So when we talk about digitally enhanced, I'm saying what are the goals that we have in education? And we need to pause and say, well, we want children who can think critically, who can collaborate, who can self-regulate in their learning, and so on. So if we look at what are those central goals that we want in learning, then we ask ourselves, so when will technology, where does it actually enrich and enhance those goals? I think the thoughtful use of technology, we need to ground that in what is the learning that we want? And I can tell you, many, many jurisdictions across North America and internationally have lists of competencies that they think the 21st century learner needs. So the OECD has a list, you know, BC has a list, we all are talking competencies. They can become our criteria. Is this technology helping to create more collaborative learners? Am I using this in a way that allows kids to, you know, for example, using Poplet where students can go online at night and add ideas and I can open that up the next day and go, oh, look at all the ideas people have contributed. Now, individually, you could write a response. So we collaboratively think we individually respond. So are we creating those collaborative communities? Are we using the technology to access a variety of sources? So instead of a text that has a limited scope, I can suddenly access resources that might give me multiple perspectives. So can I enrich the perspectives represented? Can I enrich the modalities being used so that I can reach more learners? If I can use more modalities, chances are I can engage more children.  We know from the research that multimodal learning is far more powerful. So can I increase modalities? Can I increase collaboration? Can I increase the perspectives represented in the learning? Can I increase opportunities for self-regulation where students are able to monitor their own learning? So if we use the kind of competencies as our guide and ask ourselves, so how is that technology helping advance the competencies we think are important for children, then I think we have a good place to, we should always going back to saying how is that helping with that competency? And if it's not, do we need to rethink how we're using it?


Claim your name, seven trusted domain name registrars for small businesses. Hey guys, in this video, I'm gonna talk about the top seven domain name registrars and you're gonna wanna listen to this one because this can save you a ton of money, a ton of time, and it can actually help you build your business. Okay, so first off, don't forget to hit the subscribe button down there. Hit that subscribe button. Don't forget to ring the bell.  That's right, I do go live. So that's something else. So what is a domain name registrar anyhow? That's kinda complicated, isn't it? Well actually, no, it's not. All it is is the name of your website. I own survivalkfood.com. I own jrfisher.com. Matter of fact, I own about 50 of these domain names, but they actually are the building block of your business. So it's really important that when you register one of these, everything is taken care of properly. Now the first step, most of these registrars require that you buy it for at least a year.  It's not that much more money. And when Google searches sites, if they see that your site is expiring in six months or eight months, they may not be likely to bring your site back as a result for somebody searching on Google. So always make sure that you have about a five to seven year period. Next thing you wanna look at is privacy options. When you're registering a website, the privacy options are very, very important. Initially, they will put your name and address and phone number and everything in there. And somebody can go to whois.com, put in your domain name and they can find out everything about you. They can find your street address if you put it in there. So it's really interesting that people would do that, but the important thing is you wanna mask that. You wanna have private domain registration. That is a little extra fee you'll need to pay, but I highly recommend you do that. And it's also important to pick a good company because if you don't, you've gotta transfer it to somebody else and then you're gonna have to pay domain transfer fees. So your initial picking of that domain registrar should be the one that you plan on staying with. It's not saying that you couldn't move it later, but try to pick one you're gonna stay with. Okay, my number one pick is Shopify. Shopify is also the number one e-commerce platform in the world. I like Shopify a lot. Let's check that out real quick. Shopify is probably one of the best ones. If you've never set up your own online website before even if you have convenience is huge. There's an overwhelming amount of tools and platforms you need to set it up, every component of your website, especially an online store individually. You're gonna need the domain registrar, a server to host your domain, a website design, and most likely email and customer functionality. It's possible to do all this individually, but speaking from experience it gets complicated and it's easy to fall down a rabbit hole pretty quickly. That's why Shopify allows you to get all of this in one place. And when you register your domain with Shopify, it's connected to your online store immediately, which means you won't have to worry about third-party connections or complicated technical setup. I know I have to go through all that because I use GoDaddy, but I have a whole lot of other stores and reasons for doing that. Okay, let's check out my number two pick and that is Namecheap. Let's go over to Namecheap real quick. Namecheap is a good one. It's an affordable option for domain registration. It offers nice web hosting bundle despite touting great customer support on its website. Customer views say that the server is in the best, so you may wanna keep that in mind. It's a really good option if you're looking for a fair price and a package that includes who is, protection, and web hosting. Number three on the list is Hover. Let's go over to Hover and check them out real quick. Hover has been around since 2008, so they're really not that old. I mean, they've been around a little bit now. What's great about it is its simplicity. If all you need to do is register a domain without all the bells and whistles, this is a great option for you. They allow you to connect essential services like Etsy, Facebook, Shopify, and all that. The prices are pretty low. They started like $12.99 a year for dot-com domains and they renew at like $14.99 a year, so they're pretty cheap and they're pretty dependable. The next one you may have heard of, number four is Google Domain. You've heard of that company, Google, haven't you? Let's check them out real quick. Here's your Google Domain right here. They really are no frills, okay? It's really simpler than Hover if that's even possible. A pricing is affordable. It costs less than both Shopify and Hover for dot-com domains. The best part about using Google Domains as you register is that the ease of putting out paid ads. If