Simple question. What must parents do with baby toddlers and young kids? Just think for a second so we can crowdsource the answer and you can tweet it to CCTN or you can talk to us later. Crowdsourcing is what Mobius Noodles is all about. If you ask children what they want to be, they will tell you something like an astronaut, a ballerina, a Jedi, because they want beautiful adventures, something to role-play now. What you want here is a mathematician. Now, when we grow up, we want to change this. Now, we believe young kids already are mathematicians. They certainly have more imagination than it takes to do differential calculus. And they can be all too literate, just like logicians and set theorists. And they love beautiful adventures in math wonderlands, if you help them travel down the right rabbit hole. Any parent we ever ask wants their kids education to be rich, wholesome, and beautiful. But if you grab random early math materials, well, they will be just the opposite. Counting shapes, basic patterns. This is really math. There are two analogies for the quiet tragedy of early young math. Just imagine, what if you never read anything but alphabet to your kids? What if you never play music or sing, just show the kids music scores? Well, it would be sad and frustrating for everyone, just like math lesson. Our dream and our goal is to build a hub where everybody shares campus school and kid-friendly math. We ran a successful crowdfunding campaign last year, and we were amazed how many parents offered not only money, but math ideas for our project. This year we worked on the first Mobius Noodle's book. It's almost finished. It is definitely a large group effort. Our thank you list has hundreds of people. And we are now in the production stage, final tweaks to the cover, copy editing, and so on. Some of our games are classic, such as Function Machines you may have played as kids. We added our own funky twists to it. Others are original, such as Yes, My Green. And they are based on many years of research and development with young kids. We have done before even starting Mobius Noodle's. Because Mobius Noodle's is a creative commons project, people are happy to share content through us. Games, photos, videos, stories, a lot of creativity and do-it-yourself spirit. Our kind of young master's inspirational and even therapeutic for grown-ups. And here is something that really warms our hearts. After talking to us, several colleagues made their books creative commons, too. And I hope you can join us in our adventures as well. So head to our site or talk to us to learn how to participate. Thank you. Thank you.


presents Danny Thomas and Anne Francis. Hollywood, the Mutual Network in Cooperation with Family Theater, presents Early Bird starring Danny Thomas. And now, here is your hostess, Anne Francis. Thank you, Tony LaFranco. Family Theater's only purpose is to bring to everyone's attention a practice that must become an important part of our lives if we are to win peace for ourselves, peace for our families, and peace for the world. Family Theater urges you to pray, pray together as a family. And now to our transcribed drama, Early Bird, starring Danny Thomas as Arnold. A-G-D-E-B. Let's try this. How's it coming? It is, and I think I'm burned out. Right out? Oh, sure. Twenty years you've been writing top songs every two weeks, you're burnt out. Well, it's going to happen eventually. Don't talk silly. I wasn't asking about that. How's the speech coming? Speech. Speech. Tomorrow afternoon at the college commencement exercises. Oh, my gosh. What's the matter? I forgot. You haven't even started it? I forgot all about the thing. Oh, it'll get weirded. It's tomorrow. Never make it. What? You gotta get me off the hook. You're the principal speaker. Your name's all over the program. I'll get sick. That's it. I'll get sick. You can't get sick. You gotta give the speech. I mean, I'll say I'm sick. No. Look, as a matter of fact, I am sick. I'm overworked. Well, baby, if I'm your personal manager, you gotta let me manage. You can't duck this thing. Everybody's coming. Your family, your friends. It's too late. I can't write a speech anyway. Arnold, you'll disgrace yourself. Look, I never went to college. I got nothing to say on the subject. They don't want you to talk about college. They want you to talk about yourself. Myself?  That's what they want. What? The secret. How did you do it? Are you kidding? Last year, they had this guy, won a Pulitzer Prize. He gets up, he tells him what a great man he is. I bet he went to college. In Jimbo's basement. Go on. I'm telling you, he doesn't even use commerce. How I did it. Yeah. How you got to the top? I can't tell him. That put me in a straight jack. Well, change it around a little. Give me old formula, hard work, stick to whatever it is, you know? Look, they throw rocks at me. All right. So don't tell him what really happened. Make it up. Yeah, yeah, make it up. I can't make up stuff like that. Sure you can. Just twist it around a little. Tell him you were a lifeguard instead of a milkman. I'll never get it done in time. The sooner that you start, the sooner you're through. And I'll send over one of the girls from the office tomorrow morning and I'll type it up. No more excuses. Sit down and do it. I don't even know how to start. Start at the beginning, only like I told you. Change it around. See you tomorrow. Change it around. It's easy to say. Tell him you were a lifeguard instead of a milkman. I was a milkman. So what's wrong with being a milkman? Twenty years ago, a kid like me was lucky to have a job at all, much less be a milkman. Milkman, it was a big deal. I liked it, too. Except for getting up so early. It was a trouble. So early, you never saw anybody but your horse. Of course, down at the dairy before I started out, I'd see some of the other drivers. Real nice guys, too. Except for Whitey. That one. Whitey Benson. Every morning he was on me like an overcoat with his big mouth. Well, if it ain't Johan Sebastian.  Written any operas lately? Nothing you'd be interested in. Say, I hear the dispatcher had a complaint from some people on your delivery route about all that off-key singing you do so early in the morning. What are you talking about? You claim you're sour in the milk. See, Whitey, with a sense of humor like you got, I don't know why you don't get on the radio. The reason I don't get on the radio is because it's more comfortable on the couch. I don't get it. On the radio, you know, like sitting on the radio. Yeah. Only I'm more comfortable sitting on the couch. Yeah. Well, that's it. That's the whole thing? Yeah, it's a joke. Gee, it's a riot. With a sense of humor like you got, I don't know why you don't get on the radio. Hey, what are you doing? Give me a bad time here. Well, me?  No, I really think you ought to be on the radio. Oh. Yeah? Yeah, instead of on the couch. I'm warning you, don't be a wise guy, see? Whitey, put down the milk bottle. You mess with me, you're going to be sorry. You mess with me, I'll get my horse to kick you. Yeah? Just don't be so funny. I won't if you won't. Okay, okay. And I know you won't. No? Not with those jokes, kid. Get up, Louise. Whitey Benson. Always give me the needle, because I'm trying to write songs. That's all right, let him needle. I'll show him.  Then he'll needle with my name all over the papers. When he finally gets nerve enough to come around, I'll tell him I'm taking my milk from another company. Boy, that'll be a good one. I'll show him one of these days. I'll be the greatest composer-conductor in the whole world. Gee, I can just hear it now. Ladies and gentlemen, we bring you the Crouse Milk and Philharmonic. And here is your conductor, Arnold Castor. For my first number, I'd like to play a composition of my own. And now I raise my baton and... Oh! Good grief, I didn't see that bottle there. Must have been the one Whitey took out, that big ape. And I'll get docked for it. Well, I'm not gonna pay for it. When I get back to the dairy, I'll tell him what's what. And if he doesn't cough up the twelve cents, I'll sock him in the nose, right in the nose. Just like that. Say Whitey. What do you want, Gershwin? Well, a little accident this morning. You know the bottle of milk you're fooling around with? What about it? Well, you didn't put it back in a case in it. Well, it got knocked down and broke. That's too bad, kid. Well, since it was partly your fault, and I'm gonna get docked for it, I figured you'd be in the right guy. You'd want to do the right thing. Take a sneak, Beethoven. I got troubles on my own. Look, if it happened the other way around, I'd split with you. Well, it didn't happen the other way around, so... Wait a minute.  