the Hours from La Giaconda. Oh, I... What was that again? Ponkele's Dance of the Hours from La Giaconda. Oh, yes, that's what I hoped you'd said. Phil, can your musicians play Ponkele's Dance of the Hours from La Giaconda? Yeah, but it'll still sound like that's what I like about the song. Well, then I don't want that. Have them play what's written. Take it, boys. Listen. For Lucky, take a pop. Take a pop. Because we know that you will never. You will like a lucky strike. Beat your level best. For Lucky, strike me, Spina. He's Dance of the Hours from La Cienega. Oh, it's La Giaconda.  That's below Sepulveda. We had Pico, but I changed it. Sepulveda's funnier than Pico, anyway. And Phil, Phil, your orchestra did surprisingly well. What are you so surprised about? I haven't told you this, but they want my band to play at the Academy Award ceremonies. Really? Phil, why in the world would they want your band to play for the Academy Award? Well, we play loud, and it'll drown out the screams of the losers. Well, Phil, I don't believe that you are your band. Come in. Oh, yes, Fletcher Markle. Hello, Mr. Markle. Hello, Jack. That's quite all right, Fletcher. This is my appearance on the Ford Theatre. Yes, that's exactly why I'm here. Good, good. Is there something about the casting? No, not the casting. Oh, is it about the rehearsals? No, not rehearsals. Oh, is it about the picture I selected to do? Yes, it stinks. What? Jack. As director of the Ford Theatre. It stinks while you're laughing. What were you going to say? Director of the Ford Theatre. I beg you, really, don't do this on our program. Now, wait a minute, Mr. Markle. You signed a contract with me, didn't you? Yes, but you tricked me. That contract said you were going to do an adventure story of the sea called Sailing Around Cape Horn. Well, you didn't read the small type in the contract. The full title was Sailing Around Cape Horn Blows at Midnight. And that's what I'm going to do. Jack, I think Mr. Markle is right. That script isn't suitable for radio. I know that, Mary, but I went through the screenplay, eliminated all the Dell stuff, and I'm just going to do the part that's sparkled. Mr. Benny, we have an hour program, not a spot announcement. I know it's an hour program. And I'm going to do the horn, blows at midnight, and it'll be a great show, believe me. Well, there's nothing I can do about it. So long, Jack. So long, Mr. Markle. Oh, by the way, Jack, when you signed that contract, there was one thing I neglected to get for our records. What's your social security number? Two. Wilson is number one. Yes, we got there early. Well, so long, Fletcher. See you March 4th. Goodbye, Jack. To be or not to be, that is the question. Whether it is wiser to use gas and smell up the house or use a pistol and mess up the rug. What'd you say, Fletcher? Nothing, goodbye. I can't understand you. Why, Don? You're so upset against your doing the horn blows at midnight. Why do you insist on it? Because once and for all, I'm going to prove to everyone that it's a great story. Now, Don, imagine this plot.  And I'm the angel.  Try to do something dignified, and everybody has to butt in and... I never saw anything... Oh, hello, Dennis. Oh, hello, Mr. Benny, congratulate me. Congratulate you? Why? I just sold my bicycle for $10,000. You what? Yeah, look, here's the check. Let me see that. Pay to the order of Dennis Day, $10,000... Oh, for heaven's sake. This check isn't even signed! I noticed that, but I didn't want to mention it. Why? When a man gives you $10,000 for a bicycle, why antagonize him? Holy smoke. Dennis, I'm afraid you were cheated. Didn't you even get the name of the man who bought it? Oh, sure, he told me his name, all right. What was it? Napoleon Bonaparte. Oh, fine. Dennis, don't you know anything? Napoleon Bonaparte has been dead for over 100 years. He has? Certainly. Then what does he want with a bicycle? Look, Dennis. Dennis, my child, look. Instead of having any more of this nonsense. Look, how about running through your song? Okay. Oh, by the way, Mr. Benny, happy birthday. My birthday was last week. I know, but that bubble dancer was held over. Good, good. Now what number are you going to do? The title song from Walt Disney's picture, so dear to my heart. All right, let's hear it. Okay. That's a temporary day where the leaves turning... Sing it beautifully. Why are you just saying that because I'm rich now? Dennis, that check's no good. You never like anything that anybody else has. Dennis, it's the last time I'm going to compliment you on your singing. Because every time I do, you start right in. Come in. Yes, sir, is there... Well, it's Jack Warner of the Warner Brothers Studio. Hello, Jack. Hello. Well, this is a surprise. What can I do for you, Mr. Warner? Well, Jack, I just heard the news about your making an