While Adam Sandler might play an average Joe in most of his movies, in real life this funny man lives a total life of luxury. Sometimes coming across as too casual or very down to earth, we can forget that Adam is uber wealthy and successful, but the man has amassed a net worth estimated at 480 million dollars. He's long called a nearly 13,000 square foot pacific palisades estate his main residence while he and his family also have homes in places like Malibu as well as over in Florida. Also Michael and I just dropped our very own house tour of our new home we moved into this year so go ahead and subscribe to our family channel if you want to see where we're living and more of what we're up to. Oh and once you watch it let me know what you think of our home. In these videos we don't reveal any addresses and even though I've done a house tour of my own place please do not show up at any private residences because it's not safe for anyone. Adam Sandler is a comedian actor and filmmaker who starred on Saturday Night Live in the 90s before going on to star in a long list of Hollywood films. These movies combined have earned over 4 billion dollars at the box office. Adam also has a 350 million dollar deal currently with streaming platform Netflix and his latest release Hussle a sports drama in which he plays a basketball scout has broken some records. Hussle has given him some of the best reviews of his career to date and over on Rotten Tomatoes it's currently ranked as the most liked by his fans and the highest rated of his career when judged by the audience score now sits at 92 percent. While I guess we'll have to wait and see how Sandler's future Netflix content measures up. Hey guys it's Kara the Vampire Slayer back with another exclusive host to our here on viewers entertainment in this one checking out where Adam Sandler and his family call home. Don't forget to like, subscribe and hit me up on Instagram to chat and now let's get into this video. We'll kick things off with the main Sandler residence, Adam's mansion in Pacific Palisades Los Angeles where he lives with his family located in an upscale residential neighborhood here between the Santa Monica Mountains and the Pacific oceans nearby there are miles of coastline hiking trails and more. Some of his famous neighbors include the likes of Tom Hanks and Rita Wilson, Steven Spielberg and more. Adam's estate also has some of its own celebrity history. Reportedly in 2004 Adam and his wife Jacqueline Sandler were at Goldie Hawn and Kurt Russell's house for a New Year's Eve celebration when his wife said she would like to buy that house. Sources claim the couple actually did buy the massive mansion from Goldie and Kurt the same year for 12 million dollars to be exact. The home was never on the open market and Adam keeps his properties quite private so we've yet to see interiors but we do know the details. The mansion clocks in at 12,860 square feet of space along with 7 bedrooms, 14 baths and other highlights like a sprawling yard packed with amenities. The Sandler mansion was built back in 1990 but had underwent major renovations and upgrades since then. Situated on two thirds of an acre of land, outside there's a lot to look at and aerial photos show how expansive the property really is. The features continue out on the grounds where aside from the massive yard there's a refreshing swimming pool with spa and terrace space for alfresco entertaining. Being the basketball fan that Adam is, he also has a sports area on his property fitted with the courts, fitness equipment and much more. If all of that's not enough, the actor also owns the home next door which spans 3,971 square feet of space with 4 bedrooms and 4 bathrooms. It's likely the size of a large family home all on his own and it's likely just serving as a guest house. It appears that the Sandlers do need sufficient guest quarters though because they love inviting their friends and family to stay over. During an interview with Jimmy Kimmel, Adam revealed that he had his entire extended family staying with him for one of his movie premieres. When asked how many family members were staying over, Adam said, Let's put it this way, I've got a big house and there's so many people in the house that my mother and my mother-in-law are sleeping in this same bed. It's that crowded. Well that is one big family. Reportedly, Adam's Pacific Palisades compound would be worth an estimated $28 million or more these days. If Adam's Pacific Palisades mansion gets too hectic with family members staying over, he and his family can always head on over to one of their vacation spots including their lovely Malibu beachfront property. While it's not drastically far from his main residence, it is directly on the water with gorgeous sandy shores to take advantage of. He's often spotted spending time with his family in Malibu and clearly enjoys some downtime here. Adam reportedly purchased this abode in 2001 for just under 4 million dollars but these days the estimated value has essentially doubled, sitting at over 6 million dollars or nearly 7 million dollars if he ever were to sell. Built in 1947, the house has been remodeled and now offers 4,000 square feet of space along with a super modern vibe which is obvious from the exterior views. A single family house, both three bedrooms, three bathrooms and plenty of ocean views through the many full walls of glass inside. Other features at Adam's vacation home include the large entertainment patio at beach level which is perfect for having guests over and summer parties. There's also a skylight that runs the length of the home inside allowing all the sunlight to reach you even indoors. His house is situated literally on the beach like I mentioned and on a highly sought after street with picture perfect views. I wish I had more to show for Adam's homes but the man is notoriously private with his properties. We do know about another place that he purchased for his mom apparently but he's often spotted visiting and staying here as well. This one is in Florida, located in the Highland Beach area of Boca Raton, Florida a town in Palm Beach County. It seems like a much quieter area of the state unlike Miami for instance so his mom probably gets some good peace and quiet here. The condo home Adam bought is 2,600 square feet of living space with three beds and three baths originally built back in 1981 and renovated since. We know from records that the condo is located in the building Parker Highland which is the boutique residential building offering only 26 units. The building offers steps directly to the sandy shores of Highland Beach and if that's not your thing the residents here can also enjoy a stunning pool and spa setup. In terms of location it's only minutes away from Boca Raton's downtown district to the south or the north you'll find downtown Delray Beach for endless entertainment options. Current estimates of the high rise place its value at over 1.25 million dollars which is a lot more than Adam paid some years ago. Back then around 1999 it only cost him about 650k.  While he's super private about the interiors of his properties it doesn't take much to conclude that the homes he owns are luxurious as you'd expect from a celebrity like him. Not to mention Adam has come a long way from his humble beginnings growing up in New Hampshire but still pays to visit his childhood home from time to time and he doesn't forget his roots. It seems like according to the last time Adam paid a visit to his childhood home not much had actually changed at all. A few years ago when he was there it was reportedly the same kitchen, the same bedroom that Sandler shared with his brother and still had a lot of his old stuff in place like his albums and his t-shirts because his parents still live there. Well after seeing what we could on Adam's homes what did you guys think? Be sure to let me know your thoughts in the comments down below. Thanks for watching, hit that subscribe button if you haven't, follow me on Instagram to chat and I'll catch you all in another video. Bye!