Like I say, Whitey, I'm a right guy, and if it was the other way around... Here, here's 12 cents. Now, you promise? Sure, I promise. I'm willing to do the right thing. Good. Here's a quart of cottage cheese. I forgot to deliver it this morning. Drop it off for me. Will you? I got a golf date. Well, wait a minute. Where do I take it? Name's McNabb, 1714 Locust. Locust? That's the other side of town. You promised, and I gave you the 12 cents, didn't I? But I'll be all day getting there.  Okay. Okay, but you wait. You wait, Whitey. I won't forget this. Make sure you don't. They call the boss up and say they want the cheese there by noon. You know, if Whitey hadn't schmoozed me in delivering that cottage cheese, a lot of things that happened never would have. You see, I drove all the way back across town, and by then the sun had been up a couple of hours, and the inside of my wagon was like a baked potato. Locust Street. What a snazzy neighborhood that was. Great big houses, wide lawns, lots of trees. 714 was one of those two-story colonial jobs with white pillars in front. I'd already climbed out of the wagon, was coming up the walk when I saw this lovely thing come out on the front porch and yell at me. Yeah, that's me. That's me. You're the one that's waiting for the cottage cheese? Yes. Can you hurry, please? Hey, hurry's my middle name. Oh, the place is all mine. Look out for the garden. Oh, don't you worry about me. I'll be... Look out! Are you all right? Yeah, but look at everything all over the lawn.  That was a terrible fall you took. I'm okay, but the cottage cheese is ruined. Buddy, it's the milkman. He fell down. Oh, he fell. I never saw such a mess in my... You all right? Oh, yes, sir, Mr. McNabbah. I'm sorry I didn't see the hose. Well, I don't know how you missed seeing it, right in plain sight. Well, I was wanting to deliver the cottage cheese. Oh, the cottage cheese. You brought it? Yeah, I brought it, but... My wife's been going crazy. We've got company for lunch, and it's got to be Krause's cottage cheese. But, sir, it's all over the lawn. Look... Oh, no! Can't eat that. Well, go out to your wagon and get some more. A wagon? That's a hurry, boy. Hurry! I haven't got any more in a wagon. You what? All I had was one quart.  What was the special order? What kind of delivery service you call this? Daddy, it wasn't his fault. You keep out of this. What's your name? Caster. Arnold Caster, sir. But I'm not the regular driver on this route. You won't be the regular driver on any route. When I get through telling Bill Krause how you're wrecking his business. But it was an accident. I couldn't help it. Oh, come on in the house, Jerry. Right away, Daddy. Caster, huh? Oh, yes. Well, I won't forget that name. Boy, this is sure my day. I'm sorry. It's my own fault. I should have seen that hose. I know why you didn't. Yeah? You were looking at me. Well, it was worth the look. Gee, you really flew through the air. I never saw so much milk in my life. I thought I stepped into an open manhole. Hey, does your old man, uh, your father really know Mr. Krause, the guy that owns a dairy? Oh, sure. They're old friends. But don't worry about what he said. He'll forget all about it when he cools off. You really think so? Sure. He'll stomp into the house now to show Mother how mad he is. And she'll be so busy trying to calm him down she'll forget all about the cottage cheese. That's the whole idea. I hope you're right. Gee, he must be kind of a big deal, your old, uh, father, man, dad.  Sure he is. You heard of Toot McNabb, haven't you? The orchestra leader? Sure. That old, your father, I mean, is Toot McNabb the orchestra leader on the radio? He's the one. Holy smoke, when I think about it, I'd give the meter guy like that. See, I'm a songwriter. Really? Yeah, and what a break it'd be if... Jerry, it's time for lunch. Yes, Daddy, I'm coming.  Oh, yes, sir, I was just leaving. Hell, yes. Well, then do it. And don't let me see you around here again. No, sir, I mean, yes, sir.  Will I ever forget those next three weeks? I went around like a man in a dream. And then a nightmare. I'd fallen in love with Jerry McNabb, and I didn't have nerve enough to do anything about it. And I wanted her old man to listen to my songs, but I knew if he even saw me around the house, he'd call the cops. The cops. Hmm. I know what made me think of that. By the time things came to a wind-up, every cop in town was after me. And it all started so innocently. The morning before Whitey went on his vacation. How's tricks, Mozart? Look, Whitey, will you take your gag somewhere else this morning? I'm not feeling so peppy. You still drive back to the dairy every day by way of Locust Street? What do you know about that? Don't get sore, kid. I got eyes. You kind of like that McNabb girl, don't you? So what? You figure her old man might be able to give you some help with those song's ears? You listen to me. I'd like her even if you were her old man. You really got it bad, huh? Will that be all right with you? Don't get sore. Just thinking I might be able to give you a little help getting connected over there. And just how would you do that? Well, it ain't much, but I'm going on my vacation tomorrow. I was thinking maybe you'd like to take over my route while I'm gone. Are you kidding? No. At least I'd give you an official excuse to stop by the place every day. Gee, Whitey, I think the boss will let me do it? Sure. If I recommend you. Uh-oh. I get it. What's the catch? No catch. Somebody's going to get the route while I'm gone. Might as well be you. No kidding? Sure. Gee, Whitey, I guess I had you all along. Yeah, I forget it. Look, if I can ever pay you back in any way. Well, leave me here just knowing while I'm away that you'll be handling things. That's pay enough. I might have known if I'd been using one half of my head I could have figured Whitey wasn't handing me something for nothing. But I was too excited. This meant I'd go to Jerry's house every morning. I was going to make it the last stop on the route, like 9.30 or 10 o'clock. I had another scheme in mind. I was going to leave my songs with her in an envelope with a note and maybe she'd get her old man to look at them. Well, I set up half that night writing lead sheets. The next morning, about a quarter to 10 after I'd made all the other deliveries, I climbed out of my milk wagon in front of 7.14 locusts, started to walk along the side of the house and led to the back porch. This place is pretty quiet for 10 o'clock in the morning. Anyhow, they left a note in the milk bottle. What's this? Dear milkman, just to remind you, we're leaving town for three weeks in the mountains. Please do not make any more deliveries until further notice. Oh, brother. She's gone. They're all gone. How do you like that? At the time she comes home three weeks, I'll be back on my old route. Hello there. Oh, hello. I didn't think there was anybody home. There isn't, I'm afraid. I just got in from New York last night and found they were all gone. Oh, you remember the family? In a way, yes. I'm Uncle Jim, Tooth's brother. Oh, what happened to your arm? I lost my footing on a train platform last week. I chipped a boat in the elbow. Oh, that's too bad. I haven't heard any more, but wearing this cast kind of reduces my efficiency. Yeah, I can see how it would. Say, as a matter of fact, since you seem to be rather sympathetic, there's a few things you could give me a hand with, if you would. Oh, sure, sure. Come on in. See, I wanted to send a note off to my wife, and I can't seem to get the stationery drawer and Ed's writing desk open. Oh, I'll open it for you. I sure will appreciate that. Right here in the library. Mark Knab's real name? Yes, I never can bring myself to call him that other thing. He's a pretty famous guy. Pride of the family. Oh, here we are. Is that the desk? The note paper and the envelopes are usually kept in the wide bottom drawer. This one? Yes. Open's pretty easy. When you've got two hands. Oh, yeah, I guess that's it. I don't see any envelopes or writing paper in here. Just ribbons and labels and stuff. Well, that's strange. That's usually where he keeps it. Would you mind trying the drawer above it? Oh, you bet. Nope. Wrapping paper, old Christmas cards. How about the little drawer up on the right side? This one? It looks like it might be a place for stationery. Oh, yeah, this is it. Oh, fine. Well, I certainly appreciate this. Oh, glad to help. All right, I guess I'll be going. Oh, don't forget your own envelope here. Oh, yeah, I was... We're going to leave it, but since they're all gone. Is it the milk bill or something? No, no, no, it's not that. I can forward it to Ed if you want me to. Well, to tell you the truth, Mr. McNabb, it's a couple of songs I wrote. Songs? Yeah, yeah, and a note I was going to leave for Jerry. I mean, Mr. McNabb. Are you the one? The one that Jerry wrote us about. The milk man who writes songs. She wrote about me? Just last week. Well, I'll be darned. I didn't even think she'd remember me that well. And these are some of your songs? Well, they're just lead sheets and some chords. I was hoping maybe she could get to...  