appearance on the fourth theater. Yes, Mr. Warner, on March 4th. And I'm going to do the horn blows at midnight. That's just what I want to talk to you about. What? Look, Jack, you made this picture for us in 1944, didn't you? That's right. It was for the Warner Brothers. Well, since then, our studio has produced the Adventures of Don Juan, Life with Father, Key Largo, John Loves Mary. And this year, our two pictures, Johnny Belinda and the Treasurer of the Sierra Madra are up for the Academy Awards. So? So we're rolling again. Let us alone. Now, just a minute, Mr. Warner. How can you say that? You yourself told me that when the horn blows at midnight was shown in Hollywood, the theater made money. That's right. We rented the balcony out as a trailer camp. But, Mr. Warner, you can't put all a blame on me. When you did that picture, you made one big mistake. Yeah, I know. We put film in the camera. Look, Mr. Warner, that's an old joke. Yeah, if I had anything newer to put it in the picture. Well, I'm sorry, but I still think it's a great story. If you didn't produce it good, what could I do? But if you'd have listened to me, if you'd listened to me while we were making it, the horn blows at midnight would have been a terrific hit. Jack, we tried everything. We made a lot of retakes. It was awful how we threw money away on that picture. Threw money away? Yeah, we spent over $500,000 for a new finish and nobody ever stayed to see it. No wonder you can't make good pictures. You fluff over lines. Jack Warner, I want to tell you something. All of that about the horn blows at midnight was not my fault. I think that picture had great possibilities. Well, no, Jack, it missed both ways. Both ways? What do you mean? Well, if it had been a little better, we might have gotten our money back in the theaters. Yes? If it were a little worse, it would be unnatural for television. Never mind television. It's going to be, I'll tell you, Mr. Warner, it's going to be great when I do it on the Ford Theater. All right, Jack, if you won't listen to reason, maybe you'll listen to this. We'll give you $5,000 not to do it. No. $10,000. I'm sorry, Mr. Warner, but money means nothing to me. Listen, repeat, show and see if he really heard that. See if I really heard that. See if I really heard that.  I know we rehearsed this. I know it. Anything else? Huh? What? Just a little shaky. All right, read just what's there. That's all. Read what's there. Here it is. Wait a minute. Here. Well, Jack... You didn't say Mary, please. I say Mary, please. You got the next one. Well, Jack, here's my final offer. My brother Harry and I are willing to take you into the firm and make you one of the Warner Brothers. No, I'm afraid not. That means I'd have to change my name. If you do the picture, we're going to change ours.  I'm going to do the horn blows at midnight on the radio, and that's final, Mr. Warner. Just call me Sam Goldwyn. What? Yeah, you heard me. Just call me Sam Goldwyn. You just got to say goodbye, Jack. Oh, how to say goodbye, Jack.  Why didn't you consider his offer of going in with the Warner Brothers? I'll tell you why, because I have my own company. I just produced a picture myself called a Lucky Stiff, and there's a wonderful article about it in the New Liberty Magazine. I know, Mr. Benny, I saw it. You saw the article in Liberty about me, Dennis? What did it say? Reading time, 11 minutes. Look, Dennis. But I made it in nine. What? I had my bicycle then. I don't care about the reading time. What did the article say about... Oh, I'll get it, Jack. New York calling? Yes, he's here. Jack, it's Mr. William Paley. They had a CBS. Oh, oh, I wonder what he wants. Well, hello, Mr. Paley. Yes, yes, this is Jack. Oh, yes, I'm feeling fine. How are you? Good, good. What? Yes, yes, that's right, on March the 4th. Yes, I know the Ford theaters on CBS. And you want to hear something funny? Fletcher, Markle, and Jack Warner came over and tried to talk me out of doing the horn. What? Now, just to... I know, Mr. Paley, but... But, Mr. Paley, but... But... Why, Mr. Paley? Sorry, Mr. Paley, but I made up my mind and I'm going to do the horn blows at midnight on the Ford Theater, March 4th, and that's final. Yes, Mr. Paley. Goodbye. Hmm. How could you talk to Mr. Paley like that? Mary, there comes a time in every man's life when he must have the courage of his convictions, when he must have faith in his own judgment, so he can stand up and face the world with pride and with dignity. Jack, what's that sticking out of your sleeve? Huh? Oh, it's a sardine. Must have got caught in my suspenders. Jack, we'll be back in just a moment, but first... Smoke! When you're feeling low, Lucky's fine tobacco picks you up, and when you're tense, Lucky's fine tobacco calms you down. That's what fine tobacco can do for you. And that's