Hi everyone, Joe for jaspyscasebrakes.com coming at you with another case of tribute baseball six box pick your team, too Some good baseball here On opening day of the baseball season. Happy opening day everybody We're celebrating 15% off the entire website. So jaspyscasebrakes.com tonight only Thanks, everybody. There's kshawn in the chat Yep, last bottom mojo cubs too. Good luck cubs are starting soon. I think cubs rangers Thanks all the people who bought their team straight up We pulled the remaining teams for those random number block serial number style breaks right here We did the video in a separate video here are the results So it's a serial number style break involving just these teams right here, and there's the fresh case. Good luck Rex you hit an alley auto earlier this morning in this I don't think we were live early earlier in the morning. So I don't know how to know how you would hit An alley Dela cruise or maybe you went to your local hobby shop. You picked up a box of tribute surely. That's what happened Harry thinks Ali's a fraud Sell him quick. He says Is he a friend? So I'm gonna slide do a slide out of the base cards first switch back to the baseball game Clemson's Clemson at Arizona in a commercial reds did play today Ellie Dela Cruz went one for two couple run score to walk in a couple strikeouts Spencer Sear much better Spencer Sear dead to have a better day two for three two RBI's one strikeout Reds beat the Nats eight to two by the way Could be a surprising team this year maybe with some of the punch that they have in that lineup Right Spencer steer Ellie Dela Cruz Fraley Jonathan in the at the top of the lineup Will will Benson I think Nick martini by the way 33-year-old Nick martini in the DH spot two for four five RBI's a couple homers So yeah, there may be some non numbered base cards that we'll do some randomizing for at the end Make sure I'm not missing any numbered cards back here And numbers are on the front. Oh, there it is 38 out of 50 Andrew Abbott I think rookie cards are relatively Not common as well. Maybe one a box Reds Scott picked up the reds outright and pick your team two We go Scott All right now your hits There's Freddy and there's Adley rushman 29 out of 99 Baseball season ladies gentlemen baseball is back Adley. How did you do almost everybody played today? Orioles beat the Angels 11 at three Adley Two for four three run scored two RBI's in a walk Love to see it There's the hologram sticker right there KV 119064 you go to MLB comm slash authentication. I think you're able to you're able to look up That series of letters and numbers and see You know and then go like baseball reference and see what if anything he did that day Nice Tyler with Adley on his fantasy team And John Smoltz is our first autograph 64 as 75 Smoltz he going to Josh and the Braves Anyone have any fancy faves fancy sleepers That they really like this year. I actually have Jake Fraley on my team and Ryan Mountcastle as well reds Orioles guys Like they could be some solid contributors Speaking of the reds, there's Joey Votto and Ellie Della Cruz 47 out of 75 dual relic Both reds both going to Scott. I want to switch the game over to Cubs Out Rangers And there's Keith Hernandez nice great mustache 44 out of 75 Mets are a number block team So if you're not familiar with the random number block format, we go by this digit right here Number four that four not the first four, but that four right there So that's gonna go to Chris Who's got number four? So Chris will get for that big group of teens right there four out of out of whatever 2014 out of whatever 24 out of whatever 34 out of whatever 44 54 64 744 So on and so forth. All right behind hosel to vey Is a triple relic Christian Yelich 174 out of 199 Brew crew, that's a random number block team So that's another one for brewers for and that is for Chris And we've got a three out of ten big Mac Mark McGuire That's a's edition and that's a random number block team So that goes to a's three That's gonna be for josh smith with that extra spot he won in that random number block nice first box That uh david is that a uh sign fell reference The funny thing i was reading the Javier by is always booed in Chicago, but when he plays Played the socks they booed him he They booed him and then he hit Single soul second scored only right of the game Uh kevin likes and carnaus, you know strand a sleeper On a team another reds player, I believe right reds youngster Should we whip around the league a little bit? Uh braves met repose phone braves philly's repose phone looks like they're gonna make that up tomorrow Uh yankies came back and beat the astros These are two base cards here. Yeah beat the astros five to four even though astros uh Put 13 hits on the board round Uh in extras our first extra inning game of the year Pirates beat the marlins and extras in miami six to five Detroit tarik scoogle Uh putting up scoreless innings Someone I heard someone put tabs tarik scoogle as a scion candidate, but But uh, I don't know how many is he pitched but he did not give up any earned runs Tigers win one nothing Tigers might be surprised to you this year terry thinks Twin to beat the royals four to one Uh blue jays beat the Oh that from from week one Oh, we got switch uniforms, right for santa u estate Blue jays beat the rays eight to two My dodgers beat the carnal seven to one podgers beat the giant six to four Reds beat the nationals eight to two Orioles pounded on the angels 11 to three Cubs rangers on thursday night baseball right now and then the late games guardians at a's rockies at diamondbacks Red Sox at mariners Almost a full slate if it wasn't for those, uh If it wasn't for those uh, what should we call it? rain out games Steel should be on scion candidate list. I think he was Was he a finalist last maybe he wasn't a finalist last year, but he had a great season last year Nice, here's a yamamoto rookie card For scott and the dodgers Got roughed up in his first start in korea Let's see how second start looks later, uh this weekend And there's joey vato 23 out of 75 scott with the reds Actually, there's justin sealer right now. I have them on fancy team rex We got a rookie. We got yo han rohas 48 out of 150 for the fighting fills fillies greg k with the fighting fills Strikes out corey cedar All right behind Julio is Calruchan jr. Derek jeter 10 out of 75 orials chris Yankees matt smith. So that'll be a randomizer between those two teams He's really I need 130s. He's like He's 180 There's otani is a dodger And tim rain 66 out of 99 That'll be uh nationals david h for the nets Oh, is that two k two straggats in a row? I did. I don't know how many of those were actually on the list of Of the number block team. So that's why I didn't didn't want people to think that those are the redemptions that they could win in the number block I did not say anything There's goldschmidt 64 s75 cardinals. That's going to be for darin Although someone said ellie is a redemption though Or is not a redemption And we've got an eddie murray for the ohs chris with the orials 11 out of 50 I think orials had the majority sale or whatever approved So that new owner Might want to look to spend a little bit in the middle of the season or this off season Which could be good for the orials. I feel like they have not really invested in their Their roster for a long time or really anything in a long time Nice ozzy albie 69 out of 150 Piece of the jersey hologram sticker josh smith with the braves Another box I know rex keep rubbing in our face that you're Getting hits from other breakers I get it Walk though There is a redemption a blue redemption All right, david and look rex is even responding. He's just very quiet about it No, no apology. No, nothing Unbelievable It's another base car grant Trout auto would be nice allen That would be nice. There's vlad senior Goldschmidt who hit a homer off glass now today Henry Griffey jr. Bregman Trey Turner George Brett And mccutchen to 25 That'd be for Ivan Of course our our fault our fault There's sal frillik rookie card for the brewers. So that's a non-numbered card. So this will be part of that non-numbered group this fyi ron otakuni jr. Ozzy albies 25 out of 25 Both going to josh and the braves and How about a john krupp? 77 out of 99 That's some old school autos in here too Greg k with the fight and fills That was a fun team back in the day That's what I thought too Kevin Clearly clearly Kevin Rex cannot be trusted There's vlad grero junior seven out of 10 That's going to go to the blue jays blue jay seven blue jays are a memorable off team. That'll be for jamie There's endi rodriguez for the pirates 29 out of 99 rookie auto for Ivan and the bucks That is false You made the statement someone bought the spot afterwards now now rex is just trying to walk it back Buying spots from other breakers Then goes oh, I don't have money for jaspies Gotta wait till payday. Sorry guys in reality Bankroll is going to other guys fragment's going to eugene palmer With the straws and the redemption behind vlad grero jr Congratulations, you are due to receive tribute to the threads autograph relic red parallel which I think is to 10 9 low number of Good luck everybody Sin Cincinnati joey vato nice Scott goodman with the reds Scott goodman joey vato is a goodman a good man Nice guy from what from what the people say Nice there you go scott. Congrats I know you did rex You can only Eat one thing the same thing every day. You can only watch one program the same program every day You can only Drive the same way to work every day You're only allowed to watch the same movie every day Eat the same thing every day No other changes Some good hustle by chris for morel here home They had a home to third in second stat. I don't I don't I have no context for that But I'm assuming it's very fast Look stats with no context They're like chris for morel went from home to third in 10.29 seconds or something like that And I was just like all right cool Like I don't know what the average speed is so it gives me no context as to how fast that is Thanks Thursday night baseball Good rex. I want you to feel that way feel shame Vato no, I don't think a holo thing Hall of very good I want to say Does 10 I mean 10 seconds. Did they say 10 seconds? But what if like the average was like You know nine seconds For average an average ball player chris says vato is in 10 out of 99 bow jackson for the royals I think we did this exercise yesterday No, I don't know. I actually did not see what did anyone remember seeing what the speed was On the Thursday night baseball I'm looking at the vato baseball reference page one MVP You know if he maybe averaged 300 this season maybe got Some more hits His similarity scores does have edgard martinez sapeta And freddy freeman has similar batters Maybe not first ballot 64.5 career war is pretty solid Maybe eventually he does get in All right Onwards the