Well, why don't you leave them here? I'll see that he gets them. I mean, could you really? Of course, I'd be able to thank you. Of course, I can't promise anything will come of it. Oh, I know that, but I mean, if he could just get them to look at it. I think I can guarantee that much. After all, you've done me a good turn. Gosh, I can't tell you. Now, don't mention it. Just leave them here on the piano. I'll mail them to you this afternoon. Okay. Do you want me to send along the note for Jerry too? Oh, no, I'll take that out. I'll give it to her when she gets back. Next time I'll see her. I understand, son. Well, I better be going. This sure is swell of you, Mr. McNam. Ah, you just forget it. Maybe someday when you get to be a big name, you'll remember me. Remember him? I'll say I remembered him, and I didn't have to wait until I was a big name, either. That night, when the cops came up to my room in a boarding house, he was all I could remember. But they thought I was just trying to be a wise guy. Look, Caster, you'll think a principal all over the place. On a desk, on a piano. I told you how that happened. I was helping Mr. McNam's brother because he had a broken arm. Speaking of broken arms, Lieutenant. No rough stuff, Al. Look, I didn't steal anything. This is your last chance. I'm innocent. Where did you hide the stuff? What stuff? I never saw any jewels and furs. Lieutenant, I'm telling you, this kid needs his memory jogged a little. Your mistake was leaving those songs behind you. I meant to leave them behind. Sure to mean he's hysterical. Maybe if I just slapped him out of it just a little. Now, for the last time, no. Lieutenant, you gotta believe me. I didn't steal anything. Okay, kid, have it your way. Come on. Where are we going? Over to the McNabb place. You can tell him your story. But I thought they were out of town. That's what gave you the idea to loot the place, huh? No, they left it in the milk bottle. The jewels? Okay, kid, let's go. Look, do we have to go over there? What's the matter?  We got an opinion about that, too, sonny. What do you mean? You should have taken his advice. Oh, I hope I never spend another hour like that next one. All the way across town, racing like mad with the siren wide open. And when we get to Locust Street, there's a big crowd out in front. More squad cars, people gawking. In the front door, like I'm public enemy number one or something. And the first person I see when they shove me into the living room is Jerry. Oh, Arnold, why did you do it? I didn't do it. Jerry, honest.  Mr. Mann, Mr. McNabb? Oh, yes, yes, he's the one. Oh, Caster, I'm ashamed of you. But I'm innocent. A boy of your promise. Your talent. Talent. Daddy was just running over your songs. Thanks, they're great. All right, let's skip the bouquets. This guy says your brother let him in the house. I don't even have a brother.  But I didn't take anything. Maybe you'll get off with a light sentence. And start with a clean slate. You can be sending me songs while you're in prison. But I've got a clean slate. Jerry, this guy let me into the house and said he was your uncle Jim and how you'd written him about my songs. But, Arnie, I don't have an uncle Jim. What about me? I told you these little guys can be stubborn, Lieutenant. Okay, I'll put the cuffs on. With pleasure. All the dirty tricks. And I thought Whitey was doing me a favor. Huh? Who's Whitey? Nobody. Regular milk man on this route. He was just trying to help me out because he knew I wrote songs and had a crush. Hey, wait a minute. I'll get Frisky, kid. I got it. It was Whitey. It was Whitey White. What about Whitey? He's the one who framed me. Whitey Benson, the other milk man. That's why he wanted me to take over the route. Arnie, do you think you can prove this? Sure, he must have been working with this other guy, Mr. McNabb, your brother. I don't have any brother. Well, does everybody stop yelling around? I mean the guy who said he was your brother. That's how he got me to leave my fingerprints all over the place. Caster, do you know that attempting to implicate an innocent person in a felony is another felony? I don't know. They could tell you at the dairy. He's on vacation. Sonny, if you're giving us a run around... I'm not. I tell you, you'll find Whitey and you'll find the stuff I bet that was stolen. At least it's worth trying. Isn't it, Lieutenant? Maybe. Just maybe. I can't really believe a boy like this would do such a thing. Well, that's a... had us puzzled all along. What a Lieutenant means is to pull on a job like this takes brains.  Ain't that what you said? Fuel out in the car and take raffles here with you.  Till we find Batson. Don't worry, Arnold. We know you're innocent. Yes, of course. I'll get my lawyer on the phone and you'll be out of jail in an hour. Oh, Daddy, you're wonderful. Gee, I'll never be able to repay you, Mr. McNabb. Oh, well, now don't kid yourself. I'll take it out of your first royalties. Royalties? Why, sure, son.  Composer. From milkman to composer in one easy lesson. Well, it's a cinch I can't tell that success secret to any graduating class. Let's see now. Yeah? Hi, honey. Not very good. You were all going to be so proud sitting out in that audience tomorrow, listening to you. We? You're bringing the kids? Of course. They wouldn't miss it for the world. You'll give a fine speech. What have you got so far? Just a couple of notes. I can't tell them what really got me started as a composer. I don't know why not. I figured I'd start off something like Mr. President honored members of the faculty, graduates, and friends. The few remarks I have to make this afternoon concern the word success. What does the word really mean, success? And after you've achieved it, how much credit do you really deserve? I asked this question, not so much of you as of myself, because when I walked up on this platform a few moments ago, I intended to reassure you that all success is the result of hard work and constant application. I still believe that's necessary, but sometimes, as my wife reminded me yesterday, a little good luck can help, too. I know that for a fact. You see, even though I always wanted to be a composer, I didn't start out that way. In fact, I started as a milkman. And that was a pretty good deal. Twenty years ago, a kid my age was lucky to have a job at all. And I liked it, too, being a milkman, except for having to get up so early. Well, this one morning... This is Anne Francis again. A listener once asked us how we managed to present such a variety of plays week after week and always end with the same theme, family prayer. Well, here's the secret. If it can be called a secret, there isn't a thing in life that doesn't somehow tie in with the family. There isn't somehow of interest to it, because the family is the basis of all society and civilization. Whether it's baseball or babies, railroads or religion, it's connected with the family, and the family theater listeners like to hear about it. In the same way, there isn't anything in life that isn't connected with God, who created and sustains the world and all that is in it. So it's never an effort to introduce the thought of prayer, family prayer. After all, do we possess anything that doesn't come from the hand of God? And isn't it only sensible that we should pray together as a family for the graces we need each day?  A family that prays together, stays together.  From Hollywood, Family Theater has brought you transcribed Early Bird, starring Danny Thomas. Anne Francis was your hostess. Others in our cast were Gloria Grant, Howard McNeer, Jack Krushen, Howard Culver and John Larch. The script was written and directed for Family Theater by John T. Kelly, with music composed and conducted by Harry Zimmerman. This series of Family Theater broadcasts is made possible by the thousands of you who feel the need for this type of program. By the Mutual Network, which has responded to this need. And by the hundreds of stars of state screen and radio who give so unselfishly of their time and talent to appear on our Family Theater stage.  This is Tony LaFranco expressing the wish of Family Theater that the blessing of God may be upon you and your home and inviting you to be with us next week when Family Theater will present. Theresa Wright and William Bendix join us, won't you? And I always suggest that you stay tuned for the latest news which follows immediately as reported by Charles Arlington over most of these stations. This is the Mutual Don Lee Broadcasting System. This is Mutual, the radio network for all America.