why it's so important that you select and smoke the cigarette of fine tobacco Lucky Strike for, as you know, LSMFT, LSMFT. Lucky Strike means fine tobacco. No wonder Lucky's are the overwhelming choice of the tobacco experts. Men who can see the makers of Lucky Strike consistently select and buy that fine, that light, that naturally mild tobacco. So when you choose your cigarette, remember that Lucky's fine tobacco puts you on the right level, the lucky level, where you feel your best and do your best. Yes, next time you buy cigarettes, ask for a carton of Lucky Strike. Feeling low, feeling tense. These great words are common sense. Jack Warner, good night everybody. Be sure to hear the CBS Sunday lineup every week, and don't miss Dennis Day in the day and the life of Dennis Day. Stay tuned for Amos Land, as y'all will follow up immediately over most of these same stations. This is CBS, the Columbia Broadcasting System.


Homotopical aspects of simplicial complex, by which we mean how to construct homotopies and what are certain basic properties, not topological in nature, but homotopical in nature, like local contractibility, co-fibration and so on. That is the topic for discussion for this module as well as the next module. Last time we did some combinatorics of links and stars and so on. All of them were what are called as closed subspaces, closed links, closed stars and so on. So, today we have a slightly different version here. Start with any simple x in a simplicial complex. Then the open star, let us say I am denoting, we try small stf. The capital stf remember was the closed star. So, this is open star of stf. This is an open subset of mod k. It is not defined as a combinatorial object, not a simplicial complex or something. It is an open subset of the simplicial complex consisting of the union of all open simplexes g. Remember, this notation is used for open simplex of g. Namely all those alphas such that support of alpha is actually equal to g. So, take all those points. Take for every g, take such a thing where g is a simplex containing f. In particular, g can be equal to f also here. So, this containment sign I am using for subset equal to also. So, the claim is that this is an open set just because you called the open star it may not be open. So, claim is that it is an open subset of mod k and it is a star shaped at every point of the open simplex f. Notice that when you take g equal to f, this open simplex f comes here. So, open simplex f is a subset of this one. Now, you take any point there, the entire set this one will be star shaped at a point is the claim. These are all easy not difficult, but you have to just verify and be sure of them. Let us all to show that any subset is open in mod k what you have to show is that its intersection with each closed simplex mod h is open in mod h. So, this should happen for every simplex. Of course, if h simplex is a single term, then this is automatic beyond that you have to verify this. So, clearly if f is not in h, then the star of f intersection h will be empty. Intersection h is empty because only things which are remember the whole space is union of such open simplexes. This was the starting point of point setcological thing that we have seen. So, here we are taking only those f which those g which contain f. Therefore, either this entire open simplexes are there or they are not there. Therefore, if f is not contained in g, open g will not intersect this at all. So, star of f intersection h is empty. So, empty set is open that is fine. So, let us take the case where f is inside h. Then I want to show that this intersection is open in mod h.  Both of them will be simplexes, h is a simplex, f is a simplex. So, f prime is a simplex. But what happens is now star of f intersection mod h will be precisely equal to mod h minus this subset namely the boundary of f star f prime and its modulus. In open simplex f remember boundary of f, the modulus of boundary of f is not there. So, you take a subset of f that will represent boundary part, right?  So, any subset here any subset here union all of them union you take that is the simplex complex here modulus of that I am taking. So, all that you have to do is f prime does not intersect h. So, that part is not there none of them are there their union will be also not there because this is boundary of f part. So, none of them will contain f completely. Therefore, all these things they are subsets of subsets of h by the way h is a whole thing is inside h right everything is inside h now. So, you have to throw away this part then this precisely equal to star of f intersection modulus all other things are there. Ok, namely all those subsets is f contains inside say some h prime contains at h there open simplex will be there ok. But instead of writing all that I can just write it