The first of two crucial vehicles for ALMA, the Atacama Large Millimetre Sub-Millimetre Array, rolled out of its hangar and passed successfully a long series of tests. The ALMA Transporter is a rather exceptional vehicle, supported by 28 tyres, that is 10 metres wide, 6 metres high and 20 metres long, weighs 130 tonnes and has as much power as two Formula One engines. These specially designated transporters are necessary to move the ALMA antennas, each weighing more than 100 tonnes. When completed on the high altitude Chaknaan Tor site in Chile, ALMA is expected to comprise at least 66 antennas, which can be placed in different locations on the plateau, but which work together as one giant telescope. Changing the relative positions of the antennas, and thus also the configuration of the array, allows for different observing modes, comparable to using a zoom lens on a camera. The ALMA radio telescopes will be assembled at an altitude of 2,900 metres. The transporter then will bring the antennas up to the observatory located at 5,000 metres. Furthermore, we do have various configurations of the observatory, and the transporter will move the antennas from the compact configuration of about 100-200 metres diameter to the extended configuration, which stretches up to 15 kilometres. Given their important functions, both for the scientific work and in transporting high-tech antennas with the required care, the vehicles must live up to very demanding operational requirements. Each transporter has a mass of 130 tonnes and is able to lift and transport antennas of 115 tonnes. They must be able to place the antennas on the docking pads with millimetric precision.  This means negotiating a 28 kilometres long, high-altitude road with an average slope of 7%. Finally, as they will be operated at an altitude with significantly reduced oxygen levels, a range of redundant safety devices protect both personnel and equipment from possible mishaps or accidents. In order to operate the transporter at the ALMA site, engines with 1,700 horsepower are installed. At this high altitude, the power output of the engine is limited to 860 horsepower. All the components have been checked to meet the requirements at this extreme conditions. The human factor is also considered, for example, the backrests of the driver's seat are shaped to allow the driver to wear his oxygen tank during driving. ALMA will observe in the millimetre and submillimetre wavelengths and this means the atmosphere above the telescope must be transparent. This requires a site that is high and dry. For this reason, ALMA is installed at the 5,000 metre high plateau of Chadnan Tor in the Adekama Desert of Chile, probably the world's driest, the next best thing to outer space for these observations. The biggest challenge was the environmental condition at ALMA site. With an altitude of 5,000 metres and a drive distance of 28 kilometres uphill with a maximum slope of 10%. To drive this road downhill, we have to install a special brake system. Also, the transporter is very user-friendly to take into account the reduced capability of the operators at this high altitude. ALMA will be the forefront instrument for studying the cool universe. The relic radiation of the Big Bang and the molecular gas and dust that constitute the very building blocks of stars, planetary systems, galaxies and life itself. The ability to move antennas to reconfigure the array is vital to fulfilling ALMA's scientific mission. The operations plan calls for moving antennas on a daily basis to provide the flexibility that would be such a big part of ALMA's scientific value. The ALMA transporters are therefore most important and the present delivery is a significant milestone.


Presently, the Eisenhower carrier strike group is proceeding with the mission to deliver air power from the Arabian Sea, remaining vigilant and maintaining conditions that enhance trust, cooperation and mutual respect. Freedom Sentinel has assisted in the drawdown of up to 44 percent of U.S. and coalition forces from Afghanistan. Over 370 air operations were completed in the last week. We are Mighty Ike.


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.  His 2018 verse on the remix of Cardi B's I Like It first brought him to the mainstream's attention, and ever since then, his gigantic social media following has transformed this 28-year-old into a superstar whose music has dominated the global market. By 2020, Bad Bunny had become the most streamed artist on the planet, and in 2022, his world tour became the highest grossing of the year, ranking in over $230 million in earnings, a new worldwide record. As you might imagine with profit margins like those, Bad Bunny has been so busy, he's barely had any time to return to his main residence, located in San Juan, Puerto Rico. There, Bad Bunny rents an apartment with his girlfriend, a guerrilla of Berlingueri, where the two spent the vast majority of the pandemic a few years ago sheltering at place with one another in this apartment as he revealed rolling stone during a profile written on him in 2020. Outside of that one media hit in the rare Instagram live clip, not much else has been revealed about Bunny's San Juan place. Instead, whenever he's asked to talk about his home, he likes to reflect on some of his favorite neighborhoods, like La Perla, where he often travels to listen to music, enjoy the nightlife, as well as play basketball and dominoes. Having traveled all around the world multiple times, Bad Bunny still considers Puerto Rico his favorite place to go, especially whenever he gets the opportunity to visit his hometown of Vega Baja. But if you're looking for more specific details when it comes to where Bad Bunny rests his weary head at nights, then we'll have to head over to Miami, Florida, where the international artist has created his very own Airbnb. After staying in a few Airbnbs while visiting Florida for work, the artist decided to take the unusual step of opening up his own in the city of Miami, which fans could then rent out for one night only at a rate of $91 a visit, a nod to the record-breaking 9.1 billion streams he achieved on Spotify in 2021. The thing is, this place, it's not a home. It's not even an apartment. Instead, Bad Bunny decided to build it inside of a semi-truck and designed the entire thing to include a kitchen, living room and bedroom, to come together as a celebration of his life and career. The 53-foot-long West Coast Customs-designed rig boast striking matte black finish across its exterior. Then, on the inside, you'll find plenty of bright colors, bold patterns, and even luxurious furniture that personifies Bad Bunny's one-of-a-kind style. When discussing why he selected a semi-truck as the location for his Airbnb, Bad Bunny said in a press release, I'm