hits. We've got mani machado for nano to tis jr 14 out of 75 Dual relic for kenneth in the Padres How about some tech jason veritech 43 out of 99? That is a number block team 43 out of 99. So that'll go to redsox 3 boston 3 That will be for josh I guess that that lifetime ops at 9 20 is pretty good though Pretty darn good He does have a lot of bold numbers though on the on the baseball reference right 40 doubles Led the league and walks a number of times led the league in ops 1234567 times Ops twice A few top three top five top three mvp finishes but It's joe musgrove I don't know about first ballot, but you might sneak in there Chris jose is no one risen. He says no for vato since he's a catcher lol, huh? Oh rex silly rex Stay in your lane rex if you want to try to get into this argument We're talking about first baseman joey vato catcher joe This goes to uh, kenneth and the padres The rex just executed a charlie brown trying to kick the football but landed on his back There's furgy jankins 144 out of 150 Speaking of the cubs that's going to go to kshawn and the cubbies All right, I think chris is with me on the rizzo right? Yeah Sorry rex. You're not going to get any sympathy from chris Isn't furgy already in or no? There's a statue. I think they built a statue for him outside wriggly There is mike trout 26 out of 150 angels are a random number block team that'll go to angels six alan murdoch With the it wasn't this the trout you were looking for I think alan wanted ink, but Rex no one's going to help you on anthony rizzo and hall of fame stop trying to you're trying to make like fetch happen Mean girls reference alan wanted ink. All right. Derek jeter reveals a brian wood 30 out of 150 That's for justin and the mariners Oh, nice. He is There you go. I've got the game on rex. Let's see how white langford does Is he wearing a debut patch? Let's look at his uni He is It's right on that front shoulder That's what I thought furgy jankins is in the hall of fame already David in the hall of fame already 1991 since time Yeah, that's what I thought too Yeah, I mean Yeah, why at lengthford did not have to spend very much time in the miners and I guess really impressed in uh Justin seals like started to walk off the mound like he struck him out But I think um, but he played college, right? I think those college players usually spend There's the strikeout. Uh, you usually spend a little less time in the miners, but 44 games is still pretty pretty short, I think So they must have really liked what they saw in those minor league games and what they saw in the off season And what they saw in the spring What position is he playing? See in the outfield Langford's a center fielder says uh, says oliver They haven't had dh today though Tavares is playing center field. That's quite a They got a pretty solid outfield. I mean Tavares, carter, diran Or a carter a dos garcia. I mean why at lengthford maybe not kicking anybody out there But pop him in a dh doesn't have to think about fielding. You can just hit We'll see how that works There's machado 14 out of 50. How is the ravine today oliver? Let's go to can from the podre's Dodgers were the nice w today Only blemish on a blast now is No seager it's batting second today Simeon seager josh young a dollars garcia. Why at lengthford heim jonaheim ezekiel diran Uh carter and to verus. It's a pretty great pretty great lineup Your sal perez 154 out of 199 It's gonna go to the royals that'll be for tyler I don't know who would usually be the dh Let's look at the bench I guess jankowski. Maybe You know josh smith perhaps Jared walsh is on this team mlb game day is not showing uh nathaniel low as a As a potential bench player, but maybe I don't know. I guess it would be Nate low. Nate lau. Nate low. Brandon lau. Nate low Your sal perez. I like the mlb hologram sticker there worth looking up mlb.com slash authentication There's anthony volby nice second-year anthony volby 48 out of 150. This is 2024 tribute baseball He's in 2023 as rookies. I want to say But still nice for matt smith and the yankees Ah low is on the il so that gives link for the opportunity And we got some dodgers clayton kershaw Saw him in the dugout mookie betz 137 out of 150 He's been looking good Scott good miss eyes homer on tv earlier today Almost like a little three-quarter swing Little pitch shot with the nine iron got it right over the left field wall Scott goodman with the dodgers There's kersh again And there's christian and carnasio strand 140 out of 150. This was kevin's sort of sleeper fancy baseball pick This is going to go to scott and the reds Oh, look at that Three out of ten carlos corea a piece of the baseball. Look at the stitching right there Is that from his mlb debut game? That's kv 157 906 This is worth a look This is astro. It's going to go to eugene palmer.  I don't know what the other The batter is not yoramah sanchez But there's carlos korea ml. Oh, no, I think maybe this was Okay, I get it now So the ball is from this play right here But the game is from carlos korea's mlb debut game First career hit game I get it now Initially, I thought it was going to be the one he hit, but he probably has that in his house Right. He's not giving that up This is from the game. He played in though his debut game That's still cool though Nice a baseball from his debut game I feel like if it was actually his his ball We would absolutely not one touch at yugin because what happens in a one touch There's a lot of room for the car to rattle around potentially damaging corners and edges during shipping Penny sleeve and top load is really the best way to do it That's why you don't one touch things to psa either Oh nice. It's got josh young mic'd up. I'd love those mlb hologram stickers I'm sure it increases the price of of a product, but I hope that in the future we'll start seeing more and more of those More and more of those hologram stickers on more and more relics You know, I don't know if we can event. I don't know if we'd get to a place where every card could be like that It'd be nice, but it'd be nice if we could see it a little more often though Yeah, now I agree oliver if glass now I don't think the stuff is ever in question, right? It hasn't really never been in question. It's Can he can he pitch more than 120 innings in a season without his arm falling off Let me get these little blank cards out of the way here Thurman munson three out of 99 Goes to matt and the yankees. Thank you gene. Appreciate that I'll be sure to pass that on to the team Final set of hits We got a two-team randomizer and then a randomizer for the non numbered base card for that group of number block teams Good luck everybody. Happy opening day We've got a couple Dodgers clayton kershaw freddy freddy 165 out of 199 Kershaw was in the dugout today I think he's throwing a little bit But I don't think he'll really ramp it up until For a couple months. I think Walker Bueller is a lot closer There's jim Palmer Six out of 75 going to chris and the oryals Yeah, I'm I'm 85 percent Joe tonight. This is my third your thursday my friday jonathan. I'm Do feel a little uh You know like like anyone at the end of their work week Probably running at 85 efficiency There's a doran lala rookie card Bobby wood jr. 124 out of 150 Yeah, so you get it jonathan you get it. All right, there's one. So how did one soto do as a yankee today? Let me go take a quick look I think verdue had the go-ahead sack fly at seventh inning that ended up being the go-ahead run One soto one for three rbi and a couple walks. All right Oh, that's nice. You get a good friday. Good friday everybody Like good friday today Oh tomorrow. Good friday friday. So today is your friday thursday Happy good friday everybody and happy belated Ash wednesdays and palm sundays and all all that fun stuff 97 out of 99 a dollars garcia rangers That's a number block team. So that'll go to rangers seven Jamie with rangers seven I get it jonathan. I'm picking it up Picking up what you're putting down Hank erin reveals a Paul golsmit 67 out of 150 ruining tyler glasnell's day This uh Is his home debut? I guess he pitched in one of the korea games But golsmit hit a solo shot off tyler glasnell Cardinals it's going to be for the porter house state and behind korea seager Is George Kirby nine out of 50 for the seattle mariners Part of that strong mariners starting rotation. That's going to go to justin We'll see the mariners in action a little bit later tonight. There you go. Okay. All right some randomizing to do Yeah, I got to take a look at that corbin burns box score Kind of hope that the dodge would have made a little trade for him, but That's right dodgers have enough All right, so there's some new dice. Let's grab a couple lists here Um First one is going to be for the non numbered cards So let's get the random number block group here So one person will get all the non numbered cards for those teams right there And then we'll do the dual relic randomizer between baltimore and the new york yankees And we'll do one dice roll for both lists and that's going to be Six times one out of five six times one two three four five and six That'll be chris chris norville. You're going to get all the non numbered base cards Um, including some rookie cards possibly in the mix there six times for the dual relic one two three four five And six It's going to go to baltimore. It's going to go to chris norville again Good randomizer for you chris And there you go gang very nice break another case attributed in the store I'm joe for jashby's case rakes.com and i'll see i guess we'll do a recon It's kind of a longer break. It's a sneaky long break. We'll do a quick recap of the hits There was the dual relic There's a quick recap Are you sure Kirby used a pitching tone? What's a pitching tone? There's the joe evado redemption Some really nice stuff for your old school guys new school guys Hall of famers current stars You know rookie cards Yamamoto in his dodger uniform All sorts of good stuff ladies gentlemen. Thanks for watching. Thanks for breaking with me. I'm joe I'll see you next time for the next tribute break. Bye. Bye


You have Supergirl in this who is smoking hot to look at. That's enough for most people right there, but on top of it, she's actually a great character. Michael Keaton as Batman hasn't missed a beat. The guy looks great still. Still has a physical presence, still has that wit and charm. Even Ben Affleck is at the top of his game here. He looks fantastic. The suit has never looked nicer on him. And I just, he feels more comfortable in the role, at least to me personally. The movie's also doing some cool stuff with time. Time displacement, showing different imagery, moving around in and out of focus, in and out of different layers. There's some really interesting stuff on display here. It's trippy at times. I don't see the hate that it's getting. I don't get it.  This absolutely delivers.