as mod h minus this one and this is a closed subset therefore, this is open. Now, take any x belonging to open simplex f I want to show that star f is star shape at this point what is the meaning of that take any point y in star f the line joining x and y should be completely contained inside star f ok. The line joining x and y makes sense inside both star f as well as mod g where g is the unique simplex for which y belongs to open simplex g remember every y will be inside some unique open simplex ok. Then x is also inside g because g must contain f ok. So, y is inside g this line segment makes this will be inside g because g itself is a convex open side. Similarly, star f will be there because of the definition of this ok. So, this x y makes sense inside star f intersection g ok where g is the unique simplex this just means that star f is star shape that is all every line segment for each point inside star f ok every line segment x y must be inside of that. Now, we make a definition which which we might not might have made it earlier, but now it is more relevant anyway. We say a topological space is semi locally contractible at x belonging to x if for every open set u such that x is inside u that is an open set v such that x belongs to v contained inside u and a homotopy h from v cross i to u such that h of y 0 is y which means the homotopy is a deformation starting point is identity. And the end function is a constant what constant precisely the point that we are interested in the only thing here this is like a contractibility of v, but if v is this is u this is a larger open set. If you replace u by v this is nothing but saying that v is contractible to the point x it is stronger than saying that v is you know v is just contractible that contractible to the point x is important for us, but the whole thing is taking place inside u the homotopy can go out of v, but it is it has this property ok that is the why this is semi locally contractible. However, if you replace v this u by v itself in in this one namely h is taking values inside v then we actually say that this capital X is locally contractible at x the same condition the only thing that you have to do is replace is u by v then you delete the semi locally contractible ok. So, that is the meaning of locally contractible if h can be chosen to take values inside v itself ok. So, locally contractibility is stronger than semi locally contractible ok. Now, suppose this happens at every point of x just like a function is continuous it is continuous at every point if it is happens at every point then we say x itself is semi locally contractible or in the latter case x itself is locally contractible. So, that is the meaning of this definition ok. So, now next theorem is that every polyadron is semi locally contractible take a simple shell complex take mod k I have to show that at every point there will be a given any neighborhood u there is a smaller neighborhood v of that point such that that v contracts to single point h inside u this is what I have to show alright. So, let us go through the proof you see this is not so difficult at all starting with x in mod k choose f such that x is inside open simplex f remember every every point is inside a unique open simplex because mod k is partitioned into open simplex s. So, that is used again and again the previous lemma says that star f is an open subset of mod k which is star shaped at x. So, this itself is a contractible neighborhood of the point x, but we want to do it inside a given open set. So, that is the catch here ok. So, how we how we go about it this is how therefore, first of all we have a contraction h from star f cross i to star f which is just given by h of y t equal to t x plus 1 minus t y this x is the star is apex of the star shape thing. So, any y can be joined to x by t times x this is the line segment this whole thing is inside star f. So, this is a continuous function put t equal to 0 this will be y put t equal to 1 this will be x ok t equal to 0 it is identity map t equal to 1 is a constant magnet x alright. Now, start with an open set u such that x is inside then look at this h h of x cross i x is x is the given point right h of x cross i if y is equal to x what is this map what is this one it is just x it is singleton x that is inside u u is an open set. So, this whole interval closed interval is mapped onto mapped inside an open set this is compact. Therefore, you know standard methods in point set topology like tube lemma there exists an open set v such that x is inside v and h of v cross i is contained inside u one single open set cross i instead of singleton x cross i will contain inside u. Now, you look at the same h restricted to this one what we get is this condition. So, given any u we have found a v such that and h is capital H here like that it