From Hollywood. The Hollywood Radio Theater. Ladies and gentlemen, this is Ken Carpenter. We've chosen the most provocative story. It's a comedy drama which vividly illustrates the importance of forgetting the past and concentrating on the future. And as our star, we have one of the finest actresses ever to win an Academy Award, Miss Claire Trevor. And now, Kathleen Hyde's story of one last September, starring Claire Trevor as Elizabeth. Ten years ago, if anyone had asked me what I wanted most, I'd have said, let me go to some picturesque spot with enough money to live on and let me paint as I please. As a matter of fact, that's substantially what happened. Ten years ago, with the small inheritance from my mother, I came to Santa Fe, New Mexico. Since then, I've had enough money to live on and I've painted as I please. Now that I look back, I wonder why I didn't add one more requirement. Please let me have talent. In fact, now that I look back, I wonder about a lot of things. Hey, Liz. Oh, now, please, oh, still, Ken, I won't be much longer. Well, I'm holding this as still as these full cows will let me, Liz. We just aren't used to having our portraits painted, these cows and me. Now, don't squint so much. Well, the sun's right in my eyes. Well, I need it there for the shadows.  Look, the cows aren't squinting. They don't know any better. The shadows are perfect, just perfect. Well, honey, why don't you just paint the shadows? Leave me and the cows be. Ken, please.  Quit it. Look, can't you stop bobbing around? I'm not painting a motion picture, you know.  You're just not in the mood. Now, what on earth got into them all of a sudden? You can check your watch by it. Those girls are ready to be milked. Oh, dear. Another afternoon with those girls and I'll go back to still life. Ah, let's take a look at it now. Well, it's not finished. Mm-hmm. Well? Um, which one's me? The one with the hat on. Oh. Oh, yeah. No, Ken, you never understand.  I just want the impression of you. And that's the impression of me? Mm-hmm, sometimes. No wonder you won't marry me. See, I could do better. Not around Santa Fe. I'm about as eligible as they come. No, I was talking about the painting. Ah, I don't know what's the matter with me. I can't paint. I can't get interested in anything. I can't even appreciate you as I should, Ken. I love you, Liz. It's a good little ranch, you know. A few more cattle every year and I'll make something out of this place.    Ken. No, I mean it. I mean it. Honey, when a rancher says that to a girl, that's love. I'll bring you things unless you'd rather I didn't come in. No, no, no. Come on in. The night sure smells wonderful, doesn't it, Liz? Mm-hmm.  Have you ever been to the rest of the country? No. Don't need to go. Everything I want's here. You must like it here, Liz. You've been here ten years. I sure have. Wait, I'll get the light. Oh, just while the easel mingles are in the corner for now. Okay. This isn't the cutest little hacienda in town. It isn't a hacienda. It isn't? No, it's a house. Just a house. Yes, ma'am. Oh, I'm sorry, Ken, but that's part of it. I'm tired of houses that are haciendas and adobes and mesquites and ranches and cattle and webloans and petrified forests and cliff-dwelling. Do you know that there's a brave new world outside with houses made of brick and wood and tall buildings and streetcars and buses? Yeah. Well, sure. Either now or the curkey. See, this or not, there are men who live out their entire lives in business use and dinner jackets and who never ever wear blue jeans. Well, these are clean blue jeans, Liz. No, Ken, it isn't you. It's me. I'm... I'm holding sick.  I want to. I want to go back to everything I left. Only, well, everyone's so spread out. Now, Dad's alone in Minneapolis, Bud and Edie and my niece are in Cleveland, and I'm here. We just never all seem to be in the same place at the same time, and there really isn't a home to go back to except the lodge. And that wouldn't be any real fun unless we could all be together like we used to be.  No, if we only could. Honestly, Ken, I... Sure, sure, honey. That lodge is yours. It's on a lake up in Minnesota, isn't it? Mm-hmm. Lake Winnebogosha. Lake Winnebogosha. Yeah. That means blue lake. Many fish. The family and I exchanged more letters than we had in at least two years, full of plans and arrangements for one last September at the lodge.   Mm-hmm. Of course, that's an old picture. You know, it's been built up a lot since then. That's really a bad angle of it. Uh-huh. Oh, inside, it's really charming. The rooms, well, they're simply huge. With great windows overlooking the lake and bare rugs on the floors. And the fireplace, Ken. I bet three men could stand in it without stooping it that big. They ever build fires, or you just stand around in it? No. Oh, wait! There, that's a picture of Dad, I think. Let me see. Yes, that's Dad. And a string of wall-eyed pike he caught in the lake. Uh-huh. I thought you said he was tall. Well, he is. He's quite tall. He's still over six feet. Sure doesn't look it. Well, maybe not. It could be the picture, you know, but, well, I know he's quite tall. He's much taller than you can. Sure doesn't look it. Oh, and the most divine sense of humor.  Not broad, you know, subtle and clever.  Everybody does. Oh. Oh. Yeah. Well, it's not so much what he said.  Oh. Sure doesn't look very tall, though. Oh. Honestly, Ken, he's my father. I ought to know whether he's tall or not. Oh, yeah. Yeah, you ought to, all right. Bud's even taller. Yeah, maybe I can find a picture of him. Oh. This crazy old picture album. I haven't looked at it in years. Hey, no, wait a minute. Let me see that one. That one. Oh, bad. What a funny looking kid. A friend of yours? A dear friend. It's me. No, it can't be. You're not not need. I am, too. Anyway, I was there. I know I have some pictures of Bud somewhere. Hey, what was that? That one, the one on the bear rug. It was that same dear friend. And I've never known why they persisted in taking baby pictures like that. Oh. Oh. Yeah. Mm-hmm. Oh, here's one. Oh, it's a Bud and me. Well, you can see for yourself. See how tall and lean and tan. You don't look so not need in that one. Oh, look at Bud. He's skinny, all right. Lean, slender, certainly not skinny. Okay, honey. How old were you then? Oh, 13 or 14, I guess. I remember that bathing suit. You know, it was all wool and scratch, something awful. You can't see it, but it had a brand new innovation for that time, a sun tan back. Oh, what's that you got pinned on you? Oh, well, that's my ribbon. Yeah, I placed third in the 50-yard freestyle at the Club Fourth of July festivity. Third? Mm-hmm. That's not bad. Well, there were just three of us in the race. Oh. You don't swim much anymore, do you? No, hardly ever, but then there's no swimming around here like there is at Lake Winnebough-Gosher. You know, I met a fellow up at Red River last year. He told me the mosquitoes are just fierce up on those Minnesota lakes. Oh, no. We never had a bit of trouble with them. Oh, he said they were just fierce. Yeah. Third day were bad enough in the daytime just biting you, but at night they dive-bombed you. And they had you backing up the air all night. He said they were just fierce. You know, you're beginning to irritate me. Why? Well, you don't have a kind word to say for my family or Minnesota or Lake Winnebough-Gosher or anything. Oh, now, Liz, that's not so. You're kind of throwing it all in my head, you know. Maybe it's a little hard for me to believe, but... Not one kind word, have you said. And remember, this is my family we're talking about. My lodge, my Minnesota, my home. Well, sir, there's one thing. What? You know that picture of your dad? Yes. Not sure a nice little string of wall eyes he caught. These days were filled with shopping and making train reservations and packing my paints and brushes to be sent on ahead. Ken was around, but we tried to keep our conversation off controversial subjects.  Better shake a leg, Liz. Train leaves in about ten minutes. Oh, I know. There's just these few things to put in the fortnight or I'll be ready. Well, is that something new? Oh, yes. I bought a couple of formals. I forgot all about the club. The dinner dance isn't all. I ran out and bought the first two that fit. You're sure pretty. Oh, they're all right. I don't suppose they'll set anyone a flame at the club, but, well, they're all right. I'd like to see you on something like that once, Liz. Yes, but we don't do things like that, Ken. I mean, it's a lot more casual here. Men, women, boys, girls, everybody's wears blue jeans.  Liz. Yes? How long you'll be gone? Well, I will be at the lake all of September. Are you coming back here then? Oh, I might stay with Dad a while in Minneapolis or maybe run down to Cleveland with Buddy and Edie. I don't know. You may not come back here then. You're not planning to, are you, Liz? Can I? I may not. Right now, I just want to go back home. It's all I want to do. Better go now, huh? I didn't want to hurt Ken, but I did want to leave. When I felt that train start and knew I was on my way home, it felt so wonderful. That afternoon, through New Mexico, across the corner of Texas, Oklahoma, that night through Kansas, the time flew at first, then dragged its feet the next day as we cut straight north through Iowa. How many times I'd visualize the homecoming in the Minneapolis station, I can't say. All I know is that I was ruthless when the train finally stopped there. We have 24 bags in a small trunk. Who could see it? Liz, baby. Are you all right? I'm sorry, honey, and you made it the hard way. For you to keep in mind, as many another American has, perhaps one of our greatest ambassadors was the humorist Artemis Ward, who did a great deal to cement the friendship between America and England. In 1866, Artemis Ward arrived in London for a series of lecture tours, although he wasn't in the best of health. After his first lecture, an English newspaper wrote, there is certainly this foundation for a cordial understanding between the two countries calling themselves Anglo-Saxon, that the Englishman, puzzled by Yankee politics, thoroughly relishes Yankee jokes. When two persons laugh together, they cannot hate each other much so long as the laughter continues. As Artemis Ward continued his tour, in his own humorous way, he criticized both England and America and the cordial understanding grew between the two countries. Although his health grew worse, Artemis Ward refused to abandon his tour, and he didn't stop until he collapsed in the middle of a lecture. Within a few days, he was dead, at the age of 33. In announcing his death, the London Observer said, Artemis Ward never used his great powers of humor for that biting purpose which is implied in the word sarcasm. He's been a man not only of humor, but of good humor. There is no man among us who does not feel that he is the better since his landing in this country, he was taken by the hand and the feeling of brotherhood between our two countries. So it was that Artemis Ward proved to all America that by helping others, you help your country. Now, act two of one last September starring Claire Trevor as Elizabeth. I've up to the lake that night was like every other trip up there we'd ever made together. Dad insisted on driving and gave her advice from Bud and Edie with a little Betty in the back seat and me beside him in the front. We made the wrong turns at the same old places, sang the same old family songs from I've been working on the railroad through there's a long, long trail of