you're planning to use the Google Ad platform to market your products, Google Domains makes it easy to set up. They also offer built-in email forwarding and 24-7 customer support. And believe me, when you're building these things in the beginning, you're gonna want customer support. I promise you. Number five is domain.com. Let's go over and check them out. Domain.com is known for speed and reliability as a web poster. It also offers a live chat function which can be deeply helpful if you're trying to start a new website, but privacy protection costs extra and pricing is better elsewhere. Domain.com is a good choice for reliable web hosting so it's good for those looking to host their site in the same place that they actually register it. Number six is Dynadot. Let's go check it out real quick. Here's Dynadot. Very inexpensive, actually. It's got transparent and reasonable pricing and simple website design. Dynadot is an all-around good option for web hosting, but what sets it apart is the amount of top-level domains, the TLDs it offers. Top-level domains are the tail of your domain like .com, .co, .shop, et cetera. Dynadot offers hundreds of TLDs so there's a good chance it'll have exactly what you need. Now personally, I like to stick with .com domains. That's just what I want. However, if you want a different type of extension, it's a good resource for you. In addition to that, they're very affordable. They're $7.99 for registrations and only $8.99 a year for renewals. Number seven is Godaddy. Godaddy. Why am I excited about Godaddy? Because I have over 50 websites registered there right now. Like Godaddy because they are dependable. They have great customer support. Let's go check them out real quick. It's probably the most well-known domain registrar. I don't know if you've ever watched the Super Bowl, but man, they used to blanket the Super Bowl with ads. But that's not what makes it the best option. Godaddy is the most expensive of all the domain registrars with high renewal fees and privacy as an add-on. If you wanna go with big, well-known domain registrar, Godaddy is the best choice. Based on the pricing structure alone, it wouldn't be my first choice nowadays, but I started there years ago. It's $11.99 for the first year. And keep in mind, even the best ones are $7.99. So we're talking $4 difference a year. So that's not that much difference. But you're gonna get into transfer fees and all that. That can be $7.99 for a transfer fee, which is kind of ridiculous. Of course, you wanna keep in mind that just registering your domain, all that allows you to do is to use that domain. So when I say I register jrfisher.com, which I actually did in the 90s, pretty smart that I did my name in the 90s. But what that does is that allows you to use that domain. Nobody else can use that domain. Maybe if I didn't have jrfisher.net, they could get that or jrfisher.biz or something like that. But I own jrfisher.com. That's my domain. That doesn't allow me to build a website. So once I've got the domain, and it gets kind of confusing, but all you're doing is buying the rights to use that name. You don't even own that. You gotta keep paying for it. So you don't even own it. But once I bought that, then it gives me the right to actually do a website. So if I wanna do jrfisher.com, which I do have, you wanna check that out. But if I do wanna do jrfisher.com, then of course I'm gonna have to pay for hosting. I'm gonna have to pay for web hosting. And that means the ability to actually put my website up every single month. So that's gonna be an additional cost there. So you've got your domain registration, and then you've actually got your hosting. Then you're gonna have emails. So if you want emails, mail at jrfisher.com or support at jrfisher.com, that's gonna cost you more money with some of these services. GoDaddy has a program, and I have a whole lot of different emails, so I'm okay there. But you wanna look at each individual one and see how much more is it gonna cost me to have a lot of emails. Now you may say, well, I just wanna have support at jrfisher.com. You'll be surprised. You're gonna end up with training at jrfisher, support at jrfisher. Then maybe you have different employees there, Bob at jrfisher and Jenny at jrfisher. Those emails can pile up pretty quickly. And as they pile up, so does the bill. So you wanna think about that. So don't just think about registering your domain. Think about all the other costs that are gonna be added onto this when you're trying to calculate all this. And also think about the ease of using these companies. Like I know GoDaddy charges me a little bit more. I'm well aware of that, but I know I get really good support. And I know every now and then they have really good deals where I get 20% off or 30% off. And that brings down the cost because I have more websites. Now some of these will have bundled in services. I would recommend you looking at all those packages on all those providers that I gave you. And are there other providers out there? Of course there are. Are there other good providers out there? Of course there are. I think these that I gave you are some of the top providers out there. I think they're ones that you're gonna be happy with. They're gonna be dependable. And I feel good about telling you about them, okay? I don't think any of you are gonna go to one of these domain registrars and say, oh my God, JR was so wrong about this. This is a horrible company. You won't find that. You're gonna find that they're all good no matter which one you select. Now you may have other questions about building your website, about making money online, about e-commerce. And if you do, I wanna get those questions from you. I wanna help you out. So just post those below in the comment section. When you post those below in the comment section, I'll swing back by there. I'll find your questions and comments and I'll do my best to answer each and every one of them. If you liked this video, do me a favor, give me a thumbs up, put a comment below. If you liked the video, put that in there. Maybe you have a domain registrar that you really like. Put that in the comments below too. I would love to see what you have. It's something I could check out. Maybe I'll add that to my list. Don't forget to subscribe to the channel. There's a big subscribe button down there and ring the bell. Turn on all bell notifications so you're notified each and every time I do a new video or I go live. Now I've done an extra bonus with this video below in the description. There's a $97 course you can grab. Look in the description, click that link and learn. Simple and easy, it's a $97 course but you don't have to pay anything for it. No credit card required. This is for a limited time so you're gonna want to do that right away. Thanks so much for watching this video. I really appreciate it and I'll see you in the next one. Hey, thanks for watching my video. Don't forget to subscribe to my channel and click that little bell right there so you can be notified every time I do a new video. Also, click on one of those videos there. Keep watching on my channel.