is a little h in the application this special H here. So, this proves that every polyadron is semi locally contractable okay very easy statement very easy proof. Hello sir. The u that you choose the open set u around x you cannot be subdivided to make the star of that point inside u then we can contract it. What do you solve you subdivided? Yeah I mean if we subdivided then it will be small can it be small enough to be inside any u? How do you do that? We know that there is always one. I mean can we can we can we can take any point. k is not a finite simplicial complex. Oh okay okay. Even the point x you have taken is not at all first of all a vertex even if that is the case that may be infinitely many simplices coming and meeting at x. Okay. So, it is not even locally finite even after that you have to make x to be a vertex etc. Okay. But but cannot we have a simplices structure that makes x a vertex? That is the next thing that we want to do. Even then you know the problem is by the bare centric subdivision that we have taken that will not do the job. Okay. That is why this one has to be here. That is that will do only if it is finite simplicial complex. Okay. Okay. Once you have this semi locally contractibility later you will see that such spaces admit a simply connected covering. Okay later. So, right now you assume that then what we have proved is every connected polyadron admits a simply connected covering. Okay. That is an application of this one. Okay. Now let us come to the second point here namely the co-fibration property. This theorem says that every sub complex that L be a sub complex of K then the inclusion map model to model K is a co-fibration. Remember that we had a proposition which says that how to verify co-fibration namely you must produce a detraction from mod K cross i onto mod K cross 0 union this mod L cross i and that is precisely what we are going to prove. Retraction should be from mod K cross i to mod K cross 0 union model cross i. Okay. So, this one we are going to get such a retraction in a inductive fashion. Put ln equal to L union KN where KN denotes the nth skeleton of K. Then by the very nature L is a sub complex, KN is a sub complex, union is a sub complex and each ln is a sub complex of K. And we have the inclusion maps here L is contained inside L0. What is L0? L0 consists of all of L and all the vertices which are already not in L. Those are extra vertices coming from K0 and so on. So, ln will contain ln plus 1 and so on. Finally, if you take the union of all of them that will contain all the skeletons therefore it will be the whole of K. So, I have written K inductively as a countable union starting with L. If K is a finite simplicity complex, there are many other methods to do this. There is no need to go through this. This is precisely to take care of infinite simplicity complexes. Once we have, you know we have seen this one several times, namely if you look at a disc cross I that will be deformation retracting on to the base of the disc union, the boundary of the disc cross I. Just topologically, there is no need for going to simple simple complexes. I have written it in terms of simplicity here delta n plus 1 is representing modulus of delta n plus 1, the underlying topological space is dn plus 1, the disc cross I to modulus of delta n cross 0, the base union, the boundary which is a sphere of one dimension lower is Sn cross I. So, there is a standard retraction. I am going to use this one now. For each n greater than equal to 0, we have such a thing. So, now just you know index the set of all n plus 1 simple complexes which are in n plus nn plus 1 obviously, but which are not in L. Don't worry about things which are in L because they are going to be here which part. So, let us index them. So, let us call them as FS. For each S there is a row S like this. That is just same row, a copy of row, but I am written by row S for each S. Then what do I do? I start R naught, there is a slight different definition. Inductively, I am going to define this R naught, R 1 and R naught. R naught is L naught cross I to L naught cross 0, L cross I, that is fine, but on L naught cross 0 or L cross I, it is identity. Either X is in L or T is 0, it is identity. Then there is a disjoint union of vertices cross I which are which correspond to all the vertices which were already not in L. All those vertices cross I, their line segments, you push them to just the base point see X cross 0. Instead of X cross T, all of them goes to X cross 0. So, X is a vertex not in L. So, this is obviously a retraction of L naught cross I on to L naught cross 0 union L cross I. That is a starting point. So, in this inductive picture, we have got a retraction corresponding to this L naught correspond to L. It is not a retraction to L naught to L, but it is L naught cross