winding. Then finally we drove up the lane that wound through the wooded wilderness up to the lodge and in ten minutes the car was unloaded and everyone began unpacking. I could carry a tune. Oh, look who's talking. It's an original tin ear. Let me take a look at you. I haven't really seen you in the light. No, we haven't seen each other really, have we? Well, you're not bald anyway. No, I... but it's really rather becoming to you, Bud. Tell Edie that, will you? Sure. You've grown up, sis. You're taller, aren't you? Yeah, isn't that funny? I mean, if you were shorter than I'd remembered. Still the same old 5'10' I always was. Well, I guess you just seemed taller when I was growing up. Yeah. Well, don't forget we're all ten years older than we were the last time we were together. Ten years makes a lot of difference. Yes, it does. Have you been wearing glasses long? Oh, a few years, I don't remember. You don't wear them yet, huh? All for reading and post-work, you know, sometimes. Well, sure as well to be together again, isn't it? Oh, it's wonderful. Just wonderful. Well, I guess Edie could use some help. See you in the morning, huh? Oh, sure thing. You know, it's funny, I know I have a hundred things to ask you about, but I... I just can't think of them right now. I know. I feel the same way. Well, we'll be here a whole month, we'll think of it. Oh, sure we will. Yeah. Well, see you, sis. First thing in the morning. Night. Hey, you're supposed to... Oh, I know. I know. I didn't think you'd remember. Say it again. Night. Sleep tight. Don't let the bedbugs bite. Night. It was late when I finished unpacking. I think all of us would like to have gone downstairs and settled down for a big reunion talk. But with the excitement and the traveling, we were all tired. I fairly sank into the feather tick and floated off to sleep. Well, not quite to sleep. When I closed my eyes, I kept seeing Bud, the way he really looks now. You know, I'd never thought of him as anything, but a bronze young man with a quiet, rather shy youthfulness about him. It was hard to think of Bud as a man in his forties. Oh, yes, and there was something else that kept me from sinking into wonderful sleep. Well, I got him anyway. What are you trying to do? Make a liar out of me? I said there weren't any mosquitoes in Minnesota. You're just one. I deserve it for forgetting about the beast. Never occurred to me to remind you last night anyone who came here as many summers you did ought to remember about mosquitoes. It's all coming back, believe me. Anyone else want coffee? Edie? Dad? Not for me, thanks. No. Oh, now, don't laugh at me, Dad. I know I'm funny. Oh, I've got some sister Nella for those bikes. Best thing for her, am I just thinking? Ten thousand lakes in Minnesota and ten thousand mosquitoes for each lake. Oh, Dad, you can do better than that. Oh, that's a corker, all right. I'll have to remember to tell Bud that one.  He always tells everyone how terribly funny you are. You know, I always do, too. Where is Bud, Edie? He and Betty went for a swim. Well, come by naturally enough, I guess. Oh, my father was a regular car. He was a regular car. Oh, my father was a regular car. I remember one time when those lampshades and the beadprings were all the go. Well, Dad was visiting us, and I'm darned if he didn't put on one of those things one night. I thought it was a scream. He was, he was. Well, I think I'll do the dishes, please. Well, I think I'll do the dishes, please. You were much funnier, Dad. You know, a lot more subtle than original. Well, different generation. You know, a different kind of humor. I guess Dad's comedy was a little broad now that I come to think of it. Yeah, it certainly was. I never thought you'd put on a lampshade. No, you never did. I was more inclined to put on your mother's hats and parade around. Garly, that's right, you did. Well, I'm going to run over to Harrowman and see if he's of a mind to do a little fishing. Harrowman, do they still come up here? Sure they do. Oh, Harrowman wouldn't miss a summer up here. Say, they'd be tickled to death. Better ride over with me. Oh, no, not this morning, Dad. Not with these mosquito bites. But tell them I'll see them soon, hmm? All right. Oh, uh, Deek's here with him, you know. Deek? I thought that might get her eyes out of you. Oh, it's not, Dad. It's just that I hadn't thought about Deek being up here, too. Oh, no. Oh, now, Dad, Deek was never more than a good, sort of funny and edgy tat, but I'd forgotten that you didn't dare encourage him by laughing. Nothing, and no one was safe then. I couldn't get over it about Deek being there, too. It was just one thing I hadn't planned on that September. Take it easy, Betty. Don't churn the water so much. Keep your head in the water. Well, what's all the commotion? Oh, hi, Liz. Hi. My daughter displaces more water than the Queen Mary. Natural born swimmer. Well, she's young this time. What do you mean, young? At her age, you had to crawl like Weissmuller. Oh, not me, brother. My style was a cross between Betty's there and the Mississippi paddle boat. You got a bum memory, sis. Well... I've had it when she wears herself out tell her to come up the house for me. Hey, what's happened to the lake? What do you mean? Same old, any big oceans? No, no, I mean the water. It's not blue like it used to be. Blue? Yes, you've forgotten. This wet thing was never blue. Oh, but it was. Sky blue. That's where it got its name. Winnebogosius. Blue lake, many fish. Sis, honey, what pipe you've been drawing on? Winnebogosius is Indian for wretched, miserable, dirty water. No. That's a more apt description of us. Well, try thinking of it as Winnebogosius. It's kinder somehow. Um... You going in for a swim? I don't know. I may. Well, if you do, watch out for the blood suckers. Oh, yeah. I will. Who you got all over you? Mosquito bites. Well, when you come back to house, I got some stuff you can put on it. Citronella. Well, I guess I didn't hear him right. Oil of cloth, the only thing for you. Oh, I'll remember that. Have you seen Deek Herman lately? Yeah. Last year, I think it was. He's supposed to be up here with his folks, I think. I know. I was just wondering, has he changed much? Nah. Same old Deek. Looks the same, acts the same. Same old Deek. Oh, that's nice. I was afraid maybe that... Well, that's nice at all. Is he, um... No, not yet. Yeah, I may be wrong, but I've got a hunch he's never forgotten you. Oh, I don't think that. That sounds like Deek and... I have a lot to remember and... Yeah, well, that sounds like you do, too. Well, see you at the house, huh? Uh-huh. Eddie, I'm afraid I've forgotten you were here. Oh, yes, your father's gone up to the house. Well, I wouldn't wonder. You take swimming very big, don't you?  Then you must hate it with a vengeance.  Why on earth do you swim? Because you did. Because I did. Well, that's no reason at all. I like to swim. That's why I did so much of it. If I hadn't liked it, I wouldn't have done it. Well, you must like it a little. I hate it. But you get it, so I swim. Hell, we'll have a little talk with your parents and maybe we can straighten all this out. When are you going in swimming? Oh, I don't know. I just thought I'd... Take a little fun for a while. Maybe swim a little later. I want to see you swim from here to the island. Oh, not me. Why, it's a good half-mile from here to the island. Daddy said you used to swim from here to the island all the time. Something's wrong with Daddy's memory, then. Do you still have your blue ribbon? Blue ribbon? For winning the half-mile race from here to the island. Daddy said you want a blue ribbon for that. Wow, what pipe has he been growing on? I've tried it. I've tried it. What's going on? I've placed third in the 50-yard dash, and as I recall it, it was a yellow ribbon. Daddy doesn't smoke a pipe. Oh, well, no, I guess he doesn't. Well, I think I'll just stretch out and get some sun. Can I watch? Yeah, you're really interested. I might fall asleep on you, though. That's all right. You're sort of not neat, aren't you? I don't think so. I do. Mm-hmm. What have you got all over you? Mosquito bites. I had them all over me at the camp last year. You ought to get some whoosh. Some what? Whoosh. You just rub it on and whoosh. They're gone. Whoosh, they're gone. Best thing I've found for Mosquito. Well, thanks for the tip. Recollection of being hauled into a fishing boat, in Boston, exactly 50 yards from the dock. I was brave to the core, though, insisted on walking the last few yards into shore in front of the house. It was there that the bloodsuckers got me. Oh, blessed Providence. It began raining a little later. You know, I'd forgotten about the rains, too. Forgotten that they could last for days in the Lake Country. But there's one thing about rains. They do bring a family close together. Well, whose deal is it this time? I dealt last time. It must be Dad's. No, I dealt last time. It's Edie's deal. It can't be my deal. Look where the cards are. Oh, all right. I'll deal. But it's not my deal. It's not Dad's. It's nothing you get upset about. After all, it's just a game. I'm not upset. Of course, it's just a game. The thing you do is relax. After all, none of us really plays tournament bridge, you know. Well, I like that. If you haven't liked the way I've been playing, the least you can do is come right out with it. No, Edie. Well, I didn't mean anything of a kind, Edie. Well, it certainly sounded that way to me. Oh, no, really, I didn't mean that. I think you play wonderful bridge. Well, I do try. After all, with a growing child and a house to keep, I don't have much chance to play like some people. Pick up your cards. The cards are down here. Say, what do you mean by that? Some people. Now says. My dear, I didn't mean any harm. And it certainly wasn't directed towards you in any way. Well, it certainly sounded that way to me. Well, really, I didn't mean that. I think you play wonderful bridge. Well, I do try. Oh, for heaven's sakes. Two spades. Two spades. Yes, partner, two spades. Oh, honestly, you two. Go on, bit, Edie. Well, you don't have to shout at me. After all, it's just a game. A game if we don't play it. Come on, Edie, it's your bid. All right. Now, wait a minute, Edie. Dad, you know how much honor count you need for a two-demand? Do I know how much honor count I need for a two-demand? Oh, dad, for heaven's sake. Dad, take it easy. I just asked. In spades, I have the ace, King Jack, ten. In hearts, I have the ace. Dad, don't tell him.  In hearts, the ace, King Queen. In clubs, the ace, King. In diamonds, the ace, King, Queen. Jack, ten. That's amazing. It's a grand slam. It was a grand slam. It was a misdeal. Huh? You've got 14 cards. Ultimately, the rain only lasted three days. And when it stopped, most of the temperaments stopped with it. It was a brand-new family when the sun shone on it. For sanity's sake, however, everyone did take the precaution to go off in different directions. Dad went fishing with Mr. Harriman. Edie and Betty went into town for the day. Bud took the motorboat to the far side of the island where he debated whether or not he would ever return. And I, well, it was so pleasant around the lodge, I decided to stay home. I set up my easel on the veranda and began to reproduce Lake Winnebogosius on Kansas. Miserable, dirty lake. Why weren't your sky blue like I remembered you? You know what you are?   Hey, watch it. You're talking about a friend of mine. Oh, my goodness. Deke. Hello, Liz. Deke Harriman. Imagine seeing me here. I only spent my youth on this veranda. You can't really be that surprised as soon as... Oh, but I am. And you didn't...  