Hello. In this video, I'm going to briefly chat about some ways that you can debug your code and how to really solve problems when you encounter them. Because a lot about learning to code is really about learning to understand errors and be able to figure out how to solve those errors to ultimately solve whatever problem you're trying to. And so today we'll just talk about some of those some tips in order to help you get started in that process. All right, so here we are in a very short code. And there are four ways that we can work on debugging our code. The first is just to Google. For example, you can type how to read a CSV into Google. As I've done here, and you can get all of these different answers of people that have written tutorials to documentation. And occasionally even stuff on Stack Overflow. So this is a really the first stop when you're coming up to issues and you're trying to debug your code. The next step is to use Stack Overflow. A lot of times Stack Overflow will come up in part of your Google search. But you can also use this link here to get to specific Python problems. And then you can sort of say read a CSV. And let them know that you are not a robot. And you get all of these answers of people that have asked the same question as you and have gotten community based answers from people that are, you know, experts in Python. So that's, you know, step two. One thing you can do within Python is use the actual help function. And so here, you can just type help, and you can give it whatever command you're struggling with. So this is a little different from Googling where you sort of ask a question. You need to know what the command you're trying to use is, but if you're, you know, you need to use pandas dot read CSV, and something's not going right. And you're not sure what the problem is. You can use the help function. You need to make sure your library is in is installed. And it will actually print out the very intensive documentation about this particular function here to read CSV. It tells you how to actually do this. It tells you that your file path needs to be a string or a path object or path. If you're scrolling through here, you can see here we've got skip rows that we used in our import videos. And it tells you how to use skip rows. And lots of other arguments that you may or may not need to use, but that's always an option to use the help function. And then the last bit of debugging options is to use the actual documentation by the library. Python libraries, especially those used in this class will have a large amount of documentation available. Here I have the pandas documentation open, which I just reached by Googling pandas documentation. And they've got a really nice website here with tutorials and everything. But you can click into their user guide, and you can see all of these different options. And they even have a search feature. So again, you can search read CSV, and it'll come up with the documentation that we just saw using the help function. And so we have these four options for you whenever you are stuck and trying to solve the problem, but you can also always email instructors if you're having a really tough issue.


Commander 788 here and this week we have a special video. This may be the specialist of all of our special videos because this week we have a special guest and that is none other than G.I. Joe Berg. Now you probably already know what G.I. Joe Berg is but in case you don't it is a great G.I. Joe related podcast out of South Africa and they also do some excellent G.I. Joe Toy Review videos right here on YouTube. So for this video G.I. Joe Berg and HC788 come together and we are going to answer the tough questions. We are going to address a burning issue within the G.I. Joe community. Who was G.I. Joe's best mountain trooper? Was it the 1985 alpine or was it the 1988 hit and run? We know you have all been asking this question. You've probably had many arguments about it during family dinners. Well we're gonna take it on right now. Arguing for the 1988 hit and run will be G.I. Joe Berg. Arguing for the 1985 alpine will be Hood and Cobra Commander 788. So I ask you all to lend your focus and attention as G.I. Joe Berg provides the opening argument. The eyes. The eyes. How many vintage Joes have the whites of their eyes painted? Just one. Hit and run. But this debate has a very important very subjective topic which is the best mountaineer on the Joe team. 1985's alpine or 1988's hit and run. G.I. Joe Berg says the title belongs to hit and run and if you're undecided then this review is aimed at you. 88 was a good year. In one year you had a big green amphibious personnel carrier and a rocket ship. You had a purple dude in a bulky suit and you had a bog standard green army man. The very basis for G.I. Joe. That green army man was hit and run. Orphaned after his family was wiped out in a car accident. Absorbed into the system until he was old enough to run away from it and he ran straight into the army where he excelled enough to make the Joe team. He's sneaky, fast, resourceful, all the attributes that make him an excellent infantryman. But hit and run is also a mountaineer of distinction and the figure is well equipped for his purpose. Rope. Lots and lots of rope. And an ingenious and unobtrusive play feature for winding it in and out of his bag. Sculpted snap links. And a big old knife that sheaths on the side. One of the first improvements I made as a youngster was to up the length of the rope and the grapple claw didn't inspire much confidence with its thin plastic. So I managed to find a sturdier substitute. That's not cheating is it? I suppose you could up the length of rope for alpine but where would you put it? Hit and Run's gear makes perfect allowance for additional rope. Advantage hit and run. Forgetting the job done he packs a Colt 9mm submachine gun. The Colt has a reputation for being far more accurate than typical SMGs with an extendable stock and a closed bolt which sets it apart from the open bolt of Alpine's stockless Beretta Model 12 SMG. The Colt weapon is newer having only been first produced in 1982 specifically for the war on drugs. Alpine packs a true relic. His gun was first produced in 1962. Did you know that the Colt weapon accepts Uzi magazines? Yep you know what that means. Hit and Run carries the same ammunition as Snake Eyes. Two man infiltration team anyone? The sculpt is damn fine too. Minimal functional. He's not wearing anything extraneous. Green BDUs with a black harness and helmet. Two black fragmentation grenades. Two pouches for mags or maybe chalk. A red lens akin to low light to acclimatize his eyes to low light conditions. And a clever play feature embedded in the sculpt something which was unprecedented in the vintage line he has a belt loop allowing you to run a rope line through. He can do simulated fast rope drops or zip lining. Awesome. Lastly he means business. I loved this figure as a child but I loved him for about a week before he disappeared into the jungles of my cousin's garden. He's that good at concealment. The camo pattern is devastatingly effective in the mottled black and green shadows of the triple canopy. I don't think there's a more sneaky looking Joe in the vintage line. Nun Chuck notwithstanding and that guy's on the ninja force. How can Alpine be a true mountaineer with gloves that thick? You need to be able to feel the rock. Ask any climber. He's got an open jacket over a shirt. What possible function could that have other than looking trendy as a stand-in on Michael Jackson's thriller video shoot? The shoulder pads would fit right in. And a peak cap. Form BX257 put it best when he said Alpine's sculpt looks like a civilian hiker done up in military greens and browns. The end result is a Joe that looks the parts in color pallets only and not design. The pickaxe is so oversized it would look more suitable in a quarry breaking rocks. And these anchors are just as frail as Hit and Run's grapple but without the handy duffel storage gimmick Hit and Run comes out on top yet again. In closing the best way to summarize the difference in these figures is embedded in their names. When I think of the word Alpine, I think of downhill skiing, glue vine by the fire in the mountain resort. Basically a Peter Stuyvesant commercial. When I think of Hit and Run, I think of death. Death that you don't see coming before it's too late and gone before anyone can do anything about it. Both figures fit their names sake perfectly. I know which Joe I'd rather have on my squad. It's Hit and Run. Well argued G.I. Joe Berg but now it's time to make the case for G.I. Joe's first and best Mountain Trooper Alpine. 