I to L naught cross 0 union L cross I. So, that is the kind of thing we are going to do here all the time. So, once you have this one, you can assume that you have defined this one for up to N and you can go for N plus 1 stage. So, what I am going to do is for R N I am defining L N plus 1 cross I to L N plus 1 cross 0 in L N cross I. It is similar method. The first part it is identity namely all L N cross I or L N cross 0 to the identity X T. The second thing when they are inside a delta N, delta N plus 1 cross I, there I take this rho S because they are all indexed by rho S on F S, rho S of X T. Rho S of X T will push delta N plus 1 cross I into the boundary of that which is already inside this one, boundary of that cross I and the base, base is here. So, take X belong to F S 1. So, on the boundary we already its identity. So, these patch up. So, there is no conflict here. So, each R N is a retraction. But R N is a retraction from one index higher to one index lower. What we want is from entire K to L. So, how do we do that? Now, take any point X in mod K. This will belong to a unique n simplex inside the unique n simplex namely X inside open F. This open F may be a singleton also. Open simplex for a vertex is the same set as a vertex. In all other cases it is different. For a, for an edge it will be a open interval and so on. So, X belong to open F. That means it is inside the nth skeleton in any case where F is dimension of F is N.  So, that you come back to all of it L cross 0 union K cross sorry K cross 0 union L cross I. So, that will be the final picture. So, really R agrees with R N. You see once a point is inside K N then it is R N and then all these are R R naught. The next stage R N plus 1 will agree with this one if it is already inside R N inside K N. K N plus 1 from K N plus 1 the R N plus 1 will already agree with R N because R N plus 1 of that part is already identity and then only R N will up to R naught R N will change it. Therefore, this compatible family of this one that is why R of X T makes sense. Restricted to each K N it is only this part if K N plus 1 if I have taken one more composition will come R N plus 1 will have come that is all. But they are all well defined and when you want to check the continuity or check the continuity on this part. And then this is the composition of finitely many continuous functions so it is continuous okay. The retraction for retraction path it follows very easily because for each point this is why this one. So, every sub complex the individual maps are co-vibration okay. What we want to do is arbitrary open subsets containing inside model. Inside that you must get a smaller open subset such that the inclusion of that pair model to you that itself is a co-vibration. So, these are the harder results all right. That is what we want to do. We would like to strengthen all these results. The first step let us have one of the exercises that we had earlier namely that every point can be thought of as a one text inside a simulation of complex. The first step is again this we have discussed several times I will discuss it again. Let mod L to SN be a triangulation mean L is a triangulating SN that is a homeomorphism. H is a homeomorphism that is the meaning. Take any point we in the interior of D N plus 1. D N plus 1 is a disk bounded by a SN. Then this triangulation extends to a triangulation L star V which is a cone. This is the the join of these two simplex is that is the another definition cone of L over V okay cone of L with V as vertex. So take the modulus of that this H hat defines a homeomorphism of that okay. This is given because given any interior point of V take any point on the boundary SN say call that as U. Then every other point has a unique expression namely it will exactly on one of these line segments U V. Only V will be in all the line segments that is why it becomes a cone okay. I am just recalling that this we have done earlier right. So this process let us call it as starring a simplex. This L is the boundary of the simplex which has been subdivided okay L could be any subdivision not necessarily delta N plus 1 boundary of delta N plus 1 that that triangulation extends to triangulation of this starring a simplex okay with whatever boundary triangulation is there that gets extended. So this process we will call just starring a simplex just temporarily if do not worry about other people may call it differently and so on. So now we can solve this problem given any polyadron x and a point x there exists a triangulation in which x is our text okay start with a simplex complex that means what I am taking about now x is now mod k right choose a point x inside mod k again choose the unique n simplex f belong to k such that x is inside open simplex f define a subdivision k prime of k as follows okay we are going