hoping that by hosting guests in my truck, I can give them a chance to feel like they're on tour with me. This truck has played such a big role in the concept of my tour and my last album that I want to share this unique experience with them. As hinted at in that statement, this Airbnb draws inspiration from some of Bunny's most popular music videos, including a bedroom decked out with flowers like in Yo Perio Sola, as well as other spaces that elude to the aesthetic seen in the likes of La Noche de A Noche. Now, of course, this was all a limited time offer and one that by now is run its course, but lucky fans who got to experience this for themselves last year also received VIP tickets to Bad Bunny's sold-out tour in Miami, as well as a virtual greeting from the artist himself when they arrived at their destination. While his ties to Miami might not have been permanent, recently Bad Bunny has taken steps towards buying his first real piece of property in the United States. A new home located in Los Angeles, California. The man that's frequently been referred to as the world's biggest star now has a piece of property that truly cements his A-list status after saying, I like it like that to an 8.8 million dollar LA estate for just high up on the Hollywood Hills above the city's bustling Sunset Strip. Though the house was never officially listed at the time Bad Bunny made his acquisition, it had previously been placed on the market last year with an asking price of $9 million. The first thing you'll notice about Bad Bunny's new Los Angeles zip code is that the property boasts gates and an enormous wall that shields the home from public view and offers parking for up to as many as six cars. The nearby garage is attached to a two-story guest house that's just a short walk from the main house and when combined both structures have a total number of eight bedrooms, seven bathrooms and just under 8,000 square feet of living space. Recently renovated Bad Bunny's main house offers whitewashed interiors and lavish contemporary finishes. Inside the main house is set up with an open plan living space that includes a sunken living room located next to a chef's kitchen that's been decked out with multiple ovens, a large Eden Island and a series of sliding doors that open seamlessly to the outdoors. Not far from there is the home's gorgeous dining room that's been decorated with a giant satellite style chandelier and also features a wet bar that blends into the nearby living room. Meanwhile upstairs the primary suite includes a private balcony as well as an oversized walk-in closet that comes complete with its own sitting area. There's also an all-white ensuite bathroom that boasts a huge glass-walled rain shower as well as a soaking tub nestled into a windowed alcove. Situated on a half-acre lot of land, this property also features something of a rarity in this area of the Hollywood Hills.  Better yet, the home also offers a breathtaking panoramic views of the city lights and nearby wildlife. Take all of that into consideration and you can see exactly why someone as successful as Bad Bunny is ready to call this place home. But before we finish our tour for the day, let's take a look at one last way that this Puerto Rican artist has put down roots in the U.S. When you're as big a deal as Bad Bunny is, at some point it only makes sense to diversify your career portfolio and I'm not just referring to this international superstar becoming a part-time wrestler in WWE. No, he's also decided to create and open a lushly decorated Japanese state coast in Miami, Florida called Gecko. In an interview with Food & Wine, the rapper explained why he chose Miami as the place to open his first eatery, stating, The culture reminds me of home. Miami is a city where I can relax. I love to go out to eat, see the ocean. It's just a great city. Just a few weeks after announcing his newest business, the front doors of Gecko were thrown open in a star-studded debut. According to a press release, Bad Bunny's new restaurant can house up to 185 guests across in-nar and outdoor spaces. It was designed by the Rockwell Group and boasts a vibe-heavy jewel-toned decor, including a custom gold and red dip-dye rope installation, graphic wall coverings, velvet drapery, as well as backlit cutouts and bars to enhance the nightclub-like vibe. And as for its menu, this Japanese steakhouse offers shared plates such as Japanese milk bread, lava and ice oysters, lobster dumplings, and a wedge salad with buttermilk dressing. From there, you can move on to raw plates of sushi and sashimi. As for the steaks, expect tomahawk cuts, olive-fed filet mignon from Japan, wagyu skirt steak, and bone and ribeye. And if you prefer your meat a little less red, there are also plant-based chicken options and spicy lobster. I gotta be honest, reading those last couple of paragraphs out loud has left me super hungry for lunch. So, I'm gonna bring this tour to a close. Thanks so much for joining me in this look into the homes of Bad Bunny. Before you leave, consider answering the following question. If you opened an Airbnb meant to encapsulate your life, where would you put it? Let me know if you'd select something even more random than a long-haul truck in the comments down below. Otherwise, like, subscribe, and turn on your notifications to never miss an episode. My name is Kara. If you liked today's tour, then keep watching as we enter the homes of Cardi B and her hubby offset. I'll see you all next time. Bye. Cardi B has been telling us since her 2018 hit song I Like It that nobody stuns half as hard as she does. And this little tidbit of information has never been truer than it is today after this freshly turned 29-year-old received a truly epic birthday gift from her husband, rapper Offset. And no, I'm not talking about their newly delivered beautiful baby boy. Instead, I'm referring to the $1.49 million investment property they just scooped up in the heart of the Dominican Republic. And today, we're going to check it out. 