And so do you. No, but just exactly. Oh, sit down, please. I don't want to interrupt anything. Oh, this eye. Just killing time. I'm afraid I'm doing a fair job of murdering the scenery, too. No, no, that's nice. Well, I can't get over it. Bud told me, but I still can't get over it. What? Well, you're just the same as ever. You look the same, act the same. Same oldie. What do you expect? Something different, everything else here. And everyone seems so different, but you're... You know, it's like old times. Yes, it is, Liz. I was wondering... Well, it's been a long time. I don't know how to ask you. Look, are you with... No, I'm not. That's good. Neither am I. I know, Bud told me. Just like old times. Just like old times. Still like to dance? Oh, I love to dance. You? Oh, great. Love. Say, this is like old times. Do you know who's playing at the club now? No, I... Zeke, you don't mean... My word of honor. Snuffy baker. And his baker's dozen. Oh. Liz, I've got a wonderful idea. When? Tomorrow. We'll spend the day at the club just like we used to. We'll stay on for dinner and the dance and the evening. Oh, Zeke, that sounds perfect. I'd love to. So would I. I'll pick you up tomorrow morning early. The early of the day. Oh, gosh, it's great to see you again. Like old times. Like old times. Oh, say, and don't worry because I always keep a supply of it at the club. Of what? Dead. Dead? Only thing in the world for the mosquito bites. There's a thought for you to keep in mind as many another American has, like Fiorello LaGuardia, the busy little mayor of New York, who found time to get on the radio and read funnies to the kids. There was a man who loved children. And through his love, saw the United Nations International Children's Emergency Fund come into being. To start such a fund, LaGuardia went to Europe, not as a representative of the UN, but as an American citizen. He traveled from country to country, investigating conditions, speaking to the people and making friends. In his native land, Italy, he helped hand out food to the needy and won their admiration. On his return to the United States, LaGuardia worked day and night to sell the need for a children's emergency fund. And finally, the General Assembly of the UN adopted his proposal unanimously, in a resolution which stated that as many children as possible to the age of 18 would receive help from the fund on the basis of need without discrimination because of race, creed, nationality, or political belief. Workers all over America donated a day's pay to the fund as did children in all kinds of schools, organizations, and churches. But LaGuardia didn't live to see the full success of the great work he helped to start. His big heart stopped the night before the UN announced aid allocation to over 3,700,000 children and mothers in 12 European countries and China.  Three of one last September starring Claire Trevor as Elizabeth. I may have said that night. I don't know. It seemed to me as if I just floated from one pink cloud to another, remembering so many things about Deacon Me and growing up together. It was as if just seeing him again had unlocked a flood from the past and big memories and little recollections came tumbling through my mind. Deacon changed. He was just the same. Just exactly the same. And I awoke to a fresh new day that was full of promise. The spray! Remember, this is the first game I played in ages. Well, now just relax. You've been tensing up a little the last few holes. Remember now, just a nice, easy swing. Don't try to hit it a mile, just... Boy, you saw what I did. All right. It's funny. I am tensing up a little. I think maybe I'm a little tired.  We've only played eight holes. I know, but I've swung a lot more times than you have. Don't forget it. It's good for you. Exercise keeps you in shape. One now. Line it up and let her have it. Uh-huh. Now, let's see now. The hole is... Hmm? I'm looking at the ball. Where is it?  I mean, where is it in relation to your left heel? Oh, oh. I should have it even with my left heel. That's right. There isn't much to remember. Line the ball up with your left heel. Keep your eye on it. Now pull back with a nice, easy swing. Shifting your weight as you do from left to right. That's not too much with the hips. Now there. Now you're at the top of your back swing. Now come down easy. Shifting your weight as you do from right back to left. Keep your eye on the ball. Let the club head and meet it. That a girl? I am the ball. Where is it? Right down there on the T. There's no harm done, Liz. We've got all day now. Let's just start over. Now relax. Don't tense up. Just a nice, easy swing. Remember, it's just you against that little white ball. Into your back swing now. Nice and easy. Uh, Deke. Hmm? Just me against the little white ball. Hmm? Set point. I thought it would never come. Well, let's see. That's six games to love, wasn't it? Not go six love, six love, six love, six love, six love. You left one out. All right, six love. Well, go on, Liz. Hmm? Well, time to change courts. I'm so sure. Rebecca said you loved to dance. That when I was very, very young. You said it yesterday. I was much younger yesterday. Liz, I've looked forward to dancing with you for so long. Deke, believe me. I meant it yesterday when I said I loved to dance, but not after speed voting at dawn, followed by 18 holes of golf and six sets of tennis. You're just out of practice. You used to do that every day all summer long. Never bat an eye. Well, I can't bat an eye now either. And it wears me out. Liz, whatever happened to you? You look the same.  You've got that wonderful dreamy look in your eyes. I'm probably running a fever. This accounts for the color and the dreamy look. I'm practically sound asleep. Well, you're not the same old Liz. She'd never sit out a club dance when Snuffy Baker was playing. No, she was a game kid. Whatever happened to her? Come on, Liz. Just one dance for all times sake. All right, Deke. And I can promise you, these times are very old. Full Liz. And I thought you'd forgotten that just relax, honey, and remember. Liz, do you hear me? Well, Liz, myself, but according to reliable sources, I slept 14 hours by the clock after our song was over. These same sources report that I opened one eye, stibled a scream, and settled back to some real sleeping. I said it before. I can say it again with more authority. Deke had not changed. He was just the same. But I had changed. I was ten years older by the calendar and three score and ten by blood count and heartbeat. But the feeling came back in a matter of four or five days. It was almost no effort at all to pick up the phone. No. Long distance, please. I'm long distance. Oh, I want to place a call to Santa Fe in New Mexico to Mr. Ken Stevens. His number? Santa Fe number. Oh, well, keep trying. Yes, ma'am. A number. Oh, but there has to be. I'm sorry, ma'am. Oh, keep trying. Ma'am, but we've had some complaints here at the phone company. You're on a party line, ma'am, and the other 13 parties have liked the use of a line. Oh, all right. I'm sorry. Just cancel the call. What's the matter, honey? Oh, nothing. Oh, dad. Dad, everything's a matter. Oh, dad, now, honey, you're all right. You're all right.  Let it go. Why don't let it all go. Acting like a little girl. It's true, honey. Come on, now. Just straighten up. Pull yourself together. Take a good deep breath. Let's have one of our old man-to-man talks. What do you say? Yes, sir.  I know I have, but I don't understand when I've done wrong. I really don't. I just wanted us all to be together the way we used to be and have all those wonderful times. Now, what's wrong about that, then? No, it's not wrong, exactly, Liz. It's... Well, like I said, it's unfair.  No, we didn't. I don't remember a single bad time. Honestly, I don't. That's just it. You don't remember. You've done what everyone does, I guess. Until you grow up a little and find out you're mistaken, you've selected a few choice memories, Liz. You've been gentle with them. More than that, you've been generous with them. You've fed them too much. I don't understand. Well, you've made them much bigger than they were in reality, much brighter, more memorable. You see, there's memories, especially of people and places you love, are more than kind. They're generous to a fault. Just as an example, Bud was a little bit disappointing to you, wasn't he? Well... Come on, I'll be honest. It wasn't quite the way you remembered him, was it? Well, I did think he was taller. And, well, I didn't stop to think that he'd be older naturally and a little bald and... Oh, but it isn't Bud, Dad. I love him just as I always did. That isn't much wrong with me. No, not entirely. I fell a little short of your memory of me, too, I believe, huh? Well, not really, Dad. Not in any important way, certainly. But I'm quite a bit cornyer than you remember I was, aren't I?  Maybe a little. But, like I said, not in any important way. No, I know, honey. And then the lake disappointed you in the mosquitoes and things like that. You see, you'd just forgotten some of the bad things. But, Dad, now take Deek. Deek hadn't changed a bit. Not the tiniest little bits. And that disappointed me after Donier made a permanent wreck of me. You've done some shielding of Deek in your remembering, too. Well, that's not all of it, but it has a place in it. Well, what is all of it, Dad? Well, I'm just your old man, but... I'd say you were homesick, Liz. Homesick? But I came home and that wasn't the answer. Honey, your home is where you are, where you live. See, this isn't a home anymore, Liz. Not to any of us, really. Well, then... What's he like, Liz? Well, Dad, he's tall and tan. Oh, no, he isn't. He's tan, but he's not especially tall. It doesn't matter whether he's tall or not, does it? No, good heavens know it. Well, Ken is sort of hard to put into words. He's, um, he's gentle. He's wonderfully nice and... Does he, uh, love you, Liz? Well, I thought so, but, you know, I haven't heard from him. And he isn't home. And I really didn't act very well the last time I saw him. Oh, don't worry about it, honey. It may be mistaken, you know.  