to define subdivision and that subdivision will have this x as a vertex that is all okay. So what we do this this vertex is in some simplex it is in complex if you work out all other everywhere else okay there is no need to subdivide and so on somewhere something near this part you have to subdivide. So what I do first of all for all simplex is of dimension less than n choose k prime to be just k whatever it is there so k prime m m skeleton is k m up to m less than n when it is n you have to be be careful what is that except this f which is also dimension n don't divide anything other than this this f keep them as it is all the n simplex is in k are in k prime also the simplex is are as as they are that means we are not dividing dividing that one okay all right so far so what you have done is you have got into a simplex you have concentrated one simplex and f p n little x is inside in the interior of that the boundary of that is not divided okay it is any whatever it is boundary is actually a simple a simple complex now you use the previous lemma star this simplex at that point instead of barycenter you choose any open any any point in the open simplex that is all okay for example this becomes a vertex and all the lines joining these two vertices of the boundary they will be the one simplex says and so on so that is the star right okay so this will complete the subdivision of k n the nth skeleton after this there may be many simplexes which contain this f which has been divided therefore you don't know what to do and so on so don't worry divide all of them by starring at the barycenter beyond beyond this n dimension star this we have done so up till here we have done the last thing is inductively as soon as m is bigger than n extend the subdivision of this k prime whatever up to m we have got to the next one by subdividing k m plus 1 okay each of m plus 1 by starring at each m plus 1 simplex at the barycenter keep starring them put them all together you had k prime so that will complete the subdivision okay achieving that the given x is a simplex is a vertex is done at the stage c okay this does not complete the subdivision so for that you have to do the stage d also this we will use next time so today we will stop here thank you


The proposed MTSS block matching algorithm is a hybrid algorithm which combines the advantages of rectangular and hexagonal search patterns. It is faster than the existing MTSS, TSS, and ETSS block matching algorithms and provides better quality of video at lower bit rate. Furthermore, the proposed MTSS block matching algorithm is also compared with the plain fractal coding and found to be superior in terms of reduced encoding time and better quality of video. This article was authored by Shailesh Campbell, now Singh Thakur, and Preeti Bajaj.


Russian drones have become the prey of German Gepards in Ukraine. One of the most valuable weapons that Ukraine has gotten to help fight Russia is a relatively unsophisticated German system from the 1970s, the Flak-Panzer Gepard. According to Business Insider, the Gepard is self-propelled, anti-aircraft gun that uses two automatic cannons to take out aerial targets. It was the first heavy weapon that Germany sent to Ukraine. Despite being close to 50 years old, the Gepard is filling an important role in Ukraine's air defense network, taking out low-flying drones and missiles that are very threatening but too cheap and numerous to justify using sophisticated surface-to-air missiles to shoot down. The Gepard's performance in Ukraine has demonstrated the enduring value of simple and mobile anti-aircraft systems in the 21st century, but Ukraine's Gepards face another problem far from the frontline. The country that makes their ammunition has so far been unwilling to let Kyiv have more of it. After being criticized for its reluctance to send heavy weaponry to Ukraine, the German government promised some 50 Gepards to Ukraine in April 2022. The first three arrived in July and were followed by another 27 by the end of the year. As of March, 34 Gepards have been sent to Ukraine with plans to deliver at least three more. The Gepards were rushed to the front lines almost immediately and proved effective at downing low-flying Russian cruise missiles and drones. They have been particularly effective against Iranian-made Shahad 131 and 136 loitering munitions that Russia is using against Ukraine's energy infrastructure. Ukrainian Gepard crews have been successful despite receiving just two months of training compared to the German standard of 18 months. The Gepards fill an important gap in Ukraine's air defense network which includes long-range systems like Soviet-era S-300 and Buk surface-to-air missile systems as well as Western-made systems like Nassams and the MIM-104 Patriot which currently arrived in Ukraine.