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. Cardi B is a rapper who first kicked off her career as an internet celebrity. These days, she's worth an estimated $24 million, and she lives with her husband Offset and their children in a gorgeous Atlanta mansion. Now, the happy family also has the vacation home of their dreams. Cardi has long been imagining herself owning a rental property, and now that day has finally arrived. She wrote on Instagram, For a hot minute now, I've been telling Seth that I really want to invest in a short-term home rental property in the DR and other Caribbean countries, but I felt like he didn't agree with me and would rather put money into other investments. While, as Cardi herself soon found out, she was completely wrong. Hey guys, it's Kara the Vampire Slayer back with another exclusive house tour here on Famous Entertainment. We're going to get an in-slide look at Cardi B's tropical paradise getaway that you too could potentially stay at. Be sure to subscribe and ring that bell for notifications because we post brand new videos daily. As always, you can follow me over on Instagram to chat, and now let's get into this video. Let's get one thing clear off the top. As unbelievably gorgeous as this new home is, you need to keep in mind that it isn't even Cardi B or Offset's primary place of residence. After all, how many Hollywood celebrities do you know like to shun their entire industry by living their days off continent on a Caribbean island somewhere? Yes, I'm aware that's basically how Sean Connery spent a lot of his final decades on this earth, but let's be honest, that's one ornery and unique Scotsman right there. And while Cardi B and Offset are most certainly unique, they aren't quite as anti-socialist Connery could sometimes be, which is why they'll still be spending most of their days living in their mansion out in Atlanta, Georgia. A while ago we took a look at that place with its very own house tour, so if you're interested in seeing inside their everyday family home, be sure to check that video out when you're through with this one. But for now, let's kick things off by walking you through how Cardi received one of the biggest surprises of her life. It all started with a birthday celebration that was thrown for Cardi a short time ago in LA, a dance hall-themed event that was full of Hollywood superstars. At one point during the night, Offset took to the stage to shock everyone in the room, announcing that since he had already bought Cardi some of the biggest diamonds in the world, next, he was going to get her something that meant even more. That's when he dropped the bombshell by revealing that he had bought Cardi her dream vacation home in the Dominican Republic. After hearing this news, Cardi B was left speechless, which I'm assuming was probably one of the first times that's ever happened to her in her life. And really, who could blame her? Just wait till you get a peek inside this place. While it's unknown exactly how much Offset wound up spending on this property, it should be noted that prior to this sale, this seven-bedroom, seven-bathroom mansion, situated on 8,000 square feet of hillside, had previously been on the market for a listed price of $1.49 million. This four-story residence is located in a private and gated community in the resort town of Elastarenas that boasts three outdoor tennis courts, private beach access, and even its own helipad. Moving on to the property itself, this home features not one but two primary entrances. Once you've stepped foot inside, you'll quickly come to realize that one of the coolest things about this home is how much it embraces its surrounding landscape by using mostly natural materials throughout the home's interior construction. The gorgeous texture of exposed wood is what especially stands out as a pop of color against the otherwise white walls of the inside of the home. And nowhere is this any clearer than when you're looking at the central staircase that simply put could take your breath away with its expert craftsmanship. Further into the home, the kitchen offers up a double fridge alongside the standard appliances like a microwave, burner, gas stove, and island bar with classic seating arrangement. The nearby dining room is connected to the kitchen and offers breathtaking views of the surrounding area thanks to its floor to ceiling windows. Moving on to the living room, things might look a little cramped in here, but there's no shortage of seating options and that flat screen television is almost always going to have some steep competition with the dramatic views going on just outside of the nearby sliding door. Of the seven bedrooms that are available, I don't think anyone is going to feel put out if one room gets claimed before another because they're almost all the same size. But I will say that whoever winds up with the bedroom that walks directly out to the lap pool is definitely the winner in my book. Not to mention up on the roof of the building, there's even more going on. I'm talking a rooftop infinity pool with built-in lounge chairs, fountains, and picturesque views of the surrounding lush greenery. There's also that smaller-sized lap pool that's available for those looking to be a little more active-minded with their downtime. I don't know about you guys, but I'm going for the infinity pool and a floaty type of vibe. Finally, there's a rooftop dining area that comes complete with a grill, sink, and overhead awning for those rare nights when that island weather just isn't cooperating with whatever you have planned. Oh, and did I mention the gym because yep, that's up there too. Last but not least, the property also features an individual one-bedroom studio that has its own living room, bedroom, and bathroom. As I mentioned earlier in the video, despite how amazing this home obviously is, the primary reason that Offset bought it in the first place was so that Cardi could finally live out her dream of having an investment property. After Offset announced his surprise gift to his wife, Cardi, took to Instagram to let the world know how happy she was that her hubby had been listening to her the entire time, even when she thought he hadn't been. She wrote, I'm so happy that you are actually listening to me and not just smiling and nodding your head so I could stop talking to you about it. You don't think my investment ideas are whack. And while it's unclear how much exactly Cardi stands to make from this new property, you can already rent the place out yourself over on Airbnb where it's available at around $836 a night. I'll do some quick math just to give you an idea for how much Cardi might be raking in. So at $836 a night, let's assume that a place this nice gets rented out at least five nights a week. That gives us a weekly haul of around $4,200 give or take. That $4,200 would turn into about $17,000 a month, which means that annually, Cardi and Offset would be clearing around $200,000, not so shabby. In about a decade or even less, this place will have paid for itself. So now we've seen Cardi be his beautiful new Dominican Oasis and that brings us to the end of our house tour. What do you guys think about Cardi's newest business idea? Would you rather just live in the stream home yourself rather than rent it out? Which was your favorite room or spot in the vacation mansion? Let me know your thoughts in the comments down below and be sure to like and subscribe to our content. As always, thanks for watching. Don't forget to follow me on Instagram and I will see you all in the next video. Bye!