Thank you all for being here. These are exciting times in the Denver Police Department because we are working very hard to identify the right folks for the right jobs. It's been a very intensive process looking at all of the strengths that our team has and again, Chief White did a phenomenal job of developing the bench for the Denver Police Department. We had a lot of qualified candidates to choose from. Really, we are going to introduce our team of district commanders and a couple of folks that are getting promoted, one of which will be taken over the Major Crimes Division, but I really want to highlight the deep pool of talent that we had. This is not an easy choice. We, you know, looked at what separated folks, and in many cases it was a very thin margin. When we're talking about ties going to the runner, the tie was always the community side of things. So you will see a consistent link, deep ties to the community with all of the district commander spots. And we're going to introduce the six district commanders as well as Major Crimes Division. Most of the other spots are staying the same. There's some consolidation that takes place. We are eliminating a couple of commander positions, streamlining a few of them, and again, the entire purpose behind this is so we can be more nimble and address the community's needs in a more timely fashion. So very excited to introduce the team. We'll start with Major Crimes. Come on in. This is All right, I'm gonna do a little bit of reading here. So Mark Chuck, who was a captain, he will take over the Major Crimes Division as a commander. He's been with the Denver Police Department for 29 years, so we're talking a lot of experience. He's previously been assigned the captain of the Traffic Operations Bureau. He was instrumental in the Vision Zero. Vision Zero is a big deal to us and the city of Denver. We really want to enhance the overall public safety, learn how to share the roads with our multimodal transportation needs that we have in this department. And Mark was at the tip of the spear when we're talking about that part of our service. Captain Chuck has 13 years of command level experience, including patrol, internal affairs, and pattern crimes. So he's a natural fit in the Major Crimes Division, and he has got a strong support team there. Next, we have Commander of Police District One. If you see me shed a tear, don't judge me. It looks like Jeff has his own cheering squad with t-shirts, matching t-shirts to go with. As you know, District One is my baby. It will always be my baby. It's my home, and this is a very important pick from my heart. Just know that. Commander Jeff Martinez has been with the Denver Police Department for 23 years. We basically grew up together in this department, previously assigned to the District Five unit or District Five for only about a month. I had to steal him away from there. He's a native of Denver. He grew up in Southwest and Northwest Denver. He's got deep community ties to Northwest Denver, long history of bridging the gap between the community and the officers. So when we're talking about the person to replace me in District One, I told the council member that I was going to pick somebody better than me, and Commander Jeff Martinez is better than me in that role. So thank you. Super excited for this, Reverend. There we go. Reverend Kelly, where's the t-shirts?  It is an honor and a privilege to promote Lieutenant Cathy Bancroft to Commander. So this is, yes, this is a big deal, and I don't know if there's anything other than to say this is a natural fit. Kathy has a tremendous amount of historical and institutional knowledge about District Two, the needs in Northeast Denver. I have seen her personally volunteer her time, change days off, in order to meet the needs of the community without being asked. And when we're talking about a 24-hour-day, seven-day-a-week job in the commanders, commanders connecting with the community, this is the perfect example of that. So Kathy served with the department for 30 years, and she was the executive officer of the Administrative Lieutenant for District Two. So she's basically moving from one chair to the next chair. Her ties to Northeast Denver date back to 1988. If any of you saw that yearbook photo of me that they showed on the news, that's when my hair looked really high. Anyway, she's patrolled the Five Points neighborhood as a brand new officer, and now she gets to lead the women and men that provide the public safety services to that same district. So we're honored. Thank you. Congratulations. Congratulations. All right. Rick Kyle. Commander Rick Kyle, we talked about changing and making ourselves more nimble here.  Thank you, Chief. I really appreciate it. No, thank you. We talked about consolidating some positions and really to, with the goal in mind of being more nimble, able to address the needs of our community in a more timely fashion. I'm really excited about Rick Kyle taking on this role in Southeast Denver. Previously, Commander Kyle ran the academy, and this is an opportunity where we get to transfer theory into practice. So Commander Kyle was instrumental in the Back to Basics program. There's a lot of knowledge that our officers have shared with the Back to Basics as far as innovations and ideas that they have to enhance our ability to deliver quality public safety services. And Commander Kyle's been cataloging those since it began. So one, we get to now move some of those ideas into action. And we're going to do this in Southeast Denver, which is very exciting, because then we can try to grow that from to other parts of the city. The other thing that I'm excited about is when we're talking about being instrumental in innovation, he's also been instrumental in an area that we are going to improve in. And that's the officer resiliency and wellness, taking care of the people that take care of the people. And he absolutely knows and gets this. And District 3 has a large contingency of officers assigned to that district, so we'll be doing a lot of creative ways and innovative ways, not only serving the people of Southeast Denver, but also the officers who serve the people of Southeast Denver. So I'm really excited about this opportunity. So Rick, thank you. All right. Anybody recognize this guy? You can see he got all dressed up for the spot. No, no, I'm with you. Short sleeves. A working person. Anyway, Mark is doing a phenomenal job in District 4. I think every single person would agree, and we are not changing a single thing about that. We're excited. Mark and Jeff work very closely together. A lot of the issues cross district boundaries there. So we're going to be able to continue that seamless approach to addressing some of the quality of life and crime issues that we see in District 4. So Mark's going to continue in that role. He's demonstrated very effective strategies for building connections and trust with young people, the Spanish speaking community, immigrant communities, and we're faith based organizations. Father Joseph Dank, thank you so much for being here. So we're very excited about continuing that and then enhancing that with the rest of our team. So Mark, welcome back. Thank you. Thank you. Thank you. Marcus. Commander Marcus Fontaine, he's got the best walk in the entire police department. We might have to do that. We shouldn't have walk up music for you, right? Anyway, the good people of far northeast Denver will get to know Marcus here. He served the police department for 27 years. He has previously been assigned to the operations support division. He has got a extensive history when we're talking about command level assignments and operations investigations patrol. He's got some community. He's led community teams, which is a great asset that we will tap into significantly. He's the right person for the job. And I do want to say the executive team, the division chiefs, the deputy chief have weighed in on all of these spots. We went through, hashed it out, thought about it, hashed it out, thought about it. And I'm very excited about every single member. Can't wait to see all of the great things that Marcus is going to do in District 5. So congratulations. And then last but certainly not least, at least for the the introductions that we're doing, this is future commander Aaron Sanchez. So Aaron will be getting promoted as well going from lieutenant to commander. This will all be effective on Sunday. And Aaron, so District 6, what a, Tammy, where are you? What a job, right? Division chief Sonier can tell you that. Tony Lopez, senior could tell you that District 6 is very unique, lots of challenges. And again, our team thought long and hard about identifying the right person for the job. Aaron has served with the department 26 years. He most recently was assigned to Intel. We talked about this a little bit earlier. When you're talking about the 24-hour day, seven-day a week job, Intel is one of those jobs. And so he's used to that high pace that District 6 will throw at him immediately. He's been in investigative support. He's helped us out with the ShotSpotter program that many of you are aware of. He also is a Denver native. Unfortunately, he went to that Lincoln high school somewhere that you may have heard of. The only thing that our executive team got wrong is that there's four Lincoln Lancers in the room. So we can get past that, I promise. So Aaron, not only did he lead a neighborhood police officer team, he was also a neighborhood police officer himself. And he's got about eight years of experience directly interacting with community members. He's helped to, on his own, organize outdoor recreation events and trips with young people to better engage relationships there. He's worked extensively with Denver Public Schools, which is a big deal, and their headquarters and some of the facilities that they have downtown. He's got a extensive narcotics background, which will help us out quite a bit, as well as leading one of the task force operations. So these are the folks, and by no means you'll get a full bio on, or you'll get a bio on all of the additional commanders, most of which are staying in their same positions. These are the folks that typically work out of the headquarters bureau. They do a great job for us. But when we're talking about the individuals that are interacting officers, or excuse me, with community members on the street, it's the six district commanders that that people typically interact with. So these are the folks. And thank you so much for being here. Thanks to all of the community members for being here and supporting these candidates. They're going to need your help. Moving forward, public safety is an every person job. And for us to be effective, it is listening to those different voices, pitching in and working together. So with that, if there's any questions, most of which I'm going to let the individual commanders answer. Very nice. Perfect. So and that's exactly what we want. We want to be accessible in order to meet the needs of our community. So great, great. Did you plant that? Okay. So there we go. She's doesn't even. Okay, good. Good, good, good. Leo. Leo, thank you. You hit the nail on the head. Us working together is how we're going to increase the safety for the people of Denver. Great point. Great point. Reverend? Reverend, thank you. Thank you very much. Director Riggs, anything you want to add? I don't know. Mr. Martinez, thank you. The chief is heading out. He has a budget meeting to go to, so he's going to make sure he can fund everything next year. Hey, I just want to add this on behalf of the Department of Safety and Mayor Hancock. These are difficult jobs that these men and women have taken. The chief is at the head, but these individuals sharing that responsibility, what I would ask you as the director of safety is remember them in your thoughts and prayers. And if something does happen in the community, you're not sure why it occurred or how it occurred. Before you jump to conclusions or before you let others jump to conclusions, these are individuals spent their whole life being open and transparent and honest. Give a call to them. Find out what the real issue is. And if there is an issue, take time to listen to why, how they're going to fix it because I have confidence in every one of these individuals, especially with the chief at the helm. So once again, on behalf of the mayor, and I thank you for being here. I really do appreciate the council members that are here and the support. I think this is a good moment for Denver. And you're right. Denver is going to lead the way in the future for police departments across this nation. Thank you for being here.