1988 was a good year but 1985 was a great year. It introduced some of the most iconic characters of the line. That year reintroduced Snake Eyes in his most memorable uniform. It introduced new character dynamics such as the relationship between Flint and Lady J. And it also introduced G.I. Joe's first dedicated mountaineer Alpine. Hit and Run is an excellent action figure. I've said so before. I think it is a top-tier figure. Although this figure came out at the tail end of my collecting experience, I did have it as a kid. It was one of the few figures from 1988 that kept my interest in G.I. Joe alive. The question however is who is the better Mountain Trooper? The answer is not a slam dunk. Hit and Run has some good points but Alpine comes out on top. Let's compare their weapons. Alpine had a Beretta Model 12 and Hit and Run came with a Colt 9 millimeter submachine gun. In reality Mountain Troopers used weapons more similar to Hit and Runs. However Alpine has also made a good choice of weapon. Mountain fighting could often be close quarters. The enemy may be just a few feet away, just around the next ridge. A short barrel light submachine gun would probably work well in that environment. Both Alpine and Hit and Run's weapons have the same flaw. No straps on the weapon. No way to sling them over the shoulder or store them while the figure is climbing. Both could have used that feature and neither had it. So they come out even in that respect. Let's talk about their climbing equipment which is important because we are talking about mountaineers and look at all this climbing gear that Alpine came with. He came with a grapple hook launcher and these do exist in the real world. This would extend Alpine's reach beyond just how far he could throw a hook. Hit and Run lacks this Alpine came with a pickaxe and this pickaxe is probably more for use on ice rather than rock which would allow Alpine to gain a handhold above the snow line. Hit and Run lacks this which would limit his climbing range. Alpine had a grapple line with hooks on both ends. This grapple line works great in conjunction with the pickaxe which has this little notch and if you run the line through that notch Alpine can use his pickaxe to zip down the line. Alpine's backpack has a space to hold the pickaxe and accessory storage like this is always a bonus. Hit and Run's duffel bag does hold his knife so in that respect they both come out even. Looking at Hit and Run's climbing equipment it is all in the duffel bag which is very large and it has to be to accommodate the hook and real gimmick but with it on the figure it makes the figure top-heavy and difficult to balance. The duffel bag has one hook and a pretty good amount of black line and it can be reeled in that's the gimmick. Now it doesn't have the double hook that Alpine has but the strap of the duffel bag could be hooked on something so you could still get the same effect. The figure does have that loop on his harness but that is not an accessory. The only accessory that he has that has anything to do with mountaineering is this duffel bag and this duffel bag as nice as it is is very large. It's so large that it can interfere with his primary duty to which every other part of the figure points which is jungle warfare and if he drops his duffel bag to fight range vipers and night vipers then he won't be climbing anything. Let's look at the sculpt and color of these figures and this is where Alpine really shines. He does wear gloves as he should. Real mountain troops do wear gloves similar to Alpine's if they didn't the ropes would shred their hands. The paint deco on Alpine is so well done it's hard to believe Hasbro even did it. It must have been very expensive including the base plastic color there are five colors on Alpine's chest. There are three colors on his head. They even painted the carabiners on his left leg silver even though there is no silver paint anywhere else on the figure. The color choices are also excellent. He has a variety of colors that would fit with the lower wooded foothills of a mountain the rocky area above the tree line and even a bit of lighter color for when he approaches the snow line. In short Alpine is colored as a mountaineer. The sculpted details also reflect his specialty. The additional ropes long across his body tells you exactly what his job is. The long sleeves for colder higher altitudes, the gloves for handling rope, the carabiners on his leg, the goggles. Everything about this figure says that he works in the mountains. In contrast Hidden Runs paint is minimalist and not suited for mountain climbing. Hidden Run is an excellent action figure. I love the figure. I want to make sure I emphasize that. I think most GI Joe fans do but he is an infantryman equipped to fight in the jungle. Nearly every millimeter of the figure is specialized for that purpose. The camouflage, the helmet, the red preservation goggles, the face paint, the knife. With the helmet and harness Hidden Run looks more like a paratrooper to me. In fact the target exclusive version of Hidden Run was a paratrooper. Alpine however has numerous sculpted details that work nowhere other than the mountains. He has multiple accessories not just one that enable him to do his job. His coloring fits his job as well. Although Alpine would welcome Hidden Run's assistance up to a certain point I'm sure he would ask Hidden Run to bring some additional equipment and a change of uniform if he wants to join the advanced class. That includes my opening argument for Alpine. GI Joe Berg, have you a rebuttal? Hooded Cobra Commander you make some formidable arguments but I have a few problems. In terms of weapons you say it's a dead heat both weapons are equally matched and yes they both fire nine millimeter ammunition both have an effective range of a hundred meters and are both SMGs but that's where the similarities end. The Colt weapon fires from a closed bolt it has an extendable stock a longer barrel and greater muzzle velocity therefore it has greater accuracy ensuring effective kills. Also Hidden Run carries two fragmentation grenades and a big old combat knife for the silent kill. I doubt Alpine could do much better with his pickaxe at taking out the guards discreetly. In terms of color you say that Alpine is perfectly suited to his role well in Sephora's he wears greens and browns that may be true but then he tops it off with a crisp white shoot me right here t-shirt then again his best buddy runs around the battlefield in a red football jersey so I guess the guys back in 85 weren't taking their jobs too seriously and production value be damned. Hidden Run only has five colors on him but they are perfect for his purpose of concealment. I still can't find my childhood Hidden Run who went MIA 28 years ago in my cousin's backyard. Huda Cobra Commander you also say that the sculpt of Alpine is perfectly suited to mountaineering yes he does have a nicely sculpted rope across his chest but there's one important missing feature something that Hidden Run has in spades and that is a very prominent load bearing harness. Finally the topic is which is the best GI Joe mountaineer. There are two important terms in that topic the first is mountaineer the second is GI Joe you cannot take the one in isolation of the other. Hidden Run is not just a mountaineer he's a good infantryman he's a solid runner he's a parrot trooper he's also a demolitions expert he's lethal and clearly proficient in concealment Alpine might be able to scale rocks but at the end of the day he's just a finance clerk playing soldier. Happy Thanksgiving everyone and have a great November. Oh yeah well Hidden Run is green now it's assumed that he is Caucasian wearing green face paint but is it true is there any evidence of that not only is his face green his neck is green his hands front and back are green his arms are green every bit of exposed skin is green I suspect that that is his true skin color and everybody knows that nothing good has ever come from someone with green skin except for this one and this one and this one and this one and this one and this one okay that may not be my strongest argument. So now GI Joe Berg you are asking us to accept Hidden Run as GI Joe's best mountaineer for reasons now that have nothing to do with mountaineering I would say you are describing Hidden Run as a jack of all trades but that is not the full expression the full expression is a jack of all trades an expert at none and I would agree with that assessment of Hidden Run. He may be competent in many areas but Alpine is great in one he is the greatest GI Joe mountaineer. Now ladies and gentlemen of the jury you have heard the arguments and it is up to you to decide who was the greatest GI Joe mountaineer was it Hidden Run or was it Alpine. Deliberate carefully and we await your decision. Thank you GI Joe Berg this was a fun idea I hope everyone enjoyed it don't forget to look for GI Joe Berg on YouTube and subscribe to their channel and listen to the GI Joe Berg podcast it is filled with GI Joe goodness and make sure you check back on this channel next week for another GI Joe review and next week I have planned something a little different something a little special I hope you like it and until then remember only GI Joe is GI Joe.