Right now, who's the best team in the American League? I mean, in my opinion, I think it's got to be the New York Yankees. Your check's still going to clear. I know. I know. And I know I'm a homer, but I'm going to give you a lot of reasons why. I like that. I like Yankees. Here it is. The Yankees bring over Adam Adovino, right? Only helps bolster the back end of that bullpen. You've got each individual slot from one through nine has the ability to leave the ballpark at any time, right? I tell you the one thing, the two things the Yankees are going to do this year, they're going to be dominant in the back end of the bullpen, so you better score early. And number two, they're going to hit a whole heap of home runs. I'm not sure how much that's going to pay off in the playoffs, but I think with where the Yankees are, guys, you've got to remember, it's been 10 years since they've won in the Bronx, and they're starting to get a little chatter. New York can get a little antsy from time to time, and I think with the team that they have put out on the field this year, they feel that they have a chance to win the World Series. I like the Houston Astros. I love their lineup top to bottom. They make you work as a pitcher. They don't give any bats away. They have speed. They have power. I love their front line of their rotation, which I think they have an edge on, especially with Severino being down in New York. But then they have the back bullpen help, the guys that they trust, Rendon, Ozuna, guys that have been there before. So they're the most complete team, and obviously they've been there. They've been competitive, and they know that they're the top dog, and they're already calling people out in Boston already, that they know to match up. But they're supposed to win the MVP initiative. They have potentially three MVP candidates in the middle of that lineup over there. So for me right now, I think Ace Town is rocking right now in a good way. Well, what we didn't mention is the defending world champs. You guys know how hard it is to repeat. And we saw that with the Astros last year. Injuries, just everything happens. Everything's going for you. But I'm curious what you think, Train, of them. Look, we know their core talent is phenomenal. Absolutely. But they lost Kimbrell and they lost Kelly. That was A and 1A in the back of their bullpen. I think that's a mistake. I think that's the one mistake. They should have got some more Dunkin' Donut money and got it to those two guys because you cannot replace those intangibles. Kelly's the guy that can get six to nine outs whenever you need him to be. And Kimbrell's been lights out since he's been in the big league. So that's a weapon that you need. And you can't just allow to walk away. Now they're going to have bullpen by committee. Let's see in the tight ball games if they're able to be successful going down there, especially it's going to be a race between them and the Yankees for top dog in the division. But I think it's a mistake to let those two guys go. Well, let's not sleep on that lineup. They got to get their monsters. Yeah, absolutely, they're monsters. Gady Martinez, I mean, Andrew Bennett, Tendi, Xander Bogart. I mean, the list goes on and on. Dustin Padroia is back. He's healthy again. Hey, and when you have a leader like that, he knows what it's like to win and that team talent-wise. I mean, those three teams, any one of those guys have a shot to make it to the World Series this year.


Hi, this unit discusses the major change in the pronunciation of English vowels that mark the transition from Middle English to Early Modern English termed the Great Vowel Shift. In the virtual session with its question sheet as default guidance, we will look at the stages of the Great Vowel Shift, we will define the phonetic principles involved and will discuss possible reasons for this major shift in the articulation of vowels. Both mastery worksheet and practical present data from Shakespeare's works, where selected Great Vowel Shift effects have to be associated with the words in the text. Our suggestion here is to do the practical first, wait for the model solutions and then tackle the worksheet, which is an admittedly difficult selection task. Your chances to pass are much higher this way round. So this unit provides the background for the phonology of Early Modern English and eventually enables you to correctly pronounce a large number of Early Modern English words.


Hey everyone, Shaber1000 here today back out here on the we're in a little channel again. We've been here before We're on a place called D River Road and I've got a new fishing rod here. I want to try out monkey got me for father's day Yep, it's right here in this thing. We're gonna try it out with This little Zebco 33 and if works okay, I'm gonna get the mini a little mini Zebco 33 They're smaller than this so that way I can put it in my pack in case I hike in somewhere. I can I'll be able to use it Like if there's some water around or a creek around I did see some fish swimming around when we got in here Now also brought my other pole my big pole. I'm just going to throw it out there and leave it set On this one. I think I'm going to tie a jig or something on to it. We'll see what pans out on that and We'll go from there. So stick around guys. Hopefully we'll catch something. Okay guys, so What I got here get this little fishing rod Monkey got for me. I can't pronounce that so but I've got my other rod in already. I just got some chicken livers on it.  We'll see what's going on. I did bring my body cams. So I Don't even have my body camera on right now. So all right, so we know they're hitting Got a big dark cloud over me But anyway, this is pretty cool so Take this off Very hot again, of course Let's just get stuck together. I hope they stay together because there's five pieces to this thing Four of them go together there Getting a hit on there So I'll have a I'll have a smaller Hook and stuff on this one Just like I said, I'll throw some jigs or something like that. I'll put you on the body cam for that I'm just putting this together Hopefully it stays together So now when I was fishing with monkeys the other night first came apart on me once so there's Kind of look down through it make sure it's straight It goes back all the way out okay Now we're gonna put our Our little Zebco 33 on here, but like I said if this rod works out nice I'll I'll get me a mini a Zebco mini and put on here Like I said, and that'll be my hiking camping And my string went inside there, of course Okay, so I'm gonna get this ready to go. I'm gonna decide what kind of lure I'm gonna use and Get it tied up here, and I'll kick you guys back on it's really hot So I'm gonna have to find something to cover the camera up with I may have to use my hat We'll see okay. I don't didn't have a chance to get my body cam on but I Got me a fish So see what we got here. Oh All right, we got us a catfish. I've been here nice channel That channel So I'm not gonna wane. He's just gonna go back in Just a little bit of chicken liver on there. So that's good them them guys are biting I Got I think it's 17 pound test line on this pole my new My new rod says four to eight pound or four to ten pounds. So I kind of went in between I got eight pound on it I Was just looking for a jig and this thing took off on me. So What is that and he swallowed the hook Okay, so let me get the hook out of this guy. I gotta get my pliers Let me get the hook out. I'm gonna get him throwing back in Cool. I mean, I've only been here well About ten minutes. So let me get this taken care of and I'll be back with you. Okay, so Catfish are biting pretty good.  