Yep Okay, all right. Okay. All right. This is world's largest single span glasshouse. These are South African plants I'm at the National Botanic Garden of Wales. I Did a video here before but I Couldn't take you in here then because it was closed But it's open today so You can see some birds over there. I am doing a Q&A at the same time as well So if you have any questions, please send a super chat and I will provide you with a detailed response to your question Who if you're a member of the channel You don't even need to send a super chat. Just ask the question and I will respond to you. I like these They feel really nice if you would like me to Provide you with a detailed response to your questions. Please send me a super chat of any amount Or if you're a member of the channel, I Will respond to your question as well. We've got fishes in you as well lots of colorful fishes. I actually went snorkeling in Thailand with loads of fishes I Remember there were thousands of them like swimming all around me It was an incredible experience. I Highly recommend it snorkeling It's definitely something I'd like to do again. It's a really nice atmosphere here With the fish and then you've got the running water there It's it's very peaceful. I am now answering questions to All members of the channel if you're not a member you can make a super chat the nation of any amount And I will provide a detailed response to your question I am also offering one-on-one coaching sessions. I do sessions as short as 20 minutes one hour or 90 minutes For more information on my coaching sessions, you can send me an email At naksifythecoaching at gmail.com. I've had hundreds of hours of experience in coaching Maybe even thousands of hours actually and I've been researching narcissism For Around two and a half years I've been making videos for almost two years now So I'm very Experienced in this area. I hope you're all enjoying the walk through this glass house. I'm in the National Botanic Gardens of Wales. These are all Californian. That's very true. The truth prevails They're not really interested in positive things But if they see that you're interested in it They might pretend that they are I hope you're all happy with the picture quality. It's a really good camera on this iPhone 11 Pro Max, but if you're going to get an iPhone I Would recommend waiting for a couple of months Because there is going to be the iPhone 12. I think they're doing an iPhone 12 Max Pro and Pro Max These are all Australian I thought that was real then Yep, there is a video on defeating the narcissist. I Believe that's for tomorrow and there are two premiers for tonight They will be starting in about two and a half hours So I hope you can all join us then So what's everyone got planned for today then? Vagabond is Sitting at home on the computer I hope that you'll be joining us in the Premiers in a couple of hours Richard is spending time with his boys Jenny is Dealing with her divorce and to his son's mental health. I hope everyone's doing well and having a good day All of these now are from Chile. I'm answering questions on narcissism to anyone who is a member of the channel and Also anyone who makes a super chat donation of any amount so If you have a question that you would like to be answered Please click the dollar sign icon underneath the live chat Where you can make a super chat donation of any amount and I will provide you with a detailed response to your question. It doesn't look real. Does it? No, it is it is real These are from the Canary Islands Mediterranean. Oh, look at that. It's kind of a cactus. I think you don't touch that Quickly it's very be around the birds Listening to the birds singing in the background. It's a very healing activity and I highly recommend it to Everyone who is dealing with this it makes a big difference. It relaxes you calms the mind Makes you feel at peace with yourself. It makes you feel good. And that's what life is all about really, isn't it? That's what matters to us more than anything. It's just feeling good and what better way to feel good Than to just be out in nature. It's available For everyone at any day anytime. Why is it so healing? It's because it's natural Just look at the name nature take off the e Put a L at the end. What do you get? Natural Anything that's natural is Going to be very healing for you Anything that's unnatural is probably Gonna be harmful What the very least it's gonna have an unintended consequence That affects you in a negative way and if we look at narcissists Everything they do is Very unnatural It's not good for you. So we should stay away from things that are unnatural things that are only good harm us and Focus on things that are more natural Now I'm not saying everything That's natural is gonna be beneficial for you as Richard says there Poison Ivy is natural of course. I mean in any point That anyone's ever made There are always going to be exceptions But if you keep focusing on the exception It's not gonna help you people will tell you Maybe there's been one or two narcissists who have managed to heal from this and They're very kind and supporting to their partners now Well, that's the minority That doesn't mean that they're all like that Which is why we should focus on what is commonly known and it is commonly known that nature is very good for you abuse is not So I believe we should all stay away from the things that are bad for us and gravitate towards things That are going to help us and nature in general Is going to help you it will be very healing for you Try it for yourself take some time out of your day today and Just go for a walk in the park win a forest it'd be down by the beach and Then come back to the common section and let me know how it made you feel Let me know if you feel better about yourself Let me know if you're still feeling anxious about the narcissist nature restores your natural balance It makes you feel the way that you were supposed to feel Anything that is bad for you will make you feel a way That you're not supposed to feel because underneath it all We are all meant to feel happy with ourselves other people and the world around us It's not about adding things to our life It's about stripping away the things that we don't need the things that are weighing us down and Then you will be able to fly You will feel free anyway, I hope this Message helps you. I hope that you have enjoyed our walk through the glass house at the National Botanic Garden of Wales and I hope you can join us in the two premiers later today Which start in a couple of hours? That's all for this live video. Thank you for joining us Hope you all have a good day and I will talk to you soon


of kT, the correlation kernel, chi bar A on this extended L2 space. So we have little L2 of n1 through nm. So that's set times set. And of course, the interesting thing is the correlation kernel. So the correlation kernel is nnj xj. Now, it's an extended kernel. Of course, it's got four components. It's minus q to the nj minus ni by less than nj. So that's the kind of easy part. And plus the summation k equals 1 to nj of psi ni ni minus k xi nj. Now, the size you know what they are, they're just over there. But let me write them out, because here k is positive. And if k is positive, then f has a negative thing. If f has a negative thing, then there's actually no pole at 1 anymore. So you can get rid of the 1. It's just a comment. 1 over 2 pi i to the gamma 0 dw over w to the k. It's maybe written a tiny bit different, but it's almost the same thing over w to n minus k. OK, great. So the size are just those things. And everything looks fine, except what are the phi's? That's the problem. Yeah, the q is that. So that's q you think of as a matrix. And you take that matrix to that power. And then you evaluate it there. And nj is bigger than ni, so you don't have to worry about the power's negative. OK, I'll have to erase that, I think. OK, but what about the phi's? Maybe I'll write it here. Here's the problem, phi satisfies 1 and 2. So the phi's are not explicit. They're bio-arthogonal functions to the size. That's the phi's are supposed to be that, and they're supposed to be polynomials. That's the answer. Well, this came because you had to invert some matrix, and you don't know how to invert it. So you encode that in these things being bio-arthogonal. So once again, your initial data is encoded in the size. There it is. There's your initial data. Then you produce these psi functions, and you are asked to bio-arthogonalize them by finding these functions, which are polynomials of the right degree, and then build the kernel out of it. And that kernel tells you what's the probabilities. And of course, I think everybody in this room can see the problem with this. How on earth are you going to do it? In fact, the phi's were known in one case. And actually, you can kind of see it already here. If you look at this thing, the first thing you're going to ask yourself is, maybe there's some case where it's toplits, right? And of course, these toplits, if the x's just look like i, or minus i, and the y's look like minus i as well, right? So of course, if x of i is just minus i, and y of i is minus i, of course, that's a toplits matrix. And then you can start solving, right? So if it's toplits, you're in really good luck. But of course, it's a special case. So if it's a toplits, so for step initial data. Well, another thing to see about the formula. Actually, this is kind of interesting. Notice the big n doesn't appear very explicitly in the formula. It's very weakly dependent on big n for step initial data. Now, it just doesn't depend on big n. That's because particles don't feel the particles behind them. So if you're asking about the first n particle, they just don't care about all those other particles back there. Now, of course, that doesn't mean that you can get away with just stopping here, right? Because these n's may not be the first couple of n's. That's not where you want to look, right? So don't get that impression. OK, for step initial data, you actually can just guess them, basically, which is what they did. And it's this, m of 0, dw over. It almost looks like the same thing as the size. Well, to the uneducated eye, you can hardly tell the difference. e to the tw, you just get that. OK. So as it was known for that case and a couple of other cases, it turns out if any time you put the particles periodically in a box, you can get a formula like this. OK, well, that's it. That's where it stopped around that period. OK, 246. So last summer, we discovered how to get the phi's, which is what I'm going to show you, probably next class.  