Oh, I remember this all right. That was pretty awful. You were just fierce. Oh, Ken! Yes, ma'am? Well, but you can't be here. I mean, uh... Oh, forgive me, Dad. This is Ken. I believe we've met. Now you two just excuse me, will you? He's sure a nice fellow, your dad. Yes, he is. Ken, how long have you been here? Oh, I drove in early this morning. Figured it'd be too early to wake anyone, but my golly, your dad was up and we had a real nice talk. No, I mean, how long have you been in this room? Well, just long enough to hear you say you hadn't been too friendly the last time you saw me. You were just fierce. I know. I'm sorry, Ken. Well, it doesn't matter now. No, it doesn't matter now. Well, sir, I was just passing through here, and I thought I'd dropped in. Drop in? Passing through? Something happened to your hearing, Liz? No, I don't think so, but... Well, Ken, what are you doing up here? Oh, traveling sort of. Never did much of that, you know, born and raised right there in Santa Fe. Figured all to move around a little, see the country. Oh, well, uh, how do you like it? What? The country. Nice. I like it. Oh, there's one thing. Yes? That fellow I talked to last year in Red River, he was right. Mosquitoes are just fierce up here, Liz. They're just fierce. But, Ken, I know a great cure for mosquito bites. Citronella? Nope, not oil of clove or dead or whoosh. No? No, the best thing in the world for mosquitoes is to go back to Santa Fe. Oh, well, now I was thinking I'd do just that, Liz. Before I do, though, I'd like to ask you one question. Ken, darling, I just want to go back home. That's all I want to do. You sure? Oh, I'm so sure. Well, sir, I'm glad to hear that, Liz. Because just before I left, I had a chance to buy a hundred head of white-faced cattle at a real good price. Liz, to tell you the truth, I can't afford van. And you, too. Let's go home, honey. I'm going to go to the Easton, Pennsylvania, anytime soon. You might run into a fellow who is studying at Lafayette College. It wasn't too long ago that he was a kamikaze pilot. He was an enemy. Today, he's getting a free democratic education because of a boy by the name of Robert Johnstone. Johnstone died in the Philippines, but the kind of heart that he had will be evident for years. You see, he stated in his will that his $10,000 government insurance was to be used to educate one of his former Japanese enemies in American ways. It was his last wish, and it's being carried out as he desired. And this took place in World War II, of course, but such acts by you and your friends even today are shaping our world of tomorrow. And now, here's Claire Trevor. Irving coming. You and I are old friends again. No, we certainly are, Irving. You know, I directed you in some of the first pictures in the Fox. Yes, and I'll bet you don't remember what the first one was called. It's a bet. It was, uh... a mad game with Spencer Tracy, wasn't it? That's right. And what's your latest picture, Claire? Well, Irving, it's a 3-D western in color, and it's called The Stranger War, a gun. But tell us why you dropped in tonight, Irving. Well, next, it's one of the season's finest motion pictures. Adapted from one of the best-selling novels of the year, the great love surrounded by mystery and intrigue, Daphne de Morier's My Cousin Rachel, and as our star, that gracious and lovely lady who created the title role, Olivia de Haviland, and playing officer, a newcomer to our stage, Ron Randell. Yeah, that was certainly got your 20th season off to a brilliant start. Good night, Irving. Good night, Claire. It was a really wonderful scene. This is the presentation of the United States Armed Forces Radio Service.


Rockowitz who asked the first question to Jay Hap moments ago. They seem like you are really aggressive from the jump. How much was that part of the game plan to attack from the beginning. Yeah, it was definitely a goal going in. You know, they had a lot of good at bats last night and we felt like we really needed to get ahead. Excuse me. Try to force the issue. You know, I knew we were also a little bit. Maybe light as far as reserves in the bullpen. So I wanted to try to be efficient as much as I could. 10s, 10 strikeouts. How were you able to get so many swings and misses? What did you feel like was really working for you? Gary was good back there. I thought we we had a good next. I think the change up was was important for me today. And Jay, just how frustrating is it? You go out there, quality start and don't get any run support from the offense. Yeah, you know, I'm focusing on my job. I'm kicking myself for the pitch to Davis there didn't execute. He put a good swing on it. And sometimes that's enough. Unfortunately, we played hard tonight. Next one goes to Eric Boland. Jay, Jay, does the 16 and six start almost feel like a season ago? I don't know about that. I think we all know what we're capable of, you know, unfortunately, we're in a little bit of a obviously a tough spell here the last couple of weeks. But, you know, that was a good couple of weeks stretch. So, you know, we know that we can turn around, hopefully do something similar to that. And starts with one game and in that game, hopefully be tomorrow. Thank you, Jerry.


A warm welcome to the 22nd lecture on the subject of wavelets and multirate digital signal processing. We continue in this lecture to build upon the theme of the short time Fourier transform and the continuous wavelet transform from the point of view of reconstruction. In the previous lecture, we had briefed about the meaning of the short time Fourier transform and the continuous wavelet transform, a generalization of the dietic wavelet transform that we had built in the higher multi-resolution analysis or in some of the other examples of the Dobash filter banks and so on. We saw that in the short time Fourier transform and in the continuous wavelet transform. The two parameters characterizing the transform or indexing the transform were continuous. In the short time Fourier transform, it was the translation and modulation parameters that were continuous. In the continuous wavelet transform, it was the translation and dilation or scale factors that were continuous. In the short time Fourier transform, the modulation and translation both extended all over the real line. In the continuous wavelet transform, the translation extended all over the real line.  Let us bring out the theme of the lecture today. So, in today's lecture, we intend to talk about reconstructing the signal from its transforms, whether the short time Fourier transform or the continuous wavelet transform. In that process, we wish to bring out the very important notion of what is called admissibility. Recall that in the previous lecture, we had raised the question of what qualifies to be a wavelet. We said the Haar function, the Haar wavelet, of course qualifies to be a wavelet, but what in general characterizes a wavelet function? We had partly given the answer yesterday. We said that a band pass character is essential, but today we need to qualify and build that concept in greater depth and that is what we intend to do. So, with that little background, let us proceed then to ask the first question. How do we reconstruct a signal from its short time Fourier transforms? So, let us recall the notation that we had used for the short time Fourier transform, the short time Fourier transform of the function x with respect to the window v. These are the secondary arguments and the primary arguments at the translation and modulation given by tau naught and capital omega naught. So, this is the question. How do we reconstruct x t from this? In order for a transform to be useful, it must be invertible, otherwise we are losing something in the transform and therefore, we need to ask this question and answer it satisfactorily. Anyway, let us therefore, recall once again what this quantity was. So, s t f t x v as secondary and tau naught omega naught as primary arguments was equal essentially to a dot product. The dot product of x t with a suitably translated window and then modulated. Recall that we in general must put a complex conjugate here and that means that we must make this e raised to power minus j omega naught t. Now, in a certain sense we realize that this is like the component. It is a dot product of x t with a translated version and modulated version of the window. So, it is like a component of x t along something, you know could view it that way. So, it is like a component and one simple way to reconstruct from components is to take each component multiplied by a so called unit vector or if not a unit vector a properly normed vector in the direction of that component and then some overall such components. We use this idea before, we used it in understanding the Fourier transform, we used it in understanding the passibles theorem and now we use it also to reconstruct x t from its short time Fourier transform. So, suppose we perform that experiment, suppose we simply use the same idea of taking this to be a component multiplying it by a unit vector in the direction of that component and summing overall such components. What should we get or in other words what should we do to get back x t by this process that is what we should now ask.  What should we get? We should ideally to within a constant hopefully get x t, they should give us a multiple of x t let us say c 0 times x t and we shall now put down conditions for this to happen. In fact, it is easy all that we need to do is to expand this s t f t and then take note of what is happening. So, expanding we have a triple integral one should not be frightened of the triple integral, it just requires a little bit of rearrangement afterwards. So, the innermost integral is essentially to calculate the s t f t and keeping in mind that we have a t outside we should give a different name to this variable of integration here. So, this is first an integral with respect to t 1 and then this whole thing is multiplied by v t minus t 0 e raise the power j omega naught t d tau naught d omega naught this is the kind of integral that we need to construct. Now, let us make an observation on this you see we will first make an observation on this part here. So, let us interchange the order of the integrals incidentally all these integrals are from minus to plus infinity. So, I have not written that down explicitly let us first look at this integral we should take one integral at a time. So, it will be easier to evaluate the integral on tau naught on omega naught first and then take up the integral on t 1 and so on. So, here let us consider the integral on omega naught because that is the easiest one to deal with after all there are only these two terms that depend upon omega naught and therefore, they would be dispensed of first considering first the d omega naught integral.  So, none of these terms are depend on capital omega naught only these two do. So, let us isolate those two terms and dispense of that integral. We have essentially e raise the power minus j omega naught t 1 times e raise the power j omega naught t d omega naught from minus to plus infinity and if we go back to our very fundamental principles of the Fourier transform this integral is very easy to evaluate after all when we combine this what we get is the following we get e raise the power j omega naught t minus t 1 d omega naught essentially if I multiply and divide by 2 pi this is a very familiar quantity you know what we are doing here effectively if you look at it is taking the inverse Fourier transform you know if you interpret this carefully it is the inverse Fourier transform of a quantity of a function which is 1 for all omega naught and that happens to be the Fourier transform of an impulse. So, this part is essentially the evaluation of the inverse Fourier transform of a function which is 1 for all omega naught. So, that would essentially be the impulse placed at t minus t 1 and of course, this factor of 2 pi is there. So, I can write that down and do a dispense entirely with the integral on omega naught. So, this becomes 2 pi times delta t minus t 1 where delta is the continuous impulse unit impulse if you like and now I substitute this. So, when I substitute this one more integral vanishes let me substitute this back into the original integral that I had the triple integral as I said that triple integral is no longer so frightening. So, all this this integral on omega naught let me underline the terms here this, this and this have been dispensed off with an impulse located at t minus t 1 and now the natural choice is to dispense of the integral with respect to t 1. How would we dispense of the integral with respect to t 1? We now dispense off the integral with respect to t 1 and doing that is essentially putting t 1 equal to t that is the way to dispense of it. After all when we multiply by an impulse at t minus t 1 what we are saying is simply