I'm gonna toss this out here and see how well it does Hope it's nothing too big because like I said, it's a pound line and it's just that evil claw stuff. So Okay Didn't come apart Excuse me in a bite on this one or I was I mean I'm sure he took my bait by now. I crossed this line over my other line I had something hit that just now bringing it in. Let me get it over here a little bit more And we'll see what happens. I still got to get my body camera on but I'm not situated yet. So Stick around guys more to come. Okay guys First fish on the new rod. Let me get my other camera turned on Okay, guys First fish on the new rod I'll stay out of there another catfish Get the net Fish number two a Nice catfish there All right guys, I Got a hit on the other poor right now. So Of course these guys always swallow the hook So let me get the hook out of him Not gonna weigh me feels like about a pound. So So the new rod brought him in Hopefully We can do better than that. So all right, let me get this taken care of right Let's go for what's that to let's go for number three now. Okay, so Let's get my lines taken care of one My new rod and this rod I just have to look out seen the Seen the line way out there. So I think I got something on it. Oh man I'll never get this one out.  I hope I don't catch a turtle. She said no no turtles today Well, it didn't work month Okay. Well, I'll point you down here. I don't know what you guys can see and what you can't but I Don't know if I'm rolling on this one Yeah We get him out of here. We'll weigh. I don't think I can get him out of here Let's see what happens one of the many reasons why he catching these turtles especially from a bank What do you guys see him probably nothing now he's stuck in these This moss and these weeds here Hopefully I can get him to go out.  I got to see what I can do here You're still rolling up here.  Yeah. No, I just my line already broke. So oh, yeah Yeah, I'll find out. Oh, yeah, I went on a lock in 19.19. All right. Thank you. There he is guys Let's put him on the scale. Let me raise you up here. Oh shoot whoa Came out of the net. I'm gonna stay in the net. I want to wait. Hang on guys. All right Here we go. Let me zero this out again Tear that didn't lock in Anyway, I just waited for a guy that was here. Oh I think this camera caught it 19.19 So almost 19 and a quarter pound and now Let's see if we can get him out of here.  All right. I Gotta take a break. I got to fix two poles now And there goes one of my drinks Okay, guys I'm back at it Man, it's a good thing. I didn't hook on to him with this little rod over here the 33 would have brought it in I Brought in heavier things than that with that with those 33s But I don't think the rod would have handled it too well So anyway I'm getting bites. I'm moving a lot of breath He bent. I'll show you later, but he bent our or our net so Find like crazy today though. So yeah, I'd just like to like to catch some fish that are them turtles, you know Yeah, I had a guy stop When I had the turtle in the net he said what's the way because I had to scale my hand I think this camera picked it up So we weighed it 19 pounds point one nine so 1919's what a way So yeah All right, guys. I'll be back with you. I gotta get you covered up I don't know if I got something on here or not. It got real tight and then it got loose Like it was coming in to me. Yeah, I feel something biting on it I think there might be something on it. What we got here nice Okay guys that's something on here nice fish another nice cat No, that's not a cat check it out.  I think it got off. No, it didn't it's on here. All right Double header. There we go. Nice little bluegill there. Yeah Of course, we're not gonna weigh him so watch him spines on the back. Come on Perfect eating size bluegill there pan fish. We call them down here. Now, let's see what we got on the other one I'm gonna fix I'm gonna fix my line on the other one Because I think this one's got teeth Yeah, sure does All right, let me get him off the hook. I Gotta use my pliers on him. I'll be back with you I don't like messing with these since that one monkey caught that one day Bit me like three times and a half a second. I probably should be using steel leader on this one Or 50 pound leader or something. All right, nice one guys About a pound pound and a half.  He's just going back in there it goes now I've got a bait the yellow one and Put a new sinker and well the hook's fine on on my other one But I got to put a sinker on it. He bit through it somehow, but So let me get these taken care of and I'll be back with you. Okay guys got another fish on the new rod here That'll be number three for this one not very big, but I'll take him I'm slaughtering him today What we got here another catfish Yeah Grab mother Grab the other camera here Yeah, guys another catfish Cool, so that's three for this rod perfect for that size.  I got a hit on the other one. That's cool that double-header earlier fish on Yeah, what's that number four? for this one And I got one on the other one Well something's hitting on it anything I don't like about these catfish Follow the hook so bad Another one. I'm okay guys. I've lost count. I Did have the tip the last The last end of this one this rod here it did come off on me one time, so I'll have to figure out a way to tighten that up All right, let me get this guy off of here and We'll check our other one and it's recording. Oh I don't ah ha ha really Shouldn't have messed around try this again Yeah, I just missed one guys. It wasn't very big, but it's they're still fun Still fun to catch Are you guys recording? It's my fault too messing around turning the other camera on because I try to keep getting both rolling when I'm Bringing in a fish, you know that way If something happens, you know at least one of them will catch something, you know catch something on film film Record just had a hit on the black Max and I got this This rag over top of my other camera back here. Yeah x 53 Keep it cool. When's blown right now.  These insects you got mosquitoes the size of hummingbirds and Got that hornet flying around here the size of a small bat Here we go. We're getting hit on this one now Somebody take it can't complain though. We've been slaughtering them today I'm pretty good on the catfish and we caught a bluegill and a turtle Yeah, that was my trash I've tried to get it with a net before I leave three cast this one Yeah, I'm into some there we go give her a little bit more to the right this time there we go See what happens here. I'll kick you back on if something Goes down time to go a monkey's here. She's gonna go up here and turn around so I'll go ahead and do my outro So anyway guys, thanks for watching. I hope you guys had a good Father's Day and Good Sunday if you're not a father, but if you have a father Hopefully you can still tell him happy Father's Day. I Mean I still have a father He's just not with me, but I know he knows so Anyway guys, thanks for watching Say bear the myth of man alleged him gone for now And in the meantime monkeys only going up here about a quarter mile and turn them around So I doubt if anything will happen between now and then I'm gonna pick my stuff up I'm gonna go home edit this video and get it up for you guys. So See you in the next one. Bye. Bye guys. Take care