What's good Josh boros back again with another video. So just finished up doing the live stream reaction of smackdown with the homie dub On the inter clutch page shout out to everyone that was a part of that stream We always appreciate y'all when y'all are in the stream with us It makes the shows that much better But we have to talk about the Cody Rose and Roman Reigns promo segment started off the show Their first time interacting on the road to WrestleMania The past few weeks has been Paul Heyman kind of relaying messages or whatnot to get under Cody's skin So Cody comes out there or whatnot crowds, you know Loving his entrance as they usually do and he's in the ring and then you have Roman Paul Heyman solo Sakura in Jimmy Uso and Cody starts the promo off saying like look You know, I'm here really to talk to you 101. I don't see why everyone else is here is needed Unless you feel like they're needed. So Roman tells everybody get out the ring. It's just gonna be Roman and and Cody Rhodes one-on-one man-to-man Promo back and forth and I appreciated that because the attention and focus was strictly on them because we all know there's another storyline going on with Jimmy and Sammy Zane and that the bloodline situation and Jay Uso That's another storyline within this storyline So I'm glad that they just focused on these two in the ring This was a great segment and I like the fact Roman laid the titles kind of threw him down on the ground But kind of just put it all out there I like the fact that Roman Started to you know, really turn up the heel persona Like he wasn't outwardly being like just you know Evil for the sake of being evil or saying you can't get the job done or whatever. It was more so of like Psychological like really trying to get under his skin and I like that he started Simply with the have you ever won this title before Tom about the WWE Universal Championship Then he talked about have you ever won the WWE Championship then he's like have you ever main event or WrestleMania and Crowd is eating this up because he you know Romans like Bob. I've been there I've done all of this, you know I'm saying and I like Cody was also trying to prove his points of you know, his his struggles are like I've always broken past the barriers that I wasn't supposed to break even mentioned him leaving WWE getting past the Stardust gimmick leaving WWE finding a name for himself again starting up a Indie wrestling company, you know pretty much mentioned in AEW out saying it and how he's always shattered through people's expectations and This is an a big a big Expectation that people don't think he's gonna be able to complete and he's here to do that And I like how Roman is just still trying to knock him down like I get that But you haven't even fought for any of these titles let alone main event of WrestleMania What makes you think you're gonna be me and then he started getting into his father, you know It's very sensitive subject for Cody and I like this. This is when Roman start getting in his real heel back He's like, I'm not gonna disrespect your dad. Your dad has done everything for me Your dad has laid out the blueprint everything that he said I will become your dad Was right on in the crowd cheering because I'm pretty sure there's a true story to that, you know I'm sure Dusty was a very pivotal point of very pivotal person for Roman Reigns Development and confidence. He even started talking like Dusty which was quite hilarious. I loved it You know try to impersonate him not in a disrespectful way, but more than like a respectful and daring way. Oh and not But things started taking a turn for worse in this promo segment because then he was like, yeah, man We talked about so many things about life everything The one thing we didn't talk about Was you and that's when That's when things started to you know, irritate Cody, bro You know, this is the story. They're always trying to bring up Dusty was a great man. You're saying you're doing this for your dad but He's the one that helped Roman get to where he is and he's never talked about you his son Huh, it's very interesting and I like that.  He's like No, that wasn't you sending Paul Heyman these couple of weeks just to get under my skin. No That's what you believe that's how you feel Well, now it's even more imperative that I win now.  There's no pull apart. I'm sure it's gonna get to that point but It wasn't there yet. They're slowly building it and I like that interaction. That was a cool Interaction nothing else needed to happen. I enjoyed all of that. It was just them two in the ring Roman playing mind games trying to get under his skin. I love it. That was fantastic I am looking forward to them Having some more interactions I'm sure we'll get that as we get closer to WrestleMania, but I like it they kept it chill at Some point it's not gonna be chill. I know that and I can't wait for it to really get personal I'm really wanting Cody because Roman's going to keep pushing that button to get get as disrespectful as he possibly can and I want to be able to see the moment when Cody says, you know what screw it and he starts getting Personal and he starts digging in where it hurts and he gets under Roman skin And now it's gonna get to that point where now it's not even just only about the championships now It's it's personal. There's more personal Beef on both sides and I can't wait to see that when if it does happen in it if it does get to that point So I'm coming down below Let me know did you guys enjoy that segment as much as I did because I know that was fantastic Great way to start off back down tonight. It was great You know, let me know how y'all feel about that segment and where do you guys want the segment? They're their future promos leading up to WrestleMania. Where do you want it to go for? Do you want to kind of stay where it is? What what lane do you guys think should be taken to intensify this feud? Let me know down below, but I appreciate all love and support you guys have shown on the channel road to 150k and I am still young to speed a YouTube wrestling champion the world and Still in the clutch world every way champion preachers. Okay, me You