substitute t 1 by t not forgetting as a factor of 2 pi there. So, in fact without rewriting all of that let me straight away do that job let me put before you that triple integral again. Now, when we take this integral on t 1 and when we note that this part has been dispensed off these three terms have been dispensed off with an impulse located at t minus t 1 what we need to do is to replace all these t 1 these two t 1 by t's and the impulse goes away. So, I shall now write down what is left that leaves only one integral now. So, that leaves only the integral on tau naught and what is that integral on tau naught x t times v t minus tau naught bar v t minus tau naught d tau naught with a factor of 2 pi. Now, that is easy after all x t is independent of tau naught. So, I can bring it outside the integral and that essentially leaves an integral of this quantity for me let me write that down explicitly. So, this becomes equal to 2 pi x t times integral minus 2 plus infinity mod v t minus tau naught the whole squared d tau naught and if you only care to make the replacement t minus tau naught is lambda where upon d lambda is minus d tau naught and as you note when tau naught goes from minus to plus infinity lambda would go from plus to minus infinity, but with a minus sign here if we absorb that minus sign and the order of the integration we can easily show that this is equal to 2 pi x t times integral from minus to plus infinity mod v lambda squared d lambda and this is a quantity entirely independent of t. Remember that when we made the substitution we did it for a fixed value of t and therefore, this process is justified. Essentially, this is the L 2 norm squared L 2 norm of v which is finite. So, we are in very good shape it is very clear that this reconstructs x t to within a constant. So, this is essentially equal to 2 pi norm in L 2 of v the whole squared times x t and this is the constant c 0 that we desired. So, our job is done that is very nice. We reconstructed x t from its short time Fourier transform. In fact, we did not have to work very hard. We did not have to do anything more than just taking each component multiplying by the unit vector in the direction of that component. Summing up conceptually I mean summing is replaced by integration here because tau naught and omega naught are continuous quantities spreading all over the real line. So, when we integrated overall tau naught and omega naught essentially added up all such component multiplied by vector we got back x t to within a constant easy. Now of course, we need to ask a few more questions and those questions are little more difficult to answer we shall not even try to do them all at once. You see here what you are doing is to take one tile with a fixed shape a tile which is constructed out of v with a spread given by the time variance of v in the horizontal direction and the frequency variance of v in the vertical direction. So, we are taking that tile and moving it continuously along the time frequency plane. And of course, as expected there is a tremendous amount of redundancy because we have a function in one variable t or time represented by a function in two variables tau naught and omega naught and we are using all possible tau naught and omega naught to reconstruct it. So, we should definitely be able to do that unless we have chosen our v very badly and as you can see all that you need is that ultimately v be a window function because otherwise you know it is spread in time and spread in frequency are not finite somewhere and that it be an L 2 r function very modest requirements. So, as you can see the short time Fourier transform was something good to start with when people started looking at time frequency or joint domain approaches, but it would be more interesting to ask can we discretize these parameters then it is useful otherwise this is by itself not so useful. We are asking for continuous tau naught and omega naught that is quite a demand. So, it would be useful to have discrete tau naught and omega naught can we take discrete steps that would be a real tiling in the true sense and the answer is yes. We can take discrete steps, but at this moment we shall not go into that issue safe to say that if you wish to reconstruct x t from discrete points on tau naught and discrete points on omega naught one can follow an approach similar to what we have just done namely put down a reconstruction based on multiplying the component by the so called vector in the direction of that component summing over all here it is literally summing because you have discrete tau naught and omega naught literally summing over all these discrete points tau naught and omega naught and then manipulating the behavior of v so that when we so sum we essentially get a constant multiple of x t. So, that would be the approach to we follow I shall not dwell further on this issue at the moment because we wish to go to the continuous wavelet transform now and before we go again to the reconstruction issue in the wavelet transform we need to answer one question that we had partly discussed yesterday what kind of tiling is possible with the continuous wavelet transform or what kind of tiling does it give you and let me try and bring that out little more clearly now. So, yesterday we saw that in the time frequency plane the short time Fourier transform essentially moves a tile of constant shape constant shape means its height and breadth are fixed as a function of tau naught and omega naught essentially you should visualize this tau naught and omega naught moving horizontally and vertically. So, it is this very tile just graphically to explains this very tile that moves like this in the continuous wavelet transform we have a slightly different kind of tile and we understand what we are doing in the C w t we shall use C w t to abbreviate the continuous Fourier transform. Now, here there is a tile of variable shape the area of the tile is fixed and the area of the tile is related to the uncertainty product or the time bandwidth product. So, abbreviate time bandwidth product by T b p the T b p of psi you know how to calculate the area of this based on the time bandwidth product of psi what happens as we change and of course, here we must interpret this tile as moving based on tau naught and s naught. Now, when one increases s naught one is essentially compressing or expanding when you know depending on if you increase s naught or decrease s naught if you increase s naught you are essentially expanding the function in time and therefore, compressing it in frequency now of course, s naught always takes on positive real values. So, s naught is indicative of frequency in a broad sense and in fact, we also noted taking the example of the Haar wavelet yesterday that when we make s naught large that means, we are essentially compressing on the frequency axis we are compressing both the band or the bandwidth and the center frequency. When we reduce s naught when s naught goes towards 0 notionally we are going towards higher frequencies. So, you know for example, is s naught is equal to 2 we have compressed the wavelet by a factor of 2 in the frequency axis, but expanded it on the time axis on the other hand if we put s naught equal to half. So, taking a value less than 1 then we have expanded the function psi in the frequency sense and compressed it in time. So, we have gone to higher frequencies in fact, with s naught equal to half we would have doubled the center frequency and the band this should be understood very firmly. So, in fact, let us make a note of this what we are saying is as follows increasing s naught means expanding in time or compressing in frequency and vice versa. So, I do not need to write that again decreasing s naught means contracting in time and expanding in frequency I do not need to write that down again. So, movement along so to speak the frequency axis happens because the center frequency is non-zero. The other observation that we have is that the center frequency and bandwidths are both divided by s 0. Needless to say tau naught has no effect on the magnitude of the Fourier transform. Tau naught is a parameter that controls only the time behavior it has nothing to do with frequency behavior. So, here also we have a separation tau naught controls movement on time or the translation parameter s naught controls movement on frequency, but in an indirect way and that is what we now need to understand. How can we reconstruct x t from its continuous wavelet transform allowing the entire range of tau naught from minus to plus infinity and the entire range of s naught from 0 to infinity.  So, when we wish to reconstruct x t from its continuous wavelet transform we shall need to make an allowance for some weighting differential weighting based on the parameter s naught. What I mean by that is the following you know now coming back to the tiling what we are saying is the following in the continuous wavelet transform larger s naught means smaller frequency and smaller s naught means larger frequency and of course, the distinguishing point is 1. So, when s naught is 1 there is no contraction or expansion when s naught is less than 1 there is an expansion in time and contraction in well when s naught is less than 1 there is an expansion in frequency and contraction in time when s naught is greater than 1 there is a contraction in frequency and expansion in time and in the time frequency plane what we are doing is effectively something like this. You see, for s naught greater than 1, we contract in frequency, but expand correspondingly in time proportionally keeping the area the same. And when s naught is less than 1, we have expanded in frequency, but contracted in time keeping the area the same. So, this is the kind of tiling that we have. And you must visualize this as happening continuously. You know what we are saying is, you should think of these tiles as either rubber tiles or elastic tiles. So, when they move along the time axis, there is no change of area. When they move along the s naught axis, then they start expanding vertically as they come down s naught. And they start contracting vertically as they go up s naught. And as they contract and expand in this manner, they reverse expand or contract in the tau naught direction, in the translation direction. So, in the time direction, I mean. So, of course, what I am saying here is, you know, you must interpret this figure. It is a little requires a little bit of thinking. What I mean by this figure here is that the horizontal is the time axis, the vertical is the frequency axis. And as we increase s naught, we are going towards lower frequency. You know, so there is a little trick here. I mean, maybe what I should do is to redraw. I have tried to overload this figure. This figure looks overloaded with more than one parameter. So, perhaps we should redraw it. We will draw it like this. We will say on the time frequency plane. So, if we take this to be the time frequency plane and not tau naught and s naught, this is time and this is frequency. Then at lower frequencies, we have broader functions in time. And at higher frequencies, we have narrower functions in time and correspondingly broader functions in frequency. This is the situation. And of course, what we are saying is, this is decreasing s naught and of course, tau naught follows the same pattern as time. So, tau naught and time align. As one moves along tau naught, one is moving along time. As one moves along s naught, for decreasing s naught, one is expanding or when one increases s naught, one is contracting. So, this is the whole situation. Now, with this little background, we see there is a bit of asymmetry. Tau naught is ok and therefore, we will allow tau naught to go as it is. We will take a Q from the short time Fourier transform. In the short time Fourier transform, we could just let tau naught go as it is in reconstruction. So, we would not worry about tau naught here either. But s naught poses a problem. So, we need to allow a weighting factor here when reconstructing using the same principle of component multiplied by vector and that weighting factor must depend upon s naught here. So, in other words, what we are saying is that the reconstruction of x t from C w t of x with respect to psi evaluated at tau naught and s naught should be and we have again a triple integral, but with a difference. So, before we write the triple integral, we will first just write down this C w t x psi tau naught s naught multiply this by the function the unit vector. So, to speak, but as I said with a weighting factor. So, the unit vector or the vector in that direction is psi t minus tau naught by s naught and we agreed that we must normalize this.  