Dear President Vladimir Putin, I'm so sorry that I was not your mother. You plan a trip. You want to go there? I want to go to Italy for four years. I haven't been able to make it because of of the pandemic.  Yeah. But at the same time, he's being compelled by God. Whenever something earth-shattering and awful happens, we've come to expect grifters, celebrities and attention whores to go to the mattresses with the dumbest and most self-obsessed takes imaginable. That's all well and good. Viva free speech. But we shouldn't stand for that from the serious people, the policy makers, activists and media figures who keep telling us how smart, informed and wise they are. That's especially true when we're talking about actual war. So here's a word to the would be wise. You don't have to serve up glib, instimatic replies that would make Joy Behar wince in the hopes of building a consensus for your preexisting obsessions. The Democratic Socialists of America whose ranks include Alexandria Ocasio-Cortez and Bernie Sanders don't have to rush into print to blame America first when Vladimir Putin defies international law. Former Wisconsin governor and perennial Republican presidential contender Scott Walker doesn't have to turn a hot war into a hot take about Joe Biden and immigration of all things. Supposedly sober journalists like NBC's chief foreign correspondent Richard Angle don't have to flow trial balloons about blowing up rushing convoys and starting World War Three. John Kerry, Joe Biden's special envoy for climate change, doesn't have to drag the topic immediately back to the melting Arctic ice pack. I hope President Putin will help us to stay on track with respect to what we need to do for the climate. State liquor authorities in Virginia, Oregon, Utah, Ohio and New Hampshire don't have to rush to ban Russian alcohol from store shelves. Democratic members of Congress like Eric Swalwell don't have to propose kicking every Russian student out of the United States right this very moment. And Republicans like Adam Kinzinger can just say no to no fly zones. That would be an absolute act of war. War hysteria is a real thing and it drives decent people to stupidity, madness, and worse. When America declared war on Germany during World War One, people all over the country killed doxins out of patriotic fervor. Walter Jones was the North Carolina congressman who got the house cafeteria to rename French fries Freedom Fries after France refused to join our 2003 invasion of Iraq. Within a couple of years, he regretted not just his culinary intervention, but his support for Operation Iraqi Freedom. And he spent the rest of his life apologizing for a catastrophic military action based on bad intelligence. War is complex and filled with unexpected outcomes. Will the rush say to cut everyday Russians out of the global economic system have its intended effect? Or will it simply increase the scale of human suffering and the resolve of the aggressor? Passwords teach us the folly of quick decisions and courses of action informed by bad information and emotion. Our elites, especially our elected leaders, can leave the glibness to the rest of us and maybe, just maybe, think before they swarm social media like Russian tanks driving deep into Ukraine.


It's DDK and I'm back today with another video. Shout out to you for tuning in. If you're not subscribed, do me a favor and smash the subscribe button. The goal is 50,000 subscribers by June 15th.  We're about to get the clapping, hey baby. In today's video, we did our highest paying bungee order, baby. The highest paying bungee order we've ever had. Are you ready to get to this bag? Let's go. Because we have an order on Rodee, Rodee, Rodee, with a dab of ranch. We are picking up 11 items that weighs 25 pounds each. We're driving four miles for a 45, 48. We're about to pull up, maybe great and get this cake, let's go. And just like that, we are here at Home Depot trying to stack this money like Pringles. Just to tell you that's over here. Hello. Yes, sir. I might have a Rodee order to pick up. This is your order number. WP678-618-09. OK, these can be down at the row, down at the other end. They're already pulled for you. You just got to go down there. So go all the way down to the end. Yeah, we're pro registrarists. Pro registrarists. OK, you'll find it. All right, thank you very much, sir. I appreciate your time. Hello. How you doing, sir? I might have a Rodee order to pick up. Maybe you do. Oh, right here. See what it is? Because I don't know if you were just a car or you're not taking it. No, I have a van, a Sprinter van. OK, all right, no problem. Is there any paperwork or anything I need to sign on? I'm OK. If you want to just take it, normally we give the van drivers the Rodee. So it's up to you if you want the paperwork or nothing. No, I'm fine. I'm going to take it like this. I just want to make sure. I just have to take a picture and I'm good to go. Thank you very much, sir. You have a great day. It is time to load these short scenes so we can go and hang it at the end. I have to put the gloves on so I don't get that splinter, baby. I got to spend that because I'm whipping that wood. You don't know how I'm whipping that wood. I want those splinters, baby. When we get to the house, I may take the doll. It depends on how far the garage is. So we'll see what happens when we get there. I'm going to get the grill to get our pockets on slow. Let's go. Here, 2 by 4. List of them right there? Yeah. OK. You want me to set them on the other ones or just on the side? OK. Thank you very much. How are you doing this morning? I'm OK, I'm good. That's good to hear. OK, here. Yup. That's one. Hey, thanks. You're welcome. I just have to take a photo and I'll be out of here. Step back in with photo your noodles. Got that? We go. Order number two. Do this bungee order. It's what we're about to do. Now, if you don't know what bungee is, bungees are apps that I use. And you usually have orders like tile shop orders or floor decor or things like that. In today's video, we are going to be going to floor and decor. Get to this money and we would not ignore it. And if you want to know how to sign up for bungee, I do have a video on how to sign up for bungee. So you might want to go ahead and go watch it now. We are picking up 25 boxes of tile. We are driving 10.5 miles and we'll receive a payment of $131.25. It's a great time to be alive. Now, the thing about this order is we are doing a dual order. Meaning it's going to be two people.  