to have to like wait in line and to like wait for all the acceptance They wanted us to like roll the red carpet out for them Oh, yeah, that's that's actually like russia could have been part of the western sort of power structure They just like refused because like, you know, fuck it. We're we're empire. We're big big boys Look where look where they got them Not true, uh, russia in the beginning in late 90s and the early 2000s several times Made overtures to the west about russia being part of nato and they were flatly Denied No, the the the the the denial was that okay the guys will leave it up. It's not going to go immediately The the the denial was that it's going to have to be a process and I'd anyway And I'd leave it up to the viewers to research them Oh, please do please do Okay, I don't know what this means. So maybe one of you guys can uh, give me some levity on this Thoughts on paul joseph watson's comments.  So I guess I don't know. It's pretty crazy, but Okay, keep going them This is from ethan steinberg and this is for justice. Should the united states stop providing weapons to ukraine question of whether or not the united states should be currently providing weapons to the to the ukraine is a little bit like the question of Whether the united states should enter world war two After, you know, the germ, not nazi germany has taken over all of europe or most of europe and has invaded the soya union The question isn't what we should we should or shouldn't be doing now The question is what we should or shouldn't have been doing to avoid this this inevitable conclusion of events in 97 in 98 and throughout the 2000s our own foreign policy establishment warned us in 97 98 99 when clinton was beginning to Escalator or expand nato that this was what was going to happen the same foreign policy establishment that helped us win The cold war warned the clinton administration that this was the inevitable result of nato expansion So should the west help ukraine uh And again, i'm putting myself into some pretty uh dames position here trying to answer this honestly, but should the west help the ukraine Rebuff the russian invasion at this point in the game I don't think there's any other option. We should look for for exits. We should try to find overtures for peace But of course, you know, you you don't want to land war in in europe obviously like that's But but on the same time it's this is what my whole point is Let's look at the broader picture and look at what led to this not not just looking this again um, it's it's the question of we should we should have tried Should the treaty of versi have been different? Um, and might might that have uh helped Mitigate the world war two or or or not allowed for it as opposed to whether or not we should get involved in world war two Yeah, careful careful with the answers. I hear I hear in fascist countries. There's a risk that you might go to jail If you say their own things on air so Next question. I missed that was that a question adam. There was breaking up. No, no Next question is from k. I what are your thoughts on western generals like mr. George bush and mr. Girls imperialist actions against russia. I think that's for justice I was mr. George bush girls something breaking up Oh, sorry. Okay. Let me say that. It's a dot meme question. Go to the next one. Okay, got it Okay, thanks question a little off topic others from vish But what do you all think this could mean for china and taiwan? I feel like the situation between china and taiwan is is markedly different like the relationship between taiwan and china and You know, whether that is like a truly independent country, which is not in the way that ukraine was from 91 onwards I just I think the situations are dissimilar. I don't think you can compare that to Yeah, I agree I think they're dissimilar But but I think that there are lessons being learned and I think to one at one at one degree The chinese are seeing What the international community's response might be to Something like a reassessing of power over taiwan And and hopefully, you know taking notes Okay, actually I just finished a video on this that i'm going to release this thursday So if anybody wants to watch that of course ilmer and stavey on youtube, uh shameless plug Next one mr. Saksake for twenty dollars. Will the real kremlin gremlin? Please stand up. Please stand up. Oh my god. He's standing goddamn hazer short amazing Next one Yes, adam is the kremlin gremlin. Oh gee snoop So slavery didn't play a role in shaping the south of today What a dumb point of view Slavery did shape jim crow and current economic segregation I think it's a real debate or something All right, okay Next is from ambient wave With rumors of putin soon to announce full-scale military mobilization How would this development affect the current conflict in ukraine? I don't think that will happen. It might there's a chance but actually, you know So, okay complicated question russia short answer russia Has had enough troubles with their trained forces already, right? I mean they fucked up enough with them like especially with the baby, you know And to to to order a full-scale country by mobilization that would take months to train up people, you know And and to to to be actually To train them up to at least to be marginally better already to send them in and then we haven't even talked about moral issues supply issues hardware etc Plus the sanctions the sanctions effect on the direction between industrial complex Also in russia the funny situation is that they would also need to train the trainers To train the to train the conscripts that they would get so mass mobilization might happen Although it's not it's not going to have a substantial effect on the war in ukraine Adam something likes to point out um russian ineptitude in the ukraine and likes to make funny posts about uh russian equipment looking funny and that's ha ha ha there's so much of it But but the but the reality the reality is is that Russia in its invasion of ukraine has made the mistake that large countries often make When they do not uh assess their opponent correctly much like the united states north in the civil war Where they thought they would have the war over in two or three weeks and it took four to five years Um, so will russia declare war on ukraine there are both pros and cons of that The pros of that would be that they could legitimately then use the their conscript army Again, this is another propaganda from the west russia doesn't have a concrete conscript army as much as it has mandatory service um and part of the the law around mandatory service is that you cannot use the people who are the the men who are in the army for mandatory service in Wars that are outside of russia's russia's borders So there's definitely a pro argument to declaring wars that you can use the vast majority of the reserves To to to advance the war in the ukraine That is the pro the con is that that would go against the the particular propagandistic narrative that is in russia right now, which is this is not a war This is a internal conflict between brother nations And this is a special military operation and it's just tactical That would would give total would totally belie that narrative. And so there's definitely a con to to doing that. I I definitely of course as the pro piece Party in this debate Hope that russia does not Declare war and does not use its its conscript army if the war drags out long enough russia will win It's it is unless the western powers actually put troops on the ground Which then will have a lot of other things to worry about than whether ukraine will fall or not Well, I guess we'll see Okay, uh, this next one is from gombi This is a hundred check krona, which either is a lot of money or not a lot of money. I have about four dollars amazing For the pro russia side, which I mean I seem means you Are we supposed to believe russia is intervening to prevent fascism? When the same lie was used before by the soviet union to justify its evasion of hungary and shekel savakia Russia has legitimate history issues with fascism.  