We have agreed that we do not need to do anything about tau naught, but we do need to use a weighting factor for s naught which is f of s naught a weight and this is integrated with respect to tau naught and s naught. Now d tau naught is inside tau naught would go from minus to plus infinity, but s naught would go from 0 to infinity there is a little bit of a difference and we need to choose this f of s naught in such a way that we can get reconstruct that is what I am trying to say. A little difficult to understand right in the initial phase, but once we go through the working you will see what I mean. Now what we shall do is instead of trying to use the time domain expressions here, we will take recourse to the passables theorem. You will recall that we had written down the passables theorem for this yesterday and we shall therefore, express C w t using the passables theorem in the frequency domain. So, the C w t using passables theorem is essentially x cap omega psi cap s naught omega complex conjugate multiplied by the square root of s naught and then multiplied by e raised to the power j omega tau naught integrated over all omega. This is what the C w t was and we had also interpreted this yesterday. We had given it a very interesting interpretation based on a filtering operation. We shall be using that interpretation later, but for the moment let us substitute this in the expression of reconstruction. So, substitute in reconstruction and when we do that let me just put before you the reconstruction integral once again for convenience. I am going to substitute this by that frequency domain expression. Let me do that. So, I have now again we get you know one shouldn't as I said be frightened of a triple integral because much of it will collapse very soon, but we do get a triple integral and we will write it down bravely. So, we have a 1 by 2 pi there x cap omega. So, I think we will have to it is a big quantity. We will have to continue this. So, we have all these terms here and then we also have and I do need to write them down separately and we have a triple integral the omega first then the tau naught the s naught looks formidable, but as we will see in a minute is not at all difficult to evaluate if we start collapsing the integrals 1 by 1 as we did in the short time Fourier transform. Let us bring out a strategy for this collapse. So, if we put flash the whole integral triple integral before you just for a minute. This is the triple integral an integral on s naught then the outermost is on s naught the next on tau naught the next on omega x cap omega times all this followed by this and these elements of integration. Now, let us observe this carefully you know as we did in the short time Fourier transform it will of course, be useful to change the order of integration because there is a collapse of terms based on the fact that only a few terms depend on specific parameters. Now, for example, if you look carefully at tau naught after all what we see is that it is only this term this term and this term that depends on tau naught here. So, in fact well actually there is a little correction I do not need this e raised to power I have repeated this term here. So, already written so, I do not need to write this twice. So, the tau naught part is the easiest to deal with it is only this and this that depend on tau naught. So, let us take care of that first. Let us first dispense with the integral on tau naught and therefore, let us identify the parameters that or the functions that depend on tau naught and those are essentially the following. So, if we identify them they essentially this and this that is all and this is a very familiar quantity here after all this looks very much like a Fourier transform. In fact, if you simply care to put t minus tau naught by s naught equal to lambda as is the standard trick you will indeed get exactly that. So, then t becomes well you have t minus tau naught is s naught times lambda and therefore, tau naught taking it to the other side is t minus s naught lambda and therefore, d tau naught noting that t is a constant for fixed t is minus s naught d lambda and we can substitute that in this expression here. So, we have this becomes equal to and once again you know if you look at the limits when tau naught goes from minus to plus infinity lambda goes from plus to minus infinity, but with that change of sign again we can rewrite this in the following way. So, I am just skipping those steps psi of lambda e raise the power j omega times t minus s naught lambda minus s naught becomes plus s naught d lambda and then we need only to isolate this part. So, this can be rewritten as e raise the power j omega t coming out s naught coming out and then we have minus to plus infinity psi lambda e raise the power minus j s naught omega times lambda d lambda and this is very familiar here. This is essentially the Fourier transform of psi evaluated at s naught lambda instead of just s naught omega instead of just omega. So, we have made a change of this variable here s naught omega and not just omega and therefore, we can now collapse the integral on tau naught entirely. Let me write that down this is essentially e raise the power j omega t s naught times psi cap evaluated at s naught omega and therefore, we can now collapse the d tau naught part. Let me put that down for you once again for reference I agree that this is a little bit of work, but it is not so difficult after all let me put that down before you once again for reference. So, what I am saying is this is all now been collapsed this and the d tau naught part here this has been collapsed. So, what is now left is the d s naught part and the d omega part. Now, let us look at the d s naught part in fact, let us write down what is left for convenience at the moment what is left is as follows the integral on s naught and the integral on omega the 1 by 2 pi we keep as it is x cap omega psi cap s naught omega bar you will note that this square root of s naught and the square root of s naught in the denominator cans. So, this goes away. So, I remove them the rest has been collapsed you have an f of s naught and then you have this part. So, I will continue that this is what is left we need to flash it once again you have an integral on s naught outside an integral on omega inside x cap omega psi cap s naught omega complex conjugate a weighting function in s naught and then in s naught the power j omega t s naught psi cap s naught omega integrated on omega and then on s naught and now we will use the standard trick again. We shall note that this this and this depend on s naught if somehow we can do away with the integral on s naught. So, well let us isolate the integral on s naught first let us now dispense.  Now, you know if you look at it carefully of course, this part is easy to deal with its mod psi cap s naught omega the whole squared s naught f s naught d s naught. Now, we need to interpret what we are trying to do here you know if we could make this independent of omega the whole objective here should be to make this independent of omega I will explain why if you look at what was left we had this term dependent on omega left this we have taken in the s naught integral this we have taken in the s naught integral this is of course, a term dependent omega we cannot do much about this we have to keep it and we want it anyway all this has been taken in the s naught integral. If all this this this and these two terms can be made independent of omega what will be left essentially a constant in omega what will be left is x cap omega times e raise the power j omega t d omega which is a very familiar quantity it is essentially the inverse Fourier transform of the Fourier transform of x. So, we are getting where we know where our destination is and we know how to get there we shall get there essentially by making this independent. So, we must make this independent make this omega independent and that is where the role of this weight function is we will employ s naught f s naught d s naught you know this part can be nicely configured to make this integral independent of omega and to do that we use the following lemma. If we could make s naught f s naught d s naught essentially equivalent to d s naught by s naught in other words f s naught is equal to 1 by s naught squared and we note that this is a valid weight function it is strictly positive you know f of s naught being 1 by s naught squared is a is a valid weight function why do we want to weight function to correct the so called magnitudes when we put the components together. And when we put a correction factor for the magnitudes that correction factor must be non negative for all values of the parameters in the components. Now, here we have a valid weight function a weight function which is non negative for all s naught.  And now if we take say s 1 equal to 1 by s naught equal to omega times s naught then d s 1 is omega d s naught and now we can divide here. So, d s 1 by s 1 is d s naught by s naught of course, assuming omega is not equal to 0. The only catch is that when omega is positive the limits are 0 to plus infinity when omega is negative the limits are 0 to minus infinity. And if we take care of that little detail we have done away with the dependence on omega naught. So, what we are saying in effect is the triple integral finally collapses to some constant multiplied by the following. So, this whole thing multiplied by some constant and this constant emerges from integral psi cap s naught omega d s naught by s naught from 0 to infinity. Which boils down mod squared here of course, which boils down to essentially mod psi cap s 1 squared d s 1 by s 1 when omega is greater than 0 and integral from 0 to minus infinity psi cap s 1's mod whole squared d s 1 by s 1 when omega is greater than 0. So, omega is less than 0. Now, we have narrowed down the evaluation of this integral to essentially an inverse Fourier transform of the Fourier transform of x, but with this condition these two integrals over s 1 need to be finite and in fact equal. And this is where the concept of admissibility of psi would come from. We shall continue discussing this in the next lecture to complete the idea of admissibility and to interpret it. What do we mean by imposing this condition that these integrals must be finite? As I said we shall proceed from this point in the next lecture. Thank you.