Let's go. And just like that, we are here at floor and decor to do this chore. Do this chore, baby. We're going to do some chores. Hopefully, it ain't critical. We're going to see. How you doing? Nice to meet you. Sorry, I never did it. It was in another delivery. No, it's OK. I think you came here early.  No, I just waited for you because it's only like, I only did like one or two of these before. So I'm kind of just learning everything. Oh, you're learning them. Yeah.  I'm still learning. Thank you. Thank you. Can you help me with that? Yeah, I'm from Bali. I have to pick up the other note for Janine, please. OK. Just give me a little bit of time for yours because I'm getting another to pick up stun right now. So let me get that to it and then I'll work on yours, OK? Oh. So just give me probably like five minutes before I get to the floor. OK. Both of us is on the same order. So each one of us is going to get a pallet. Yeah.  So, yeah. Five minutes? OK. And just like that, my God, about to pull in so we can go ahead and get it in. So he has the lighter one on his vehicle because we are going to do these together, baby. T-Work makes the dream work. You know what I'm talking about? They say this one is heavy. Whoo! Whoo! Ha ha ha! That's going to be heavy. You know what? Yeah. So you got to be able to fly. All right. Whatever is the easiest way to unload at all. Uh, strap this joint in and then you're going to let it roll. Y'all see how this joint weighs 2,000 pounds. This is not what they put on this buggy, is it? No. I thought it was Tau 4. This is just, at least it's bigger than the Tau. Yeah, yeah. But we thought it was like vinyl. Yeah. Anyway, if it's not a stick, I got a dolly. Yeah, both. I got a dolly, too. Yeah, OK. We are good to go. Yeah. Yeah, I took one. I'm going to take another one when it's strapped in so we can go ahead and get it in. Step back from the front. Ma! That's it. We are good to go. Yeah. 25 minutes left. 25 minutes? All right, see you there. Let's go. Hey, look where we at, baby.  So if this video don't make it out, hey, y'all know what happened, baby. This joint is back here, back here. Like, they have their own private roads to get back to this joint, baby. This joint is cray and zee. How do they plow this snow in the wintertime? I know that's critical. This house got to be huge back here. Let's see. Oh, yeah, this joint, nice. Look at that joint. That joint, right? I guess. And we have to deliver this inside the house. So I don't know where we're going to have to deliver it to. Hopefully, I can get some footage of that, but we're going to see what it's going to be. Let's go. Okay, thank you very much. You're welcome. Have a great day. You too. Yes, sir. Nice to meet you. I'm sure we're going to see each other again. Yeah, yeah, yeah. We're going to see each other. And just like that, in the end of the video, I hope you guys enjoyed the content. Today, we made 20673. We barely got any cheese. Hey, I'm going to say, I'm going to give you G-Words because it has been going very slow in my area. We literally only had those two orders. And I was out all day from like eight all the way until like 1.15. Because guess what happened, baby? Bungie reached out to me yesterday and changed the pickup time from 9 a.m. to 12 p.m. So I'm like, okay, cool. I'm going to go out by someone, see if I can get some orders there. That didn't happen. And I end up getting that one roti order. And that's it. The rest, everything else, I was sitting and waiting, not getting no cake, and not being great, baby. And I was trying to skate, but I couldn't, baby. So all in all, it was, today went okay, baby. It went okay, baby. We got you playing it. We maintained it. And lately, we've been, we ain't been getting that much money, baby. I'm going to tell you the truth. We haven't been getting much money. We've been getting like one some, almost days, baby. A hundred some dollars. That's it. That's all. We trying to borrow, but we can't, baby. It is what it is, baby. It's going to be ups and downs in this whole thing. All y'all thinking that green, and in between. So while you're out there, do not get discouraged. It is not going to be always great, man. You're going to have some good days. You're going to have some terrible days. You're going to have some okay days. So all you need to do is keep on grinding.  And don't ever stop. I hope you guys enjoyed this video. Again, if you're not subscribed, go ahead and subscribe. Give me a thumbs up. And do not skip those ads. Like us. I'm Marcus. This is Matt. This is Matt. This is only three. It's big bags. We'll keep on giving a reason. I'll see you guys on next one. All y'all already know it is DDK. And I'm on my way.


Welcome to this lecture on initial boundary value problems for heat equation. We are going to use method of separation of variables to solve the initial boundary value problem. So what is an initial boundary value problem? We will define and we will define what is the meaning of its solution. So the initial boundary value problem for heat equation consists of solving the heat equation u t equal to u x x posed on the domain x belonging to the interval 0 l and t belonging to the interval 0 comma capital T where capital T is a fixed number. u of 0 t equal to g 1 of t valid for t between 0 and capital T. So this is one of the boundary conditions. The second boundary condition is u of l comma t that is also prescribed to be g 3 of t. And then the initial condition u x 0 equal to g 2 of x where x is between 0 and l. So where g 1, g 2, g 3 are given functions. So we are given these three functions g 1, g 2, g 3. We are supposed to find a function which solves this equation and also satisfying these three conditions. So this is x equal to 0, x equal to l. So this is a boundary. So we are prescribed u of 0 t here and here u of l comma t and this is u of x comma 0. Of course we have put this is the t axis. So t equal to t here. We are not prescribing any condition on this. If we prescribe a condition on this line as well then it will be a boundary value problem. We are going to see in a future lecture that boundary value problem for heat equation is not well posed. So that is why we consider only initial boundary value problem. So initial condition you have x 0 or time t equal to 0 and these are the boundary conditions which are given. Of course we can consider other kinds of boundary conditions. Here we have considered what are called Dirichlet boundary conditions. We can consider other types of boundary conditions exactly like we did for the wave equation. So what is the meaning of a solution to the initial boundary value problem? Let R denote the rectangle 0 l cross 0 t. Let C h, h for heat denote the collection of all functions phi defined on this rectangle taking values in real numbers such that the functions phi, the first order derivative is phi x and phi t and second order derivative with respect to x phi xx these are all continuous on R closure. In fact we do not require it to be continuous on R closure because the conditions are prescribed only on this on this on this. We do not require