Like it's it's ready made. So of course, that's part of the propaganda. No russia did not invade ukraine to Um stop fascist brigades. It's part of the propaganda. It's part of the narrative Um, and it's and it's part of the whole issue that you have to they have to take into consideration The main reasons for the invasion of the ukraine was legitimate security concerns advancement of nato um, western western orienting, uh Countries in the buffer zone around russia. That is that the issue Okay, the next question is this is from a ukrainian Uh stanji Uh justice you are simply wrong. All battalions were either disbanded or integrated into the ukrainian armed forces by 2017 I said out as I countered two destinies in 2014. They were integrated. I said they were not integrated in 2014 My dear friend, I'm not right. I'm right. Sorry that I'm speaking russian. I would speak ukrainian if I could What? He called you a beta destiny Going to the next one. This is from streaks Uh, 10 dollars can just as common on the Wagner group the russian pmc group that is essentially an unofficial wing of the russian military Whose leader is a confirmed neo nazi? Uh founder, I believe. I don't know if they're the current leader still Being being an uh pro russian ethnic natural nationalist is not the same as a historical neo nazi That's a but b the Wagner group and its That's reason is like the entire war undistifiable And russia is playing with with with fire with its own with its own stuff. So this is absolutely You know, it's bad bad bad bad stuff. No no no good Okay, next one dorth else. Can I ask a personal question? To hell. Yeah, uh, jesus Okay, I think a lot of people have seen like the the tattoo, right? Like the I don't know. I'm a clean. No, not not you about About the founder of the Wagner group with you know, the s s and the In the in the swastika with them with the nazi eagle. What is that about? Um, what a good question That is uh a good question. Um, but again, I would say that it's it's about, um I oh wait a second. I was just not aware of that. I guess. Um, so it's an it's in the five foot on that one It's an it's an it's an it's an ironic as a symbol So there's a very famous there's a very famous photo of the leader of the Wagner group with a shirt off and you can see A swastika with the nazi eagle and s s it do maybe I mean, I've seen it like a million times Maybe it's it's a fake and I have not been made aware of it But I I I've been Dylan told me about the reason why people the reason why people don't usually take it One second one second This is a private okay, right? Yeah, I would like to hear the swastika. I'm genuinely curious by the way I'll two two points on that one wrote number one. I'm not here to defend the Wagner group and the Uh work of paramilitary groups in the Dumbos Dumbos and Lugansk region Russian paramilitary groups. That's unjustified I don't I don't justify it. I don't think it's right number one So I'm not so familiar with their history and and and the the exact political Um affiliations of that of that group although I'm aware I even had unfortunately acquaintances who went to fight in the Dumbosk and Lugansk region So I'm at least that to that degree familiar with with situation number two It feels like you are definitely violating the neutrality of the platform I'm sorry, I was just I was just definitely curious. I I did I did I didn't want to ask before if I could ask I'm very sorry, but we can continue. Um to do um This is from darth elsa again considering russia has uh completely surrounded at mario poll maybe the threat of the asov nazi super soldiers is exaggerated Um, we've kind of been over this like a million times. Can we just keep going? I think we can just keep going, right? Yeah, keep going. Okay. This is from translucent man Isn't russia to blame for the anti russian sentiment in many of the ex-soviet countries does decades of subjugation of those nations to people not count as russian provocation Absolutely, and so eastern european and central european countries who want to be part of a security alliance or security uh regime of some kind that can keep them out of The militaristic influence of their largest neighbor is understandable I'm not i'm never have I said that those that those motivations are not understandable But again, we have to look at both sides of the motivational schema in order to make sense of the world and not repeat Mistakes that cause more conflict and not less The next one's from a u studios justice said russia is not fascist But he's worried about being arrested for not being strongly pro-war slash pro-putin if they weren't authoritarian Why would they arrest you when? You're are clearly 99 percent there. I don't know what the end of it says but you get to just the question Fascism fascism In general nazism in particular Which is the allegation that's floated most often in modern debates about Anything you don't like it's kind of the go-to whipping boy Is a particular form of right wing tyranny and authoritarianism russia currently is far more authoritarian than uh Our western counterparts. Uh, there's no question about that. Uh situation, uh, of which i'm not necessarily happy, of course That's that's that's unfortunate that that that's the the path that we have chosen the russian people have chosen to To follow and to go down.  There can be left-wing authoritarian regimes There can be centrist authoritarian regimes so forth and so on Okay, um now we've gone through all of the questions that have been pre approved Do we have any time left for the uh other questions like as? Um, there was one more from cup of tea. They were just asking. What does everyone's take on the Uyghurs? um I mean, it's it's a genocide by china. He fought right regime justice it's a genocide by china much like the war in yemen uh supported by the united states is a genocide against the yemenis All right, okay, uh Okay, should we go on to the warning given questions or? um Well, destiny is not here. I think we should just go ahead and uh close it out. Um, Yeah, wonderful. Yeah, um, I just want to let everybody know. Uh, thank you so much for joining us tonight. Um All of our debaters are linked in the description below whether you're listening on podcast So please check them out if you like what you heard tonight and please uh subscribe Because we have more juicy debates coming up for your enjoyment And uh, just want to let you know that want to thank dylan for helping us put this all together Getting adam something justice together. Uh, hopefully we have more Collaborations in the future. Thank you to justice adam something destiny and connor for being here Um, so again like it if you loved it share if you want to spread it and subscribe We have many more debates coming your way. Um Thank you. Have a great night and keep sifting out the reasonable from the unreasonable and have a great night everybody Fantastic, we're out of here. Good night everyone. Thank you guys and best wishes to james Nice job, uh justice for hanging in there for the two and one at the end. Yeah. Yeah truly All right, so thank you for what?