Welcome to this lecture. In this lecture, we are going to discuss boundary value problems for Laplace equation. The outline of the lecture is as follows. First we derive what are called Green s identities which are derived from the divergence theorem and then we introduce boundary value problems associated to Laplace equation and study the uniqueness properties of solutions to those boundary value problems. So Laplace equation in d space dimensions, there are only space dimensions here there is unlike the wave equation. So Laplace equation in d dimensions is given by u x1 x1 plus u x2 x2 plus up to u xd xd equal to 0. The operator on the left hand side is called Laplacian u and denoted by capital delta u that is a standard notation for a Laplacian. This is nothing but the trace of the Hessian matrix of the function u. Hessian matrix recall is the matrix of second derivatives of u. So this is the trace of that that means some of the diagonal terms. Any solution to the above equation is called a harmonic function. Laplace equation in 2 dimensions we are going to deal with in some of the lectures which come later on. When d equal to 2 the independent variables x1 x2 are denoted by xy that is a standard practice and we write boldface x sometimes that denotes the tuple x comma y in R2. Thus Laplace equation in 2 independent variables is nothing but u xx plus u yy equal to 0. The non-homogeneous problem where there is a right hand side not 0 function but a general function f. This equation is called Poisson's equation. So in particular if f is equal to 0 get back this equation. So the difference between Poisson equation and Laplace equation is kind of blurred for us we simply call anything as Laplace equation. Actually Laplacian is this operator which is on the left side of both the equations. Here it is a homogeneous equation here is a non-homogeneous equation and the non-homogeneous equation is usually called Poisson equation. So let us discuss what are the green identities and how to derive them. Greens identities play an important role in the analysis of Laplace equation. They are derived from divergence theorem. So what is divergence theorem? It is stated for a domain omega which is bounded and piecewise smooth domain we are going to use in this form. Let psi be a function which is R2 valued function that means psi has 2 component functions psi 1 and psi 2. It is defined on this domain omega. Omega is assumed to be bounded and piecewise smooth domain we will come to that as soon as we see the formula. And we assume that the psi i's have this property that it is c1 of omega bar intersection cf omega bar. It means the derivative is continuous up to the boundary of omega. So omega closure involves omega union boundary of omega. That means derivative of this function psi i is meaningful on the boundary of omega also. And cf omega bar means the functions are meaningful on the boundary of omega. In fact the function psi i's are continuous on omega bar. Then we have this integral omega domain integral. On the right hand side you have a integral on the boundary. So this is divergence of psi dx equal to integral over boundary of omega psi dot n d sigma. What is n? n is a unit outward normal to boundary of omega. And what is d sigma? It is the surface measure which is coming to the boundary of omega from the omega from the usual measure on omega. So in order that the right hand side is meaningful we need to assume that domain is space wise smooth domain. We will not elaborate further on this because you would have learnt this already in course on multivariable calculus. So green signatories they are derived using the divergence theorem on making specific choices for psi in this conclusion of the divergence theorem. So choosing different size will give you different identities. And omega is assumed to be a bounded piecewise smooth domain. n denotes the unit outward normal to boundary of omega. The functions uv appearing in green signatories are assumed to be of this type c2 of omega bar intersection c1 of omega bar. We have already seen this space on the previous slide. This means that the functions second derivatives are also continuous on omega bar. That is the meaning of c2 of omega bar. To make sure that the integrals appearing in them are meaningful. First of all this integral to make sense it is enough that inside the integrand is continuous on omega bar. And that is guaranteed if psi is c1 of omega bar. Later on we are going to see an integral which features second derivatives as the integrand that is why we are assuming c2 of omega bar. It will be very clear from the context. These hypotheses are made so that it is valid for all greens identities. For some of them we do not require all the hypotheses. So now if you apply a divergence theorem with psi equal to grad u what we get is called greens identity 1. So divergence of psi is now divergence of grad u that is Laplace in u. And grad u dot n is precisely the normal derivative of u dou n u. So this is greens identity 1. Now if you apply divergence theorem with this choice of psi we get greens identity 2.   In fact this identity we have used many times before which we called it as integration by parts formula. If you notice here the derivative on v is shifting to the derivative on grad u which is giving you Laplace in u. Grad when it goes to the grad u it becomes divergence. So you get v Laplace in u and this is the boundary term. So let us now discuss some of the boundary value problems for Laplace equation. On a bounded domain omega with piecewise smooth boundary boundary of omega it is boundary of omega is denoted by dou omega. We will consider the following three boundary value problems. What are they? Dirichlet boundary value problem. Unknown function is prescribed on boundary of omega here and then we have a Neumann boundary value problem in which normal derivative of the unknown function is prescribed on boundary of omega. Then we have a Robin boundary value problem which sometimes is also called as third boundary value problem. In this a mix of the unknown function and its normal derivative are prescribed. So linear combination of the unknown function and its normal derivative is prescribed on boundary of omega. So let us see what is a Dirichlet boundary value problem. So this is BVP1 boundary value problem 1 Dirichlet problem. Given functions f and g Dirichlet problem consists of solving the boundary value problem. It consists of solving the Poisson's equation Laplacian u equal to f in omega and u equal to g on the boundary of omega. That means the unknown function should agree with the pre-prescribed function g on the boundary of omega and Laplacian u should coincide with the prescribed function f in omega. This is what we want. What is the meaning of a solution to Dirichlet boundary value problem? Here we have to start assuming something on the data. Let f be a continuous function and g also be a continuous function on omega and boundary of omega respectively. A function phi which is c to omega intersection c of omega bar you will actually see why this hypothesis is coming here on phi. It will be very clear once you see the definition. It is said to be a solution to Dirichlet boundary value problem if Laplacian u equal to f in omega and u equal to g on boundary of omega. This is a statement of the Dirichlet boundary value problem. If the function phi is a solution to Laplacian u equal to f it means that for each x in omega Laplacian phi of x should be equal to f of x. So in order that the left hand side makes sense we are assuming phi is c to of omega that is why c to of omega. Now there is a second condition that is a boundary condition. For each x in boundary of omega phi of x should be equal to g of x. If phi is continuous up to boundary that is c of omega bar then values of phi for points x which are in the boundary is meaningful and asking that it is equal to g of x is meaningful. So that is the reason why we have to put these spaces naturally. Now let us look at the second boundary value problem called Neumann problem. So here given functions f and g Neumann problem consists of solving the boundary value problem Laplacian u equal to f in omega and the normal derivative equal to g on boundary of omega. G and f are prescribed they are given to us. What is the meaning of a solution to the Neumann boundary value problem? Assume that f is continuous on omega and g is continuous on boundary of omega. A function phi which is c to of omega intersection c 1 of omega bar is said to be a solution to Neumann boundary value problem if the function phi is a solution to Laplacian u equal to f that is why we are going to assume phi is c to of omega because of this Laplacian phi at every point x in omega should be equal to f of x. Now we have the boundary condition which involves the normal derivative. Normal derivative is nothing but gradient u dot n it is a directional derivative in the direction of the normal. For that reason we need c 1 of omega bar that means derivatives are also continuous up to boundary and hence this is meaningful. So dou n phi of x is meaningful because phi is c 1 of omega bar and hence we can ask that it should be equal to g x that is why the notion of solution to Neumann problem is like this phi should be in this space. Now let us look at the third boundary value problem which is also known as Robin problem. Given functions f g and alpha is a real number Robin problem consists of solving the boundary value problem equation is Poisson's equation in omega boundary condition is a combination of u and the normal derivative of u u plus alpha times dou n u equal to g on boundary of omega. What is the definition of a solution to Robin boundary value problem? Let f be a continuous function on omega and g be a continuous function on the boundary of omega. A function phi c 2 omega intersection c 1 of omega bar function phi in this space c 