So first of all, we know that if we start with step initial data, then we're supposed to get the area process, right? And that's how the area process was obtained roughly. OK, so if we take the area process, I won't go through the details of the calculation, but I'll tell you roughly the results. So we take the 1, 2, 3 scaling limit. So 1, 2, 3 scaling limit. So if you remember, that's h epsilon tx is epsilon to the 1 half, h 2 epsilon to the minus 3 half t, 2 epsilon to the minus 1 x, plus epsilon to the minus 3 half t. So we want to take that scaling limit of the h's. Now, are things written in terms of x's, but that's OK. h tz is just basically the inverse function of the x's. So x's and z's basically the same thing, just in terms of inverse functions. And here, h 0 x, well, that's just you have step initial data. And so your initial thing looks like this wedge. And now under this rescaling, the wedge collapses down to a narrow wedge. And h epsilon 1 x is supposed to go to the area 2 process minus a parabola. OK, so well, I won't go through the details. You have to ask what's the probability that this is less than some things, which turns into the probability that x's are greater than some complicated expression involving epsilon to the minus 3 halves. And you plug in that, and you make this sum, which can actually be done and turned into one contour integral. And then you take the limit of the kernels, and you do it by steepest descent, and you get the following formula. You get the probability that the area 2 process, equal to g1, is equal to the determinant of i g l 2. And with this extended area kernel, and the extended area kernel k extended is equal to integral 0 to infinity. And now this is the area function. That's a lambda, this integral d lambda. And this is for u1 bigger than u2 and minus. OK, so this is just another one of these awful formulas. OK, so that's the formula for the area process you get by just taking a limit of this particular special solution which they found. And yeah, that's what I said. So let me just repeat. Think of that special initial data, and then this thing just looks like i, or n minus i, or something like that. And then suppose you just ask for the y's, also to be like that special type of initial data. Then you see this matrix become topolets. And then you have all the topolets checks. You solve it. And then after the fact, once you see this machine, of course, that's Johansson's calculation. After the fact, once you see the machine, you can guess what's the phi's, which make that work. So if you have a finite box and you put the particle periodically in the box, this trick works. And you can get a phi like that. So you can also imagine how to get the area one process. So you put the particle periodically every two in a box. And then you do the scaling, and you let the box go to infinity, and then take the limit, and you'll get the area one process. So that's why those are the only ones they could do, because it's really the only phi you knew. But I just want to end by showing you that there's another version of this. So there's a problem with these type of formulas. And the problem is that the formula's complexity depends on n, or maybe m.  It's the same thing, right? So we would like a formula. Remember, I wanted a formula for the probability that the area process is less than a function everywhere. And it's obviously going to be kind of hard to do it from this formula, because the dimension's increasing. So there's a better formula you can use for that. You'll see why next class, why you want a formula for it. It'll become extremely apparent why you want a formula for something less than something. So there's a path integral version of this. It's also equal to the determinant i minus k area. Now, k area, this is the original area kernel, which is just what you get here if you put u1 equals u2. Well, u1 equals u2, then you're in this formula. This guy goes away, and you just get the area kernel. So that's called k area. And now I have chi bar. There's reason these things get inverted. e to the x0 minus x1h, g2 bar, e to the x1 minus x2h. And you keep going. gn, e to the xn minus x0h, k area. But now this one's just on L2 of r. That's the beauty of this formula. Now, this formula, actually, this formula, strangely, this formula predates this formula. Because the way I'm telling things isn't completely historically correct. The formula, this thing was happening at roughly the same time that Prahafer and Spam were trying to derive the area process. And the way they did it was different. And actually, they arrived at this formula. But then they heard about all this other stuff, and so they rewrote it like this. And then everyone forgot about this formula. But this formula is the good one. Why is it good? Oh, here, h. I didn't say what's h. h is the area operator. Everything's called area here, sorry. Minus d squared plus x is the area operator. So this is a format. Now, one thing about this formula, you notice a cyclic in the variables. You say x0 minus x1, x1 minus x2, et cetera. But then it goes back. And then this one, this one's funny. Now remember, they're in order, right? So x0 minus x1 is negative, right? And this is minus d squared. So this is a fair game heat equation applied to this, right? This is the heat equation in the right direction, because x0 is less than x1, but then there's minus d squared. So that's fine. But then, at the end, something goes horribly wrong.  And that's just intrinsic in this problem. Well, of course, you were told when you were a baby, you were never allowed to solve the heat equation backwards. But as I showed to you last time, if you apply the heat equation to airy functions, it works both ways. And this thing's being acting on airy functions inside here. So it's fine. Actually, this is just fine. You can just compute this, OK? So although it looks bad, it's perfectly legal. And the thing that makes perfect sense is a kernel. All right. So I have, like, five minutes? OK. OK. Now, the other nice thing about this formula, so we'll mostly see this next time, I guess, is it's kind of clear how to take a scaling limit of this formula. By a scaling limit, I mean, I want to know what's the probability the airy process is less than some function g. So maybe if I'm in some box, I want to know what's the probability the airy function is less than some function g. OK. What you do is you just put in lots of fine points, x0 through xn into the box. And you solve this. And then you make your mesh size go to 0. But look at what this thing's asking. It keeps cutting off all the values higher than gn and then applying this e to the delta now, h. So you cut off high values, everything above gn. And then you apply e to the delta h. And then you cut off high values and apply e to the delta h. But what is that? All that is, is it means you're solving, along this box, you're solving the equation dtu equals minus, sorry, etu equals minus hu with a boundary condition of g, with Dirichlet boundary conditions there. That's all that it means. So you can just look at this equation. You can immediately see what the limit is with a fine mesh of x's. chi bar just means cut off everything less than the value. OK. So here, once again, I'll say it.  So you're basically just solving a heat equation with this potential x, well, the potential becomes minus x. You're holding heat equation with the potential minus x with Dirichlet boundary condition on your function g. That's actually what you're doing. So that's what's beautiful about that formula. The other thing, and this is where I'll end, is that this is a very, very, very general fact. So these extended kernels pretty much always have representations like this. There's a way to go back and forth. So I'll write you the version here, as Craig said. That's because you're free for me on. I defer to you, yes. Do you want to explain that? Well, you wouldn't have formulas with which to make this in ASAP. So I'm saying you wouldn't get either side naturally in ASAP. So the only claim I'm making is that extended kernel formulas like this coming from kernels like that can very generally be written like this, where you just have to say what's the thing that goes here and what's the thing that goes here. So let me show you the one for a TASAP. So in other words, all I'm claiming is that this formula, just like that one, there's a way to rewrite it on L2 of z in the following way. So let me just write it. So just write equals determinant of i minus k T n m. Now this is just that k T. This is k T n m dot n m dot. You just put n m's in both positions. And then you call it that. The one point version of the guy. And then you get, OK, you have to believe me that these two things are roughly the same. But q to the n 1 minus n m a 1, I called it a here, yeah. q n 2 minus n 1 chi a 2, q to the n m minus n m minus 1 chi a m. But now it's on little l 2 of z, OK? So the propagator here is this q, remember what's q, OK? And that thing plays the role of this guy. And you've got your k T at the one point distribution, which is this analog of this k area. And then you keep cutting off here. You cut off above, OK? So you're solving it above with a boundary condition at these a's, OK? Now q may not look like you're solving the heat equation or anything. q is this indicator function of x greater than y. But there's a trick. And the trick is that if you conjugate the formula, remember the formulas are invariant up to conjugations. If you conjugate the formula by 2 to the x, then the q just becomes 2 to the y minus x, there. That's all the result of conjugation by 2 to the x, OK? But now it's a probability transition function. This is now just the transition probabilities of a geometric random walk jumping down, OK? So now this thing looks like heat equation.   