imistic in terms of moving into something even more close to normal, even though it's not. But I have every confidence in the licensees in our employees that they will do exactly what they want and more to make sure that this is as safe as it can be as a reopening. But it is not without risk. And there is a huge part of it that is contingent on the patron compliance and mutual empathy and respect and compliance. And I hope that everyone does keep that in mind in terms of attending and enjoying themselves at the casinos. We will, of course, be ever vigilant if there is a need for us to come back and amend. But I think that this body, the people that work here in the licensees, has done a tremendous amount of work to put us in a position when the governor's made the decision that it's safe to enter this phase that we were in the best position we can be in. And I wish everyone a good, long, 4th of July weekend. Conscious step-ins. Not much more to add, but a big thanks to our team and to my colleagues. This has certainly been a unique experience, which Commissioner Zuniga pointed out and one we don't want to go through again. But a special thanks to our MGC team. And I would say we've had a great deal of cooperation and collaboration with our licensees. And we can't thank them enough. And we know the work they have ahead of them. As Commissioner O'Brien just touched on, this isn't going to lead to a safe reopening if we don't have the collaboration and cooperation of the patrons as well. All three of us need to be cooperating, abiding by guidelines, enjoying the experience. But now we need their collaboration to make this reopening safe and successful. I think we did our work. And when I say our, I'm looking at all the names of the team, the standards that have been adopted by this body are as strong as guidance that can be given in this very difficult circumstance. And I just wish Monday that all goes smoothly. And then the following reopenings will stay tuned. Karen, I believe we have a gender setting meeting then on next week, correct? Correct, correct. And Mary Ann has already sent the notes around for that. That's right. So we'll be looking forward to seeing how each, we'll have some time to watch the reopenings and monitor progress. As I pointed out, the guidelines, we are really hopeful that the public health conditions and the virus metrics trend in the right direction. And that when we revisit these conditions, we're revisiting to give some leniency and as opposed to having to in any way reverse. So let's cross our fingers, knock on wood. And again, thank everyone, the licensees, all of our employees and of course the patrons for their full cooperation. And with that, I guess I probably need a motion. I move to adjourn. Thank you, commissioner. Second. Thank you, commissioner. Any further comments or questions? Commissioner Cameron. Aye. Commissioner O'Barrion. Aye.  Thank you, everybody. And commissioner Stevens. Happy 4th of July, everybody. And I can I say the same. Enjoy the long weekend to all my fellow employees and let's reflect on what the 4th of July means. So, happy 4th. I vote yes. Thank you. Bye, Sarah.


In Boston, a lot of them come from Berkeley College of Music. Apart from their famous alumni who have already made their name out there, a lot of young aspiring musicians are still working hard to fight for a music career. I mean, I was a photo tech, so I was just spending all day just developing pictures. Bobby Etienne, a bass player at Berkeley, is transitioning to be a full-time musician with his band, NWA Soul Project. It's recording their first album, and Etienne is mapping out the future for the band. Although the CVS job pays the bill, he is not lingering. The long run, it's better for me to be out there, that way I can get more work. Because working at CVS kind of takes away a lot of my free time. Many musicians come to the number one Hollywood Music School, ranked by Hollywood Reporter, for a leg-up in today's competitive music industry. When not everyone is able to rise to fame like John Mayer or Psy overnight, the struggle might seem like a failure for some of them. It doesn't have anything to do with Berkeley. It's just the business in general. It's a struggle. There's not a lot of jobs. There's a lot more people that want the jobs than there are jobs to fill. Most of them for the first few years are not high-paying jobs. This Tobin has worked with many Berkeley musicians as a Berkeley business graduate for a couple of years. Being on the other side of the table, bookings shows for bands, and now for Brighton Music Hall, Tobin says Berkeley does give you an edge, but it matters more that young musicians use the tools after they walk out of Berkeley. Figure out what's going wrong with you personally, and I think most of that doesn't have to do with Berkeley. It has to do more with the person. And Berkeley will help prep you for that. You do lots of projects. You do things that force the weaknesses out of you. We're really what the goal of the school is. Besides, not everyone is trying to be as known as John Mayer. Bobby Eaton is happy with who he is. Yeah, because I'm doing the music, and I'm supporting my family, I'm good. Reporting for BUTV, this is Alex Shee.