2 omega intersection c 1 of omega bar this is natural because this is coming from the requirement of Laplacian phi equal to f and this will come because the requirement of the boundary condition. So this function phi is said to be a solution to Robin boundary value problem if the function phi is a solution to Laplacian u equal to f which means for every x in omega Laplacian phi of x equal to f x. This is the reason why we assume phi is c 2 of omega and second condition is the boundary condition for every x on the boundary of omega the equality phi x plus alpha times dou n phi of x equal to g of x holes in order that the left hand side is meaningful we have assumed that phi is c 1 of omega bar a remark as per the definition of Dirichlet boundary value problem the problem is posed on a bounded domain omega. In fact, the other two boundary value problems are also posed on bounded domains by definition the domain omega is bounded means what enclosed by the boundary of omega boundary of omega omega. For this reason Dirichlet boundary value problem is also called an interior Dirichlet problem. Similarly, the other notions we can say interior Neumann problem, interior Robin problem. On the other hand there are boundary value problems which are posed on domains which are complements of bounded domains Laplace equation is to be solved on a domain omega which is the complement of a bounded domain. For example, my omega is here this is my omega it is a complement of this set which is inside and this is my boundary of omega the this curve is my boundary of omega I am drawing the picture in the plane obviously. So, omega is outside this outside region which is here this is boundary of omega. So, we need to solve Laplacian u let us say equal to f outside and here we are prescribing u equal to g for example, on boundary of omega unknown function is prescribed on the boundary of omega. Such boundary value problems are called exterior Dirichlet problems. Similarly exterior Neumann problems exterior Robin problems. Other kinds of boundary value problems are also possible on different parts of the boundary you prescribe different boundary conditions. For example, this is my omega then I consider this part let us call this boundary of omega 1 omega part 1 this is boundary of omega part 2. So, here I can ask u equal to g here I can ask dou n u equal to some function h. I want to solve Laplacian equation in omega in a equal to f in omega and I ask that on this piece u must be equal to g that is the Dirichlet boundary condition on this piece dou n u equal to h that is a Neumann condition. Such boundary value problems are called mixed boundary value problems we will not be studying such problems in this course. Cauchy-Kowalski theorem this is a comment about the initial value problem. Cauchy-Kowalski theorem guarantees that a solution to an analytic Cauchy problem for an elliptic equation in particular Laplace equation exists and the solution is unique locally. This problem is not always well posed we are going to see towards the end of our discussion on Laplace equation an example of a ill-posed Cauchy problem for Laplace equation. Now, the Neumann problem that we have namely Laplacian u equal to f and dou n u equal to g this is in omega this is on boundary of omega. If this problem has a solution f and g are tied up with some relation. So, you cannot have arbitrary f and g for which the Neumann boundary value problem has a solution. In other words if the Neumann boundary value problem has a solution f and g must be compatible with each other in the sense that we are going to soon describe. For a Neumann boundary value problem to admit a solution the data f and g must be compatible that is written in the form of a lemma it does not mean it is a big result it is a simple observation just to remember the observation so that we can recall whenever we want we record them as a lemma or as a theorem in this case it is a lemma. Let f be C of omega bar we have to assume something more so far in the definition of the problem we needed f in C of omega only but here we are asking for C of omega bar because the tools that we are going to use will involve the continuity up to boundary. In fact, we are going to have certain integrals for them to make sense we need this assumption. And g in C of boundary of omega if you belonging to C2 of omega bar is a solution to Neumann boundary value problem on omega then integral over omega of fx equal to integral over boundary of g g is defined on the boundary so integrate on the boundary f is defined in omega so integrate on omega both of them must be equal. So integrate both sides of this equation Laplace in u equal to f on omega what we get is this okay I have exchanged the sides so integral over f integral of f over omega equal to integral of Laplace in u over omega fine. Now applying Green's identity this right hand side integral can be written as this integral over boundary of omega dou in u d sigma the integral on the right hand side thus becomes this okay same but now I know what is dou in u since u solves the Neumann problem dou in u is g and Laplace in u is f therefore we have this equation reducing to an equation where f is here and g is here so thus we have proved the lemma. In particular if f is 0 that is you are looking at the homogeneous Laplace equation Laplace in u equal to 0 that means f is 0 then integral of g over boundary must be 0. Now let us discuss some uniqueness properties of solutions to the three boundary value problems that we have just introduced the Dirichlet boundary value problem has at most one solution remember here we are not saying about existence at all we are just saying it has at most one solution that means if it has a solution then it has exactly one solution that is the correct conclusion from here. And if you solve the Neumann problem then any other solution looks like u plus c where c is a constant in other words difference of any two solutions to Neumann problem is a constant and for the Robin problem we have to assume something on alpha if alpha is greater than or equal to 0 then the Robin problem has at most one solution recall that Robin problem reduces to Dirichlet problem in alpha equal to 0 so proving 3 is enough to prove 1. So what we do is we just prove 3 and then we prove 2 as I have already mentioned Dirichlet boundary value problem is a special case of Robin boundary value problem it is enough to prove the uniqueness result for Robin boundary value problem please note I am calling here uniqueness result by this what I mean is that this problem admits at most one solution. So assume that u1 and u2 are solutions to the Robin problem the uniqueness proofs always proceed like this you take the difference of u1 and u2 and show that that is 0. So therefore you define a w which is u1 minus u2 we want to show that w is 0 observe that w satisfies Laplacian w equal to 0 because Laplacian u1 equal to f Laplacian u2 equal to f therefore Laplacian w is a difference of Laplacian u1 and Laplacian u2 both of them are f therefore Laplacian w is 0. In other words w is a harmonic function in omega and w satisfies the boundary condition w plus alpha dou n w equal to 0 because u1 satisfies the boundary condition with the same g and u2 also satisfy with the same g therefore the difference will satisfy 0 because this is linear in w. So this is the boundary condition satisfied by w. So using the green side t3 with this is the green side t3 we are going to use this with u equal to v equal to w. So left hand side will be integral omega grad w dot grad w that is mod grad w square. Right hand side Laplacian w is 0 so this term is not there so what we have is this term that is w into dou n w d sigma using the boundary condition we get integral over boundary of omega w dou n w equal to this quantity. Please check that. In the last equation the left hand side is non-negative because integrand is non-negative. Right hand side the integral is non-negative alpha is non-negative therefore the product is non-negative but there is a minus sign so it is non-positive. So we have a non-negative quantity equal to a non-positive quantity which is possible if and only if both the sides are 0. That means what grad w equal to 0 on omega dou n w equal to 0 on boundary of omega. On the last slide we have proved that grad w equal to 0 in omega and also the normal derivative of w equal to 0 on the boundary of omega. Since gradient of w equal to 0 in omega w must be a constant function. Now, since the normal derivative is 0 on boundary of omega if we use the boundary condition we get w equal to 0 on the boundary of omega. But what is w? w is continuous up to boundary w belongs to C of omega bar w is a constant and it is 0 on the boundary therefore w must be 0 in omega also. In other words we have Robin problem has at most one solution. Now let us look at Neumann problem. Let u1 u2 be solutions to Neumann problem. Consider the difference u1 minus u2 call it w. Look at the problems satisfied by w. Laplace in w will be equal to 0 and normal derivative w equal to 0 on the boundary of omega. Now using once again the gradient identity 3 with u equal to v equal to w. We have integral over omega mod grad w square that is coming from the left hand side on the right hand side as before Laplace in w is 0. So, second term drops out what you have is this term which I have written here w dou n w d sigma. But dou n w is 0 therefore this integral is 0. This means grad w equal to 0 in omega that is all we have information nothing more. If grad w is 0 w must be a constant function that means u1 minus u2 is a constant function therefore u1 equal to u2 plus constant. Let us summarize what we did in this lecture. We have derived Green s identities using divergence theorem. Introduce 3 boundary value problems for Laplace equation. Discouraged if there can be more than one solution for each of the 3 VVPs. Thank you.