Okay, welcome to theCUBE. We are live here in San Francisco, California, for VMworld for now three and a half days of live coverage. We had three days. Jeff, we added them today because one, we're still here with the setup and we're going to, you know, extra mile and talk to some great guests. We've got the Vanessa Alvarez going to come on shortly but really this is a wrap up of VMworld. Great show. Everyone's kind of winding down. All the booths are gone. It's just the last minute, last keynotes, last sessions. I'm John Furrier, the founder. I'm looking at him. I'm joined by co-host Jeff Frick because Dave Vellante flew back. He's back in Boston and he's been ready for his next trip. Jeff, great to see you again. You too. Yeah, I talked to Dave this morning. We actually had a meeting. So, you know, kudos to you and Dave for going strong for three days. Plus we also had the bonus event with the NetApp coverage Tuesday night at AT&T Park. And I know the official VMworld party was last night at AT&T Park. Part of the thing that goes on at these conferences obviously is the sessions but also there's all the vibe afterwards and there's been a lot of events. So give us a little update on how that went last night at AT&T. Party was great. None of the parties that these shows, you know, EMC, World Oracle, Open World, it involves music, a band of some sort.  But they held it at AT&T Park where we had the NetApp VIP event with a cube there. And they had the entire field turned into a carnival live stage. And it was just a great event. I mean, probably the best ever for VMworld, no doubt. People were happy. They were smiling, dancing on the field, their shoes off. I felt like I was in Golden Gate Park and, you know, and everyone's like chilling out. It was really a great event. Carl Aschenbach kicked it off with great speech. VMworld's growing, their business is growing. And obviously it's the 10th anniversary so the party was extra special. And I had a chance also to hang with Carl and the former CMO, Rick Jackson and all the senior execs in the private area yesterday got to see them at a personal level and it just reaffirmed some of the things we've already known, which is great culture at VM where great people, and again, their customers and their partners were all happy last night. Really the best party I've been to for VMworld, certainly. And it rivals up there with any of the events I've been into at these conferences. That's great. Yeah, and the guest list we've had has just been phenomenal. So I look forward to going back as the coaches say and looking at the tape. Pat Gelsinger yesterday morning kicked it off at 9 a.m. I think we're going to have that tape up online so you can watch it. But even though the theme here has been a lot about 10 years of VMware and how things have changed so much there's something like 23,000 people running around here in all the Moscone centers and all over the streets of San Francisco hotels. But Pat said really, or not Pat, but Peter said it's really a renaissance today. So it's almost like a new beginning and really the infrastructure and the pieces of virtualization as they move from simple computing to networking and storage is really enabling, I think as Pat said, to let businesses move at the speed of business and not at the speed of infrastructure. Yeah, I mean to me, it's really what highlights some of the key points here that doesn't get reported is the little positionings, the little gestures, the positioning of the vendors and some of the conversations and I took a break from theCUBE yesterday afternoon to go down to an event Intel had near the ballpark, AT&T Park. Jerry Rice was there. They were talking about big data and fantasy football and really what's happening is a new user experience is developing Jeff and I think if you look at what VMware is doing here it's about being positioned to have the engine, the engine of the car, if you will, or the main technology to power, essentially new user experiences. So the Intel event with Jerry Rice was talking about how fans are changing their experience based on data. Fantasy football was kind of like the highlight and it's really, it's about new experiences and I think to me, I've always said in theCUBE is it's about new expectations users have and IT and the technology leaders haven't always thought about that. So we're in a new modern infrastructure, new modern time where a new user expectation is going to evolve and new user experiences. The companies that can enable that and provide that will be winners. Certainly we talked to Sanjay Poonan here who's heading up the end user experience. He came from SAP which spent a lot of time on enterprise collaboration, but also in the iPad hit, SAP rolled out some really elegant and nice mobile solutions. So I think that's a telegraph from Pat Gelsinger and Sanjay's formal role that you're going to see VMware really enabling kick butt mobile experiences, not as a device but as an enabling technology and that's where virtualization will play well. Yet another part of the scene that's going on that not everybody can see overtly is all the analysts presentations that I've been able to sit in on a couple of those, John. And quite a few of them said, and as Pat, another great quote, we're in the age of APIs not UIs and there were a number of companies that we met with in the analyst briefings. I said, you know, a couple of years ago they couldn't do what they're doing today. Leveraging this infrastructure and leveraging the APIs of lots of different applications in new and innovative ways and then bringing the horsepower to provide business insight and enable people to use technology to do things that they could never do before. Yeah, and it's really about, you know, really being on the track with the right technology. And I was just reading an article, I just got a note from Beth Parasu from Tech Target who's a great reporter. She just reported that VMware is splitting vCloud director into vCenter and vCloud automation. And that's really an indication, a small little piece of news, but very relevant because you look at the orchestration and some of the things that make those end user experiences happen is about putting the right technology in the right group. So that's a significant announcement because management software and the management and the automation is the critical differentiator for a lot of these vendors because if infrastructure as a service gets commoditized, if there's a thousands of new servers out there in a new way and you've got open stack evolving, management is the key to success. And we've been hearing that throughout the show that management and moving up the stack has made critical. So again, VMware is doing extremely well. The growth of the conference has been fantastic. The people there are all jazzed up, they're all lined up, they're excited, they have very simple metrics, software defined data center, hybrid cloud and end user computing. So the vibe is very positive, customers are happy, the employees are focused. And I think that is the thing that I walk away with. Yeah. And two, I just want to take a moment, you know, VMware gave us a great setup here. If you're still at the show, come by, we're in Moscone South at the west end of the lobby. But I want to give a quick shout out, John, to some of our sponsors who enable us to come to these type of events and provide the coverage that we can. And obviously VMware for really helping us out, Newtonix, Fusion IO, a long time friend of the cube, Barracuda Networks, SimpliVity, EMC. Obviously we cover EMC's world and I'd love to cover their events. Commvault, IO data centers, a new sponsor for the keep talking a little bit about kind of the next generation, because as you said in your interview with them, there has to be something somewhere at the end of the line, right? There are actually boxes in power. Viridant, Juniper Networks, HP, Service Mesh, and Rackspace, again, without the generous underwriting support of our sponsors, we couldn't take our show out on the road. And again, if you're in the neighborhood, come by, you can see it's a nice production that we tried to put on here at the cube. And I think that's the key to the success of the cube. We want to go out and get that signal from the noise and report. And I don't mind doing 13 hours of straight interviews like we did Tuesday.  And we've got some more news. Vanessa Alvarez is going to be up next and she's awesome. She's now at a company, but she was a long time analyst. She's one of the early Clouderati analysts. She's been there from day one on the cloud explosion. Knows everyone. So we're going to get her perspective. It's just on a personal level, what she's doing now. So this is just a great show. Without the generous support of our underwriters, we can't bring the cube and bring our perspective here and make people uncomfortable. Even Pat Gelsinger was a little bit uncomfortable when I, you know, I didn't know he's going to react that well to some of my comments, but it shows Pat's got the fight in him and he is believing in his vision. So it's really exciting. Again, Gelsinger etched about all the top executives were here inside the cube, as well as customers, entrepreneurs. And you know what I'm excited about? The entrepreneurs, right? We had Peter Levine, who's a big-time venture capitalist from Andreessen Horowitz. Jerry Chen is a big-time venture capitalist now with great luck. He was at EM's VMware for many, many years. And we had Steve Herodon, who was the CTO of VMware now at VC. So you're seeing the VMware mafia, the V mafia is being called out there. Early employees out making investments, starting new companies. So VMware, I predict, will have an impactful market in Silicon Valley in the tech industry. And if this continues to track the way it is, they're going to be a very, very large growing company. Yeah, and as you said, I think in a couple of interviews, it's all about the ecosystem. So not only are there formal ecosystems with partners and technology partners that they foster in the company, but as people leave the company and go out and do their own ventures with that background and that connection, I think you'll see it, I've seen it before, the other Silicon Valley success stories like PayPal, et cetera. So it is about the people and that's really why we like to cover the tech athletes because there's technology at the end of the wire, but it's people that put it in play. Let me ask you a question. You've obviously been here in theCUBE, you've been watching all the interviews and reading the guests and also going out and scouring, looking for stories and putting your ear to the ground. What's your perspective? What did you learn and what was your observation? Shared the crowd because you were behind the scenes, doing a lot of hands shaking, meeting people, talking to people. What's your perspective? To me, I think it's the juxtaposition between it's been here for 10 years and we started at this really simple way of trying to virtualize things. I remember back the early ASP days, the application service providers where everyone was afraid of the internet and shared infrastructure and this and that and now it's completely morphed and I think it was Levine that talked about going completely the opposite way with these itty bitty pieces of compute power, networking power and some of the briefings we were in talking about, this real move of virtualization from the very simple compute into networking into storage and really it being kind of the start of another chapter. So it's not closed at 10 years. It really feels like this was just an enabling step and now the momentum is really going. Yeah, I mean, one of the things I'd like to share with the folks out there from my observations, again, getting into under the covers a little bit is that a couple points of that I'm excited about. One, VMware is transformed and positioned for the new generation and there's a lot of bets that they're making and VMware is smart. They will make the adjustments if their bets are a little bit off and we might see that with hybrid cloud or however that morphs but I think that's a great track. Two, the talent and the technology guys that VMware has, they maintain that cultural tech mojo. VMware is known to have tech geeks and real guys who care about technology. So it's like having a good mechanic work on your car. You want guys building good technology. So what you're afraid of when you see companies change with either new leadership or market shifts is they lose their soul, right? I mean, and VMware has not lost their soul yet. They have that tech mojo and people complain about some things that they do but at the end of the day, they have that leadership and then at the executive level, they're focused on the business. I mean, for the first time I've seen VMware, really flexing their management muscles saying, hey, you know what, we're serious. We want to grow and build an economic engine. Carl is straight up, straight on his metrics. They have good sales, go to market focus and again, they'll make the adjustments. So that was one point. Two, the technology people that are here understand that virtualization is going to change and enable and that was really exciting and you saw that through a lot of the other interviews. So those are the two things that I say that were really, really highlight. And again, the ecosystem, I think that's going to be a conversation. I mean- What was the biggest surprise, John? You've been coming to these things forever. I mean, what's the thing that, was there anything that just kind of blew your doors that you did not expect at all? I think what I was expecting was more of the same as the VMware, hey, we're tech geeks and I think what I was really blown away by is the growth focus of the management and they always done that before but here it's like, they see specifically a real big prize at the end of this and I think Pat's focus of execution and he knows technology and you got Sanjay Pune and he knows tech, Carl knows tech, I mean these guys are tech guys so Intel had that same concept. If you were in marketing at Intel, you were an engineer. You didn't just get hired out of business school as a marketing degree.  So again, VMware is, that was the biggest surprise. On the interview side, I was really surprised on the Andreessen Horowitz really aggressively going after Pat on the vision and Mark kind of, you know, needling him a little bit on the whole vision of the compute side and I think that's an open book. The management, the systems management and what the compute will look like. I think it's still an open book and those bets are going to be one or lost. So to me, that's a simple answer. You're done on one side of the street or the other and VMware is not on the right side of the street. It's going to hurt their growth. Yeah, well, and we've talked about many times with a lot of these larger tech companies that are continuing to try to maintain growth rates like a smaller startup or unnecessarily a smarter, but a smaller company. How do you maintain that excitement? How do you maintain the growth on such a big and growing base? And I think the other kind of insightful thing that Pat said was, you know, we still have a long way to go in mobile, right? There's a lot of companies out there now that are mobile first that are developing the mobile applications. You know, I think it's obviously the extension of the, of the bring your own device, but they do, they do have the excitement. They do say, you know, we're tech people that are excited about tech. I think you had a car analogy in one of the interviews talking about if you want mechanics that are excited about cars, right? You want car people building cars and making cars because they like cars and there really seems to be a positive mojo where people are excited about the opportunities that they could do things before that they couldn't do before.  They don't really like that. I don't like that word, but they have a very passionate fan base and their customers and their partners. I mean, people, I mean, because they're tech guys and they see virtualizations like magic really has changed the game and technology and it's still going to be that disruptive enabler.  I think that virtualizations is our generation's enabler and I think the wealth creation and the opportunities are going to be fantastic. So I think VMware smells that and they see it and they see cloud as the next data center and I think the hybrid is the right approach. Again, I still think it's still not a done deal. Hence my comment about being a way station or halfway house, I still think IT still needs to get muscled up and every CIO that I talk to always talks privately around yeah, the past 15 years has been consolidation, consolidation, consolidation. I haven't been working out, I have no muscle and now they're asked to essentially move mountains. So they need to have, you know, build up their strength and IT and technology and execute. So it's just not that easy. So that's kind of like what I'm seeing is the big trend. Yeah, but it still seems to be a lot of horses for horses with the hybrid approach and using the different types of flavors of cloud if you will based on the workload and whatever the logic is around the business process. So let's get back to it. We got a few more guests lined up today. We're excited. We're going to go to VMworld 2013. We're going to go finish the last wall, right? Our wall to wall to wall. We'll be right back with our first guest after the short break.