it to be continuous here. A function V belongs to C h is said to be a solution to the initial boundary value problem and R if V satisfies the heat equation and satisfies the 3 conditions. The 2 boundary conditions V 0 t equal to G 1 t and V l t equal to G 3 t and the initial condition. A remark since the equation U t equal to U xx is linear and homogeneous the principle of superposition holds for its solutions. Thus a solution to the given I B V P may be obtained by a superposition of solutions of 3 I B V P's where each one of these I B V P's feature exactly one of the 3 functions G 1, G 2, G 3 and other 2 functions are 0 functions. We are going to describe separation of variables methods when G 1 and G 3 are equal to 0. So, given a function phi defined on interval 0 l find a solution to the homogeneous heat equation U t equal to U xx for x in 0 l and t positive initial condition U x 0 equal to phi x for x between 0 and l and the boundary conditions which are directly boundary conditions we are considering U of 0 t is 0, U of l t is 0. In other words what we have done is we have set U equal to 0 as a boundary condition here and we are considering only this to be general function. So, this is just one of those 3 I B V P's that we mentioned earlier. If you notice here we have t positive of course we can consider t in the interval 0 comma capital T for some capital T positive that also we can consider. So, as a consequence if you want to define what is the solution of this it must be in CH for each capital T positive. We are going to discuss more about the solution of this in a future lecture. So, we describe a separation of variables method to solve this I B V P. So, what are the main steps involved in separation of variables method? The I B V P features homogeneous heat equation and 0 Dirichlet boundary conditions. So, step 1 2 families of ODE is obtained from heat equation. How do we get that? We look for solutions to heat equation in the separated form u x t equal to x x into t of t. The heat equation will give rise to 2 families of ODE is indexed by a single parameter lambda one for x and another for t that is one family of ODE is for x and one family of ODE is for t. The Dirichlet boundary conditions in the I B V P will yield boundary conditions for x. The step 2 is obtaining non-zero solutions to the 2 families of ODE's. The B V P for x turns out to be what is known as eigenvalue problem. So, it turns out that only a countable number of B V P's from the family indexed by lambda n n belongs to n will have non-zero solutions. For each of these eigenvalues we need to then solve for t at the end of step 2 we have a countable number of non-zero functions x and x into t and t. Step 3 proposing a formal solution as a superposition of this non-zero function that we have obtained. So, superposition of the function x and x t and t is proposed as a formal solution to the I B V P and then it remains to check whether this formal solution is indeed a solution what are the conditions that are needed on the initial data file. So, step 1 heat equation gives rise to 2 ODE's and one of them there is a boundary value problem. So, method of separation of variables looks for solutions of the form u x t equal to x x into t t of course x in this finite interval 0 L and t positive. So, substituting this ansatz in the heat equation gives us this equation and dividing both sides of this equation with x x and t t and rearranging terms will give us t prime by t equal to x double dash by x. So, in this equation t prime by t equal to x double dash by x the LHS is a function of t only and RHS is a function of x only. Such an equation can hold if and only if both the functions are identically equal to a constant function. It means that there exists lambda in R a real number lambda such that t prime by t equal to x double dash by x equal to that lambda. One of the tasks is to find all possible lambdas which are coming from the separated solutions nonzero separated solutions. So, this equation which we have got on the last slide gives rise to 2 ODE's 1 for t and 1 for x what are they x double dash minus lambda x equal to 0 and t prime minus lambda t equal to 0. It is not surprising the heat equation has 2 derivatives with respect to x variable. So, therefore the equation satisfied by capital X is second order and the equation satisfied by capital T is first order because the heat equation features only the first order derivative with respect to t. Now, we have 2 boundary conditions using them we will get a boundary value problem for x. So, using the boundary condition u of 0 t equal to 0 what we get is x of 0 into t of t equal to 0 for every t. So, this can mean that either x of 0 is a 0 or t is identically equal to 0 function. We are not interested in t of t identically equal to 0. Therefore, x of 0 is 0. If t of t is identically equal to 0 we get nothing because as we saw in step 3 we are going to propose as superposition of x n x and t n t as solutions. If t n t is 0 for some n it does not make sense because it is 0 there is no term like that. Therefore, we cannot admit t to be 0 and hence we conclude x of 0 is 0. Similarly, using the boundary condition u of l t equal to 0 we get x of l equal to 0. So, we got 2 conditions for x x of 0 is 0 x of l is 0. So, we have the boundary value problem for x given by x double dash minus lambda x equal to 0 and the values of x at x equal to 0 and x equal to l are 0. So, the ODE for t is t prime minus lambda t equal to 0 we do not get any condition for t unlike what we have got in the wave equation case. So, finding nonzero solutions to BVP for x that is a step 2. So, this is the boundary value problem we are interested in solving and we are looking for nonzero solutions of this boundary value problem. Of course, as you see x identically equal to 0 is a solution for every lambda but we are not interested in that. So, the lambdas for which the BVP admits a nonzero solution are called Eigen values and the corresponding nonzero solutions are called Eigen functions. So, let us start our search for Eigen values and Eigen functions. Note that the lambda which is a real number it can be 0 positive or negative. Therefore, our search we will conduct in 3 steps lambda equal to 0, lambda positive, lambda negative. Why is that? It is because of this equation that we have x double dash minus lambda x equal to 0. If lambda is positive we can write the solution in terms of exponentials, general solution. If lambda is negative the general solution is in terms of the sine and cosine functions and of course, if lambda equal to 0 x is A x plus B x of x equal to A x plus B. So, the nature or the form of the solution changes depending on whether lambda is 0 positive or negative that is the reason why we are going to divide our search for Eigen values and Eigen functions into 3 steps. So, let us take lambda equal to 0, then the boundary value problem for x becomes x double dash equal to 0 and x of 0 equal to x of L equal to 0. General solution of x double dash equal to 0 is A x plus B for some constants A and B real numbers. But now we are going to look for those solutions which satisfy these