Some of my viewers sent me a request to speak on the concept and legitimacy of a situation where an arrest can be made by a private citizen who is obviously not acting as a police officer or a law enforcement officer, meaning arrest to whether an arrest can be made by a private citizen in a situation. In common law jurisdictions, this practice came into being long ago in England when sheriffs urged common citizens to help capture and detain law breakers. This concept of citizens arrest, meaning arrest by a citizen, exists in almost all progressive nations, including India. India does recognize that every citizen is a police officer without uniform and every police officer is a common citizen in uniform. That's a great way of explaining. Every citizen has been given the power of a police officer to ensure law, peace and order in the society. A common citizen is expected to provide support in the maintenance of law and order in society. In India, section 43 of the Code of Criminal Procedure, in short known as CRPC, section 43 of CRPC enables rather empowers a private person, private citizen to arrest or assist in arresting any person who in his presence commits a non-billable and cognizable offence, or even a proclaimed offender can be detained by a private citizen. The citizen is expected to ensure without unnecessary delay to take or help in taking the said detained, accused criminal to a police officer or in the absence of a police officer taking such detained or help in taking such detained person or arrested person to the nearest police station. There are other provisions as well which you know obligate and encourage I would say the common citizens in the process of policing. For instance, section 37 of the CRPC Criminal Procedure Code requires a common citizen to assist the police in certain cases as well as to provide information to the police regarding commission of various crimes offenses which have come to the knowledge of their concerned citizen, private citizen. Section 39 of the CRPC Criminal Procedure Code specifies a variety of information to be given to the police by the common citizens, more specifically where the concerned citizen is aware of the commission of or the intention of any other person to commit any offence punishable under the specified provisions of the Indian Penal Code including offences affecting life, meaning causing death, hurt or restraint, offences of robbery and decoyty, offences relating to criminal breach of trust by a public servant, offences of mischief against the property, the offence of house trespass, then offences relating to currency notes and bank notes, offences against the state, offences against the public tranquility, offences in regard to illegal gratification, offences in relation to adulteration of food and drugs etc. Anyone who makes a citizen's arrest can also land up in some issues, they can face possible lawsuits and criminal charges such as charges of false imprisonment, unlawful restraint, kidnapping or wrongful arrest. If the wrong person is apprehended or a suspect's civil rights are violated, that's the flip side of it. This is especially true when police forces are attempting to determine who an aggressor is in reality. Private citizens do not enjoy the same immunity from civil liability when making arrests or detentions on other private citizens as do police officers. Police officers have an element of immunity in that sense. The level of responsibility that a person performing a citizen's arrest may depend on a particular jurisdiction. For instance, in France and Germany, a person stopping a criminal from committing a crime, including crimes against belongings, is not criminally responsible as long as the means employed are in proportion to the threat. Well, we can leave that for another time because this needs a wider discussion. We can even talk about the concept of bounty hunters in India, US and UK in subsequent video talks. Thank you for today.


It's DDK and I'm back today with another video. Shoes out to you for tuning in. If you want to win, go and subscribe one. Give me a thumbs up and do not skip those ads. If you want to see me get to these bags.  And I'm going to tell you guys right now, I have a confession to make. I made a horrible, horrible, critical, critical mistake. But hey, when you're trying to be great, sometimes that's what it takes. Look, we are going to be dropping off 29 orders of pre-cooked meals and we'll receive a payment of $200. I don't know how many mouths it's going to be, but if I think over here and get busy, let's go. This is how it looks upstairs, baby. Y'all see all these orders. Now we've got to find which one is ours. I think it's probably seven or eight. Let's see if this is it. I know we got 29 delivery, that's 26. Maybe this one? Nope, what is that? I think it's going to be over here. I bet you it's this one. Nope, this can't do it like me. Is that one? That's 29? Oh, this is the 29 right here. This is our route number eight for those who hate. Let's go. 25, 24, 23, 22, we're here, fantastic. I mean, it's kind of a gift-having curse. 17, 16, so we're going to put in a whip ski, so we can get fancy. Guess what? We are here to drop off order number one that they had just begun about to have us some fun and get these funds. Drop like it's hot. Man, doorbell. Stay back with the photo, you all know that. Smack, let's go. Number two, getting to this money is just what we do. How you going from this joint? Please give me some directions. They give me no directions. Guess what we're about to do? About to squeeze it around, put it up like this. Drop it right here. Drop it like it's hot. Stay back with the photo. Smack, let's go. Order number three, we just pulled up. It's time to get busy. It's time for us to get busy. I'm glad you guys are with me. Drop like it's hot. Stay back with the photo. Smack, let's go. Order number four, guess what? I'll be seen in the coming sessions. Some of y'all be hating on the low. Some of y'all be hating. I'll be seeing. Ha, ha, ha, it's all right, baby.  We gon' keep on doing our thing. Hit it with the photo. Smack, let's go.   Drop like it's hot. This house is nice, too. Stay back with the photo. You already know, though. Smack, we gone. Order number six, guess what, baby? We are here in the mix. I gotta put that G14 classified code in. Game, coach. I'm gonna sit and put it in the lobby. You may just wanna stop me. Stay back with the photos. Smack, let's go. Just like that, we are here at order number seven. Order number seven. Three of them, order number seven. They out of pocket. I wanna show y'all this. They are out of pocket for this. Look at this. Look at all these