Community members in Bemidji met with area law enforcement officials to thank them for their commitment to the job. The event held at Library Park tonight was meant to show appreciation for the many men and women who keep the community safe. LaPorte resident Virginia Overton says that targeting of law enforcement officials last year in Dallas and other areas prompted her to host the event. Officials gave a drug sniffing dog attack presentation before receiving a blessing and also a sweet treat for their work. I want to keep this community the way it was when I got here, which was respect law enforcement. So this is my way of saying thank you and bringing the community together to do so. Officials from the Minnesota State Patrol, Beltrami County Sheriff's Office, Bemidji Police and Leech Lake Tribal Police all attended the event. If you've enjoyed this segment of Lakeland News, please consider making a tax-deductible contribution to Lakeland Public Television.


AFL can knock your whole body around. Concussion can affect athletes at all levels of sport, from school children to full type professionals. A recent study revealed there are five concussions every thousand player hours, but we don't know the knock-on effect. That's why the Brain Foundation has teamed up with the Swans to help fund research into how the game can affect players later in life. The research will also help almost 1,000 of your sons and daughters currently enrolled in the Swans Academy. We have a responsibility to these young athletes to protect them from possible injuries because there's nothing tough about playing with concussion. As president of the Brain Foundation, I'm really excited by this innovative new research which could dramatically reduce the long-term effects of concussion. It'll help us boarding heroes of today and importantly of tomorrow. I've always paid to win, so let's win this one together. This website or call this number and give generously to the Brain Foundation's concussion initiative.


You talk about summer trade and folks, anemic volume is extraordinary and that didn't matter whether it's equity market, commodity market or the debt market. Right now in the NYSC you get 517 million. I suspect that's going to get up to about 650, but bottom line that is anemic volume. We go into the NASDAQ composite, composite right now running at 1.7, that's not bad inside the composite. Gold contract, gold contract that traded up a couple dollars, you're at $14.25 an ounce. Silver up 23 cents, $16.42 cents. Silver continues to get a really sweet bid folks. What happened to the silver market last week, it took off like a rocket ship. I reported out here this morning is that the money managers did come into the silver ETFs. Now it's much better when they come into the ETFs versus the futures and this is why. The futures are rolling over every three months. You go into the ETF market, bottom line, as you buy the ETF, the ETF sponsors have to buy the futures. That is telling me that they're going into this for a longer term expansion and you can see why the silver medal as well as the equities took off like a rocket ship because the next thing that they will do is that it's so cool about this. So you see, first you can simultaneously go into the ETF for the metal. You go into the ETF for the equities that's so small, most times any money manager would poly one to the other to the other. It's driving force, driving momentum. Notes and bonds continue to want higher price. We have the 10-year up two ticks, $127.17, 30-year up four, $154.28. Right now the 10-year is yielding 2.03. I expect with game here is a 1.71. We know that the end of the month, 30-31, fed you got 100% probability that they're going to reduce the rates by one quarter percent. Bottom line in September, that'll be the next meeting. And in September right now, the probability for another one quarter percent is 64%. Now that's been going down. That was bottom line was 68% this afternoon. We had last week the half a point rate cut. This coming meeting actually got over 50%. Bottom line of Fed Governors this week have to keep them out shut so the bottom line you're not going to hear anything until a statement on the 31st. Stay right there folks. Come right back.


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