boundary conditions. That means what we get is A equal to B equal to 0 because if you see look at this the graph of this is a straight line and that is supposed to pass through 0 comma 0 as well as L comma 0 and the only straight line which passes through 0 comma 0 as well as L comma 0 is x of x equal to 0, y equal to 0. So, A equal to B equal to 0 the line is the x axis itself. So, this lambda equal to 0 is not an Eigen value. What about lambda positive? Are there positive Eigen values? So, because lambda is positive we can always write down as lambda equal to mu square it is in the interest of non complicated notations where mu is positive equation is x double dash minus mu square x equal to 0 boundary condition remains the same. Now general solution of the ODE x double dash minus mu square x equal to 0 is combination of exponentials. So, A e power mu x plus B e power minus mu x. Imagine if you are not set lambda equal to mu square we would have had here square root of lambda instead of a simple looking mu. That is the reason we always do like this. The moment something is positive we write it as mu square for mu positive. So, now we have to check if there are solutions which satisfy these boundary conditions and nonzero functions. So, applying the boundary conditions x of 0 equal to x of L equal to 0 it is easy to see that we get A equal to B equal to 0 it means no nonzero solutions exist for this boundary value problem. So, no positive number is an eigenvalue. So, now what remains is to look for eigenvalues which are negative. So, since lambda is negative we can write down lambda to be minus mu square where mu is positive. So, the BVP for x becomes x double dash plus mu square x equal to 0 and the boundary condition is x 0 equal to x L equal to 0. So, general solutions of the ODE x double dash plus mu square x equal to 0 is x of x equal to A cos mu x plus B sin mu x. Once again if you are not set lambda equal to minus mu square what we would have had here is square root of minus lambda x which is confusing that is why we choose to write lambda equal to minus mu square. Now, applying the boundary conditions x of 0 equal to x of L equal to 0 what do we get? Suppose we use x of 0 equal to 0 then what we get when I set x equal to 0 A plus 0 equal to 0 that means A is 0. When I put x of L equal to 0 I get this condition but note A is already 0. So, essentially the condition is B sin mu L equal to 0. I do not want B to be 0 so the condition is sin mu L equal to 0. So, mu must satisfy sin mu L equal to 0 that is what we are going to see on the next slide. So, since we are interested in non-zero solutions to the boundary value problem at least one of the constant A B should be non-zero but we already have A equal to 0. Therefore, to have been on 0 we must have sin mu L equal to 0 which means mu n equal to n pi by L where n is a natural number. So, whenever mu n is of the form n pi by L for some natural number n sin of mu n L is 0. So, let us summarize what we have got on eigenvalues and eigenfunctions for the boundary value problem for x. The eigenvalues and corresponding eigenfunctions are indexed by n belongs to n, lambda n is minus n square pi square by L square remember lambda equal to minus m square lambda is eigenvalue. So, lambda equal to minus mu square and x n x is sin mu x. So, x n x is sin mu is n pi by L x. Now, we need to solve the ODE for t that is a step 2 that is very simple. So, solve the ODE for t with lambda equal to lambda n for each n in n and call it t n. What is the equation? t dash minus lambda n t equal to 0 that is t dash equal to lambda n t. So, answers will be in terms of the exponential. So, it is a constant times exponential of minus n square pi square by L square into t. These are the solutions. Now, let us propose a formal solution to the initial boundary value problem. So, we propose a formal solution using superposition principle as this this please note this is not really superposition principle this is infinite superposition that is why I have put them in quotes. So, u x t is proposed to be this this is x n x this is t n t and we take a combination of them and propose that as a solution to u x t. Of course, each one of these terms solves heat equation that is why we believe that the sum will also solve, but we have infinite sum. So, some justification is required. Since we are not doing the justification currently, we just put this symbol. Now, what are the coefficients b n they need to be determined but we have one condition that we have not yet used which is u x 0 equal to phi x. So, put t equal to 0 you get u x 0 equal to phi x when you put t equal to 0 this term will not be there because this will be 1. So, what we have is phi x equal to summation n equal to 1 to infinity b n sin n pi by L into x. So, thus b n are the Fourier sin coefficients of the function phi. See phi of x is given in terms of only pure sin series that is why this is called Fourier sin series for the function phi and b n's are the coefficients. So, we need to determine them. So, extend the function phi to the interval minus L comma L as an odd function with respect to x equal to 0 because phi is given on the interval 0 L. So, let the extended function be still denoted by phi then the Fourier series of phi it takes this form because now the extended function is odd function it will not feature cosine times the Fourier series will not feature cosine terms it features only the sin terms. And the coefficient b n is given by 2 by L integral 0 to L phi x sin of n pi by L x dx. We already discussed this in the context of wave equation. So, a formal solution to the IVVP is given by this the only difference is we have substituted what are the coefficients b n. The coefficients b n are the Fourier sin coefficients for the function phi. So, when is this formal solution defined above is indeed a classical solution. So, answer is on the next slide. So, theorem let phi be such that the Fourier series of phi converges uniformly to phi. In fact, the Fourier sin series of phi converges uniformly to phi and phi of 0 equal to phi of L equal to 0. Then the function defined by this is a solution to IVVP. If you recall we have defined the solution to IVVP only when t is restricted to finite intervals 0, t. But here the IVVP we have posed is for all t positive. So, what it means is that this function belongs to C H for every capital T fixed that is the meaning of the solution. A comment on the uniform convergence of Fourier series because is what we have assumed in the theorem. There are conditions under which the Fourier sin series of phi converges uniformly to phi in the interval minus L to L or equivalently on 0 L. Once such set of conditions is that phi is continuous and the integral 0 to L phi dash square dx is finite. And of course, phi of 0 equal to phi of L equal to 0 that is also required. So, let us summarize what we read in this lecture. IVVP for heat equation was solved using separation of variables method. A formal solution to IVVP was proposed. In a future lecture, we will prove that the formal solution obtained by separation of variables method is indeed a classical solution. Thank you.