Sino attended the Western Pacific Naval Symposium, or WPNS, in China April 22. Leaders voted to endorse the Code for Unplanned Encounters at Sea, or CUES. CUES is an agreement between participating nations for safety procedures, basic communications, and maneuvering instructions for ships and aircraft. This should decrease the chance of collisions and misunderstandings that could lead to conflict. Navy Voluntary Education celebrates its birthday May 14, and the Center for Personal and Professional Development is asking for your help. In May, you can submit photos and success stories to their Facebook and Twitter pages using hashtags NavyVoled or Mi5Words. Voled programs include Tuition Assistance, Navy College Program for a Float College Education and U.S. Military Apprentice Ship Program. For more information about the celebration, contact CPPD. Read more about these stories at Navy.mil. From all hands, thanks for watching.


The mysteries present on this planet are shocking to us because they are stretching the boundaries of our common understanding to the breaking point. Overwhelming and everywhere we look, we are observing things that are not only out of place, but they are also not making much sense to us. These certain things would suggest to us that we have completely missed something astonishing. And the only logical answer is if we consider a civilization of advanced technical abilities that was present on this planet before we were. From Malta to Cuba and stretching from Japan to Scotland, we are confronted with structures and artifacts that would suggest that the builders had advanced knowledge of maths, astronomy, geometry, and new of earth positions in the solar system. Not only this, but the ease in which these ancient things appear and the complexities involved in their existence would suggest that they were built and left here deliberately, almost as if an ancient civilization left us a message and we do see major signs of a countdown or a warning clock at Giza, which by the way is the central location of all the earth landmass and the coordinates of its location on the earth just happens to be the same as the speed needed to travel at to reach the speed of light. Coincidence? We think not. Our obsession with the Giza Plateau in particular, the Great Pyramid, is only gathering momentum. From the first glint of interest in these ancient marvels regarding our ancient past, it has only gathered more momentum, more of us are awake, and awakening to these very anomalous clues that overwhelmingly suggest that we have completely missed the point all throughout our struggle to understand history. The crediting of these things to a primitive people is an insult to human intelligence. One such anomaly that is out of place is in that of the Laos stone jars. Wait till you hear this. Laos is a strange place. Despite the country's potential it has remained poor. This is largely because of the power struggles between communist and royalist forces as well as in their participation in the Vietnam War. A series of bad judgments has kept the country from thriving but have tried to make real efforts to open the doors to the world and a few curious minds have made the trip. Of course among the beauty and seeming serenity there are still things left behind from the conflict in the 20th century and this makes Laos a minefield, literally. To be discovered here is a megalithic architectural landscape that is not repeated at any other location on the earth. The plain of jars is so desperately stunning that it is as if it is trying to tell us something. The usefulness of the jars is very apparent. Of course they could have been used as flower pots or as storage chambers for just about anything and nothing significant has ever been recovered that reveals its true original purpose so this leaves our speculation on the rampage. In an effort not to get carried away as to what was taking place here we must first establish an overview and not make immediate connections to ancient batteries like the Baghdad batteries though it is interesting to note that Laos has ambitions to be the battery of Asia. Fueling speculation about its ancient past and in fact if this country knows the true purpose of the plain of jars. These jars aren't secluded to a few hidden jars. They are clusters where they are found and they are still being documented in their hundreds. This is one of the three most enduring archaeological mysteries in the world and it does suggest a previous civilization was more advanced than the current civilization that has emerged with these stone jars already in existence and get this. A researcher working on the site has stated that the site in Laos has seldom ever been visited by humans. In fact he says there is no evidence that a human being has ever investigated the site at all up until now. These sites were first reported on by western archaeologists in 1935. It wasn't until the 1990s that researchers were able to begin systematically exploring the massive region particularly due to the high volume of unexploded bombs left over from the US Air Force offensive during the Vietnam War. The 15 newly discovered jar sites add to more than 90 previously recorded locations indicating the distribution of these artifacts was more widespread than archaeologists assumed. The new sites reveal another 137 massive stone pots found in Mountainous Forest in the Shangguang province. Very little is conclusively known about the people that created these large stone jars or the jars true purpose. The artifacts are thought to date back to sometime between 500 BCE and 500 CE. The general scientific consensus at this stage is the jars were used in ancient burial practices although others suggest they may have been used to store water or even brew alcohol. These new discoveries expand the mystery of the jars by revealing their presence across even larger spans of land and it is completely unknown as to who built these things or what the exact purpose is. It could be that in the very remote past that Laos in fact was producing battery power to power what? Well that is no doubt the stumbling block. There is nothing present today that they would have been powering but consider a site that can recharge something. If a vehicle already had a battery then perhaps this was a recharge site for some sort of vehicle. Just a thought. Where these giant jars are found there is no evidence of occupation at all not even ancient settlements however there is evidence of them having been quarried at a site several miles from these final resting places. All found at the top of hill points weighing several tons each and with no apparent reason for their existence these things are dated to 500 years BC and is one of the strangest mysteries in all of Asia regarding our ancient past. A number of intricately carved large stone discs were discovered alongside the jars at some of these new sites. The imagery on these discs varied from human figures to geometric designs. The archaeologists hypothesized these discs to be decorative burial markers. However the kind of decorative carving recently uncovered at these new sites has not been consistently seen at other sites making it difficult to interpret their purpose. These new sites add to the intriguing mystery of Laos Plain of Jars affirming there is still much more to be discovered. Stifling progress has been the slow clearance of all the unexploded munitions in the region limiting movement for archaeologists however the scientists believe there are probably more undiscovered sites and heavily forested unexplored regions so many clues are yet to be revealed. More than 90 jar sites have been identified within Shanquang province. Each site has from one to four hundred stone jars. The jars vary in height and diameter between one meter and four meter and all are hewn from rock. Their shape is cylindrical with the bottom always wider than the top and identified as granite and limestone rock. These stone jars are undecorated with the exception of a single jar at site one. This jar has a human frogman base relief carved on the exterior. Parallels between the frogman and the rock painting at Hua Shan in Guangxi China have been drawn. The Chinese paintings which depict large full frontal images of humans with arms raised and knees bent are dated to 500 bc the same date estimate. Since most of the jars have lip rims it is thought that the jars originally supported lids although a few stone lids have ever been recovered. This may suggest that the bulk of the lids were fashioned from perishable material. Stone lids with animal carvings have been found at a few sites such as Ban Fuqiao site 52. The base relief carvings are thought to depict monkeys tigers and frogs. Between 1964 and 1973 the plane of jars was heavily bombed by the U.S. Air Force operating against North Vietnamese and Patet Lao communist forces. The U.S. Air Force dropped more bombs on Laos primarily the plane of jars than it dropped during the whole of World War II. This included 262 million anti-personnel cluster bombs and an estimated 80 million of these did not explode and remain a deadly threat to the population to this day. Lao legends tell of a race of giants who inhabited the area and who were ruled by a king named Kung Cheung who fought a long and ultimately victorious battle against an enemy. He supposedly created the jars to brew and store huge amounts of Lao huai. Lao means alcohol, huai means jar. So Lao huai means rice beer or rice wine in the jars to celebrate his victory. Another local story states the jars were molded from natural materials including clay, sand, sugar and animal products in a type of stone mix. This led the locals to believe that the cave at site one was actually a kiln and the jars were fired there and are not actually hewn from stone. Another suggested explanation for the jars is to collect monsoon rainwater for caravan travelers along their journey at times when rain may have been seasonal and water was not readily available on the easiest footpaths. Rainwater would then be boiled even if stagnant to become potable again a practice long understood in eastern Eurasia. The trade caravans that camped around these jars could have placed beads inside them as offerings accompanying prayers for rain or the beads might simply have been unassociated lost items that were found nearby. Whatever the true purpose may be whatever reasons these things exist the fact we don't know but are trying to understand would suggest that yet again these things are the workings of a lost civilization and we will probably never know what these things were and the fact is that at some point someone will come along and say they are this or they are that and the majority of us will believe this at face value no matter how flawed the theory may be. What do you guys make of these very ancient things? Comments below and as always thank you for watching.