So you can get examples any time you have any kind of force and indeed every time you have a force You can do an example of Newton's third law. Let's start with if you're lifting a book So if you lift a book, there's a force by you to push up the book and there's a force by the book pushing down on your hand It's a handbook interaction or you might have a plane if you have a plane's propeller There you have the plane pushing on the air the propeller pushes the air and also the air pushes the plane That's how the plane goes forward It's a propeller air interaction. You might have a swimmer in water The reason the swimmer pushes on the water is in order for the reaction the water pushing on the person to push them forward It's a person-water interaction or you might have a cricketer and if the cricketer hits a cricket ball Then the ball experience is a big force, but the bat experiences an equal and opposite force And that's why she has to hold it very firmly and also hit the ball right in the middle So that doesn't twist in her hands due to the bat ball interaction So in each of these cases all you have to do is figure out what two things are interacting and that tells you how to Apply Newton's third law and the force between the two things is going to be equal and opposite


T.F.N.N. The Tiger. Financial News Network. Update. Everyone, Basel Chapman, on this Thursday, the 10th of February, we're looking at our wild opening, the futures. I couldn't understand it all night. The futures were acting really well in this morning. They were up. The Dow was up almost 90 points in the futures. The S&P was up. And then, of course, you got the news about the inflationary aspect and Kibum CPI number comes out hot, and we smashed to the downside. The Dow trading with a high yesterday at the $35,000, oops, I forgot to write that in, $35,824 level. This morning we went all the way down to this trend line support right there to $35,479. And then, by the dips, we're down only 88 points now, $35,676. If you see tears in my eyes, that's because we had an add-to in the Dow long position. I wanted to get the diamonds and we missed it by 13 cents. Well, the day's young, but I don't think we're going back there. So yeah, we got S&P. S&P is down 21 at $45,656. Trying to pierce the Chamberlain Insight Track repellent zone. If by tomorrow or Friday we're actually able to trade above $45,95,41, that would be very positive. And we've got to see, but it's got to hold, I would even say even 20 points down from here would be a problem. So the $45,50 area. We're looking at the QQQ. I'm quite a bit weaker, but it's now down 260. It was down much more with G64. It actually went down to the $359 area. So this is a good comeback to the 200-period moving average. The IWM is really down just 10 cents at $206.51. This is the one that's starting to show some better relative strength to the others. I'm kind of liking the IWM for now. It needs to get to the 209 area to say, hey, I'm back in the running. Let's go to gold. Gold was very weak, and now let's come back a little bit, 30 down, 3. But it's the dollar that's acting so well. Look at the dollar. The dollar is up 38 ticks at $95.93. We were watching Trudeau very closely. Trudeau is still finding buyers. It's up a dollar at $90.72. The real story here is, and we talked about it in my show coming up. Here's the TLT. Bonds are down 137.54. Down 89. Those rates are going higher. I'll be back in a moment. The dollar is now only down 21. We'll be right back.


Yet yet another Sort of I'll call it a cool stuff found Redox, but which we've had now this will be the third one I believe this week, but yeah, I mean, it's been three to ten years since we've mentioned some of these So it they are well-due for a remention so Yes, so the the first one Did you know that your Mac is doing all sorts of things in the background to protect you from militia software But you may not even know that's happening. Well, here's how you can know what's happening It's an app called silent night from our friends at eclectic light And it basically gives you the The low-down it will show you versions of various security things It'll tell you with system integrity protection and what is SSV? I don't even know what that is X protect, which is there, you know show you the version that and actually And the versions of some other things, but what's really nice the version of gatekeeper What's really nice is it'll show you if a database is not up to date and this actually happened on my system It was like, oh dude your X protect databases is old you want to update it and it's like, okay That's cool Yeah, so something you want to run every now and then just to make sure that all your security stuff is The way it should be I now remember We looked before the show to see how recently we had mentioned this and it had been three years I now remember exactly when I found out about this John and you were there and in fact I caused you and me and and a bunch of other people great headache There's nothing wrong with this app. I highly recommend using this app Don't do what I do what I did though. We were at max stock in person in I believe 2019 and Someone on stage I'm gonna guess it was Brent terpstra, but who knows I don't know who it was Someone either on stage or in the room mentioned silent night. I'm like, this sounds amazing So I ran it and it said updates available and I clicked install all updates Because I wanted to be up to date, you know Uh and it said well for one of these updates you need to restart your Mac. I'm like, I don't want to restart right now That's fine, but whatever later that day you and I had a live in-person Mac geek up to record and We Through the process there was something wrong with the USB interface that didn't want to work and so I'm like Well, let me just restart my Mac and so I restarted my Mac For the first time since clicking the install all updates button earlier that day in silent night. Oh, oh 20 25 minutes of updating macOS happened with a room full of people waiting to see a podcast About how to how to properly manage There was so much irony it was dripping from the ceiling folks. Yeah, so You know, it was fine. It was one of those things where it humanized the whole thing which we love to do here I mean we have problems with our computers just like you in fact It's the fact that we try new stuff all the time and break things that allows us to be able to do what we do For all of you like it. This is part of how it works And and you've got to see the sausage being made in a in a fantastic way, but yeah So that's some that's when I learned about Silent Night John


Hi and welcome to the session I am Shashi and I am going to help you to solve the following question. Question is using elementary transformations find the inverse of each of the matrices if it exists. The given matrices 2 minus 3, 3, 2, 2, 3, 3 minus 2, 2. Let us find the solution now. Let A is equal to matrix 2 minus 3, 3, 2, 2, 3, 3 minus 2, 2. Now to find the inverse by elementary row transformation we will write A is equal to I where I is the probability matrix or we can write matrix 2 minus 3, 3, 3, 3 minus 2, 2 is equal to matrix 1, 0, 0, 0, 1, 0, 0, 0, 1 multiplied by A. Now to make this element equal to 1 we will apply on R1 row operation 1 upon 2 R1. So we can write applying R1 row operation 1 upon 2 R1 we get matrix 1 minus 3 upon 2, 3, 2, 2, 3, 3 minus 2, 2 is equal to matrix 1 upon 2, 0, 0, 0, 1, 0, 0, 0, 1 multiplied by A. Now we know diagonal elements of the identity matrix are 1 and all other elements are equal to 0. So we have to make these elements equal to 1 and all other elements should be equal to 0. Now to make this element equal to 0 we will apply on R2 row operation R2 minus 2 R1 and to make this element equal to 0 we will apply on R3 row operation R3 minus 3 R1. So we can write applying row operation R2 minus 2 R1 and R3 row operation R3 minus 3 R1 we get matrix 1 minus 3 upon 2, 3 upon 2, 0, 5, 0, 0, 5 upon 2 minus 5 upon 2 is equal to matrix 1 upon 2, 0, 0, minus 1, 1, 0, minus 3 upon 2, 0, 1 multiplied by A. Now to make this diagonal element equal to 1 we will apply on R2 row operation 1 upon 5 R2. So we can write applying on R2 row operation 1 upon 5 R2 we get matrix 1 minus 3 upon 2, 0, 1, 0, 0, 5 upon 2 minus 5 upon 2 is equal to matrix 1 upon 2, 0, 0, minus 1 upon 5, 1 upon 2, 0, minus 3 upon 2, 0, 1 multiplied by A. To make this element equal to 0 we will apply on R1 row operation R1 plus 3 upon 2 R2 and to make this element equal to 0 we will apply on R3 row operation R3 minus 5 upon 2 R2. So we can write applying R1 row operation R1 plus 3 upon 2 R2 and R3 row operation R3 minus 5 upon 2 R2 we get matrix 1, 0, 3 upon 2, 0, 1, 0, 0, 0, 0, minus 5 upon 2 is equal to matrix 1 upon 5, 3 upon 10, 0, minus 1 upon 5, 1 upon 5, 0, minus 1, minus 1 upon 2, 1 multiplied by A. Always remember that row operations will take place simultaneously on the matrix A on the left hand side and the matrix I on the right hand side till we obtain the identity matrix on the left hand side. Now, in this diagonal element equal to 1 we will apply on R3 row operation minus 2 upon 5 R3. So we can write applying R3 row operation minus 2 upon 5 R3. So we get matrix 1, 0, 3 upon 2, 0, 1, 0, 0, 0, 1 is equal to matrix 1 upon 5, 3 upon 10, 0, minus 1 upon 5, 1 upon 5, 0, 2 upon 5, 2 upon 5, 1 upon 5, minus 2 upon 5. But we write by A. Now to make this element equal to 0 we will apply on R1 row operation R1 minus 3 upon 2 R3. So we can write time R1 minus 3 upon 2 R3, applying on R1 row operation R1 minus 3 upon 2 R3. We get the matrix 1, 0, 0, 0, 1, 0, 0, 0, 1 is equal to matrix 1 minus 2 upon 5, 0, 3 upon 5, minus 1 upon 5, 1 upon 5, 0, 2 upon 5, 1 upon 5, minus 2 upon 5 multiplied by A. Clearly this is the identity matrix of the order 3 into 3. So we know identity matrix is equal to A inverse multiplied by A. Now comparing the two expressions we get A inverse is equal to this matrix.  So the required inverse is given by the matrix minus 2 upon 5, 0, 3 upon 5, minus 1 upon 5, 1 upon 5, 0, 2 upon 5, 1 upon 5, minus 2 upon 5. This is our required answer. Hope you understood the session. Take care and goodbye.