Well, for random walks in terms of heat equation, OK? But of course, once again, you've got this weird guy, right? He wants it to solve it backwards in time for time n and minus n1 at the end, OK? But you can do that. So why don't I stop there? And next time I'll show you how these two pieces can be put together to solve the whole thing. Oh, this, this. Yeah, yeah. So this equals that. And just like that one equals that one. It's just the same sort of thing. Yeah, well, OK. So first of all, I would rather do this, right? Because then it looks like Brownian motions, OK? But actually, to change, to get rid of that potential is very easy because it just turns into a parabolic shift of the Brownian motions up to some trivial factors, OK? So after you do that, the propagator of this is just Brownian motion. So all I have is a Brownian motion path, OK? But it gets killed whenever it goes above g. You see, it propagates the Brownian, well, forget that. Sorry, you have to close your eyes there for a second. So the Brownian motion comes in and goes backwards in time for a while, OK? Don't worry about that. And then it gets killed when it goes above g, propagates a bit forward. Just think of these as deltas, right? So I solved Brownian motion for a little bit of time, killed when it goes above g. And as you take this delta, it goes to 0. All it is is a Brownian motion killed when it goes above g. So you can just take this formula and you can just rewrite it explicitly in terms of a kernel, which is just Brownian motions starting at some point and ending at some point and killed when it goes above g. Well, g plus a parabola, OK? Yeah, so here, what the formula tells you is that if you want to know the probability of the TASAP is bigger than AJ at some points, all you do is you take a geometric walk, right?  Being killed when it goes below this a, OK? And then it goes backwards in time for a while. Don't worry about that. And that's what the kernel is, OK? Does that mean, is that what you were asking? Well, chi a is the indicator function that x is bigger than a.  If I didn't write it, I wrote it last time, I'm sorry. So chi a, chi bar is just 1 minus chi, yeah. So you kill, you only let things be around when x is bigger than a and then you propagate a bit. And then does that make sense? OK, anybody else?


This video represents the very end of our unit on commentatorics in our lecture series math 3120 now Just to be clear that does not mean this is the last video in our lecture series about commentatorics In fact this very last topic we're going to talk about we're going to spend Well time in this video of course, but we'll also spend some time in lecture 18 talking about it as well And that's because the topic itself can be very difficult for people as they first learn about it And this is the topic of a Commentatorial proof and after all this is the second video in lecture 17 So by our tradition this should be about something logical and as such We're going to connect it to the topics of commentatorics We've been studying and so what is a commentatorial proof well It turns out that the proof technique is fairly simple to explain Imagine that you have a set a set a and typically in commentatorics what we want to do is we like to Compute the size of a set so we're trying to compute the cardinality of some set a and let's say that after some calculations we argue that the That the set the cardinality of set a is equal to some function f of n Where n is some parameter and a itself is a set that will depend on and its size is dependent upon some variable in play here So let's say that we show that The cardinality of a is equal to f of n But let's say we also can show that the cardinality of a is equal to g of n Well as we have two different formulas that then count the same set We would have to then conclude that the two formulas are equal to each other and This is at the heart of what a Commentatorial proof is we count the same set in two different ways And then we can conclude that the two different formulas we came up that counted the same set have to be equal to each other Simple, right? Well, why am I spending so much time on such a simple proof technique? Well, it turns out commentatorial proofs are easier said than done It turns out that in order to get good at commentatorial proofs because the concept is simple It's coming up with the right with the right set to count that is difficult And honestly experience and gut feelings are what we want to develop right now So let me begin with an example here a very classic example Let n be a positive number Although honestly this statement is true if you take n to be zero as well But we'll just focus on on just positive numbers here really doesn't change the proof that much And let's prove the identity 1 plus 2 plus 3 plus 4 plus 5 plus 6 up to Plus n is equal to n times n plus 1 over 2 Now this right here is a classic. I'm talking about classic formula and typically this shows up students to see this for the first time inside of calculus 1 Which for SUU students this one sides with math 1210 and this is typically introduced as students are preparing to learn about our or already learning about Riemann sums an object we would study before we introduced the definite integral as a Limit of a Riemann sum is where we get the integrals at all now it depended upon your Calculus 1 class you may or may not have seen a proof to this formula Again, it depends it depends on the curriculum and the university or the school whatever you but oftentimes when this formula is presented Whether it's in calculus or whether it's presented to another setting It's proven using a take a technique known as mathematical induction And we will actually start learning about induction in our very next unit of this lecture series about integers Induction is a very very important Mathematical tool and so mathematical induction deserves our attention Mathematical induction is very very useful at proving formulas like this. It's really good at it It's super nice super clean But one problem about induction is it doesn't really explain why the formula holds It just proves that it's true which is something we'll talk some more about in the future, right?  It can prove that a statement is true, but it's true Who knows why I mean we know it's true But it doesn't give us any idea where the formula came from like how did you discover it? conversely a combinatorial proof actually provides a Why it gives us some intuition of where the formula comes from and so the reason I want to start with this proof right here Is there's a classic myth that 10-year-old Gauss Gauss is you know known commonly as the prince of mathematics? Frederick the Carl Frederick Gauss here it's supposedly as a 10-year-old proved this formula and Basically the proof that Gauss is attributed to have given is essentially the proof that we are going to provide right now Although we will provide a more modern flair to it using the notion of a Ferris diagram But it's essentially equivalent to it and again in addition to proving the formula We also know where the formula came with because as the legend goes Gauss is supposedly in his grade school math class given the task of adding 1 plus 2 plus 3 plus 4 plus 5 all the way up to 100 I believe was the number you're supposed to do supposed to add together the first 100 integers and so while most of the grade school students Then we're like 1 plus 2 is 3 plus 3 is 6 plus 4 is 10 plus 5 is 15 going through that's easy when you have single digits But eventually start getting double digits triple digits additions. You're likely gonna make some mistakes along the way There's a lot of places where human error can happen But Gauss actually took a combinatorial proof approach Which he that that is he he decided to count this number in a different way Which was equivalent to what he had to do it ended up being this formula right here And times n plus 1 over 2 and therefore he could very quickly count the number and also is the only student who didn't do it Erroneously so let's see how it's true. So to prove this formula I first want to introduce the notion of a ferris diagram now a ferris diagram You can think of as a matrix array. It's a two-dimensional ray consisting of dots and omissions So if you want to think in terms of a matrix, you could think of like a zero one matrix where a one Represents sorry a dot represents a one and no mission represents a zero something like that But the rule is that the dots are always to the left of the blinks So if you are if you think of a zero one matrix the ones always appear to the left of the zeros inside of it And that makes this ferris diagram and typically it's not required But typically if they have different sizes, you're gonna put larger sizes downward like so So the following diagram the following ferris diagram you see right here This is the type of object. We're interested in right now So you have one row two rows three rows four rows five rows So if you could count the size of a ferris diagram with five rows That would then be the thing you're looking for now I'm gonna slightly change my fight ferris diagram here I'm not gonna look at this diagram But keep this in mind because we're gonna find it again in a moment what I'm gonna do Instead as I'm gonna consider the ferris diagram, which is an in plus one in plus one square That is each row has in plus one mini dots and each column Likewise has in plus one mini dots in it as well And what I want to do is I want to count the area of this ferris diagram Well with the first attempt this is a square with in plus one rows and in plus one columns So the standard area formula comes into play here that a squares area is just going to be The side length squared so the number of dots is going to be in plus one squared in plus one times in plus one So that's the that's our first account of how many dots are in this square now I'm gonna take the same square. I'm gonna count it slightly differently Notice this time. What if I just look at the diagonal elements of my square How many are those going to be well? There's one exactly one diagonal element for every row of the square There's in plus one rows, so I'm gonna get in plus one mini dots along the diagonal So just consider the diagonal for a moment now consider the elements which are above the diagonal You're gonna look at this object right here now. This is no coincidence this set We have right here above the diagonal is exactly the sum of one plus two plus three plus four all the way up to Plus in that's why we had a square of in plus one So when we remove them we actually get the thing we're counting before and likewise look at this ferris diagram this sub diagram I should say this is the thing we were interested in counting Okay, let's call this number s for the moment that is s is the number of dots in the Triangular figuration this these are sometimes called triangle numbers Triangular numbers because they count the number of dots in one of these ferris triangles So if you then re Redistribute the things here. We have in plus one mini diagonal entries So let's mark that there. We have in in plus one mini diagonals We have one of our triangle diagrams right here. There's s many of those We don't know if that is yet, but there's s many of them and then over here you have an identical triangle I'm sure it's been reflected, but still there's s of those So when you put these together, you're gonna have two s plus in plus one mini dots In this in plus one square, but like we saw before there's in plus one Squared many dots in this as well. This is where the combinatorial proof came into play here I counted the square once using just in plus one times in plus one But then I counted the square with a different a different scheme And I got a different formula two s plus in plus one notice s is the number I'm interested in this equation is valid because the counted the same set twice So I can solve this equation for for s right here to do that. I'm actually going to Foil out the right-hand side. You're gonna get an n squared plus two n plus one So then we can subtract the n and the minus one from both sides minus n minus one So we're then gonna get something like this right here. You're gonna get two s is equal to n squared plus n which of course you can factor the n squared plus n as n times n plus one Divide by two and you end up with s equals n times n plus one over two That was the formula we were trying to prove right there And so this is our first example of a combinatorial proof We counted the number of dots in this in plus one square in two different ways That gave us then a formula this formula right here As as the number I actually cared about was embedded inside of the formula We're able to manipulate that equation to get the formula we were looking for. It's a really really nice method It's very clear to understand But of course, how did I know I should be counting n plus one times n plus one size squares? That's intuition that is a little bit difficult to just Tell to you one as you get better and better with combinatorics You start to get better intuition a better gut feeling on the things you like to count But ferrous diagrams are a good place to go to try to count many of these combinatorial formulas Let's provide another example a slightly harder formula one that's far less intuitive This time let n be a positive integer Then if we take one plus three plus five plus seven plus nine all the way up to two n plus two n minus one So this is this on the left hand side is the sum of consecutive odd integers In the previous one we were adding consecutive integers one plus two plus three plus four plus five Now we're only adding together consecutive odd integers one plus three plus five plus seven etc I claim that is equal to n squared. That is if you take the sum Of consecutive odd integers. This always gives you a perfect square and you'll get the next perfect square by adding the next odd Integer it's fantastic. It's it's not very intuitive at all that that would be the case But it is it is the case. It's a pretty cool thing Now when I look at this formula right here If I was trying to prove this using combinatorial proof um, the previous example might have been a little mysterious how we did it, but this one has a has a super super Obvious blues clues that we need to be looking for that tells us what we're going to count when I look at the right hand side I see n squared and much like because I'm thinking of the last proof Squares count the area of a square. That's why we call it square We don't typically say n to the second power We say n squared because of its relationship to the area of a square So when I look at the right hand side, I'm thinking I should be counting dots of a square So that you want to grab the low hanging fruit? Oftentimes the formula has a low hanging fruit and that's what's going to motivate the combinatorial proof So in this situation, that's exactly what I'm going to do I'm going to prove the formula using combinatorial proof And I'm going to do that by counting the dots in the n by n ferrous diagram Which you can see illustrated right here now like we saw in the previous proof It has n rows and n columns So the the number of dots in that square is going to be n squared Okay, that gives us the right hand side We often start with the low hanging fruit in that regard So what I have to now do in order to count The the the left hand side that is in order to prove the left hand side is equal to the right hand side I have to consider this square again But then in such a way that I can produce these odd integers And this takes a little bit of guessing and checking as one tries to write the proof But notice the following idea here. I have one dot right here. I have three dots right here I have five dots right here I have seven dots right here and then nine dots and then 11 dots Like so so if I look at these l shape configurations You're always getting these odd numbers and that's the relationship that we're going to be looking for in the second attempt I've now color coded to make it a little bit easier to see here On the other hand, let's partition the cells in the in the cell based upon their address So we're going to go back to linear algebra here Notice that this is this right here is the 1 1 position 1 2 position 1 3 position 1 4 So the first index is the row that you live in the second index is the column This would be the 2 1 position the 2 2 position 2 3 2 4 2 5 This will here be the 4 1 position just just you're aware that each of these dots It is a matrix after all each of these dots has an address I'm going to partition the cells I'm going to partition the dots in this matrix based upon their address But we're going to follow the rule that you belong to the group Where you have the larger of the indices, right? So the 1 1 position is all by itself because the max of your indices 1 and 1 is itself 1 And that's the only position where the max of the two indices will be 1 On the other hand, if you take the positions 1 2 2 2 and 2 1 The largest coordinate in those addresses is 2 and so they're all going to put together in the 2 group So you get the one group which the one group was this one right here You then get the 2 group, which is this one right here The next cell in our partition would be the 3 group We're going to grab all of those addresses for which the maximum coordinate is 3 That would include 1 3 2 3 3 3 3 2 3 1 which gives you these ones right here Then the next group would be the 4 group. You're going to get 4 1 4 2 4 3 4 4 You'll get 1 4 2 4 3 4 and then 4 4 again And let's just do one more example for completeness here if we look at the 5 group You're going to get 5 1 5 2 5 3 5 4 5 5 You'd also get 1 5 2 5 3 5 4 5 and 5 5 So each of these groups that if you think of the kth group I claim that the kth group is going to contain 2 k minus 1 many elements And why is that because of those elements you're going to end up with 1 k 2 k 3 k All the way up to k k And that coincides with considering these right here this column, but you're also going to end up with k 1 k 2 k 3 All the way up to k k again Okay, that grabs the things in the row. All right, so notice how many things are in this list The first this first consideration There's k objects here because the first coordinate is allowed to range from 1 to k with this second list You again get k objects because the second coordinate is now allowed to range from 1 to k But you also should notice that you counted the element k k twice So by the inclusion exclusion principle, there's k things in the first row There's k things in the second row, but there is an overlap of 1 the intersections 1 So each of these l shape objects is going to have 2 k minus 1 many elements in it And that's that itself is just the arbitrary odd number. So when k Is 1 you get 1 when k is 2 you get 3 when k is 3 you get 5 when k is 4 you get 7 etc etc etc up until 2 k 2 in minus 1 Okay, so what this now shows us is that we can count the objects Um as of the square by n squared We can also count the objects by taking the sum of things of the form 2 k minus 1 As k ranges from 1 up to n which the left hand side is then the is the sum of all of the odd integers And so by the method of combinatorial proof these things are equal to each other and thus proves the formula We're going to do some more examples of combinatorial proof in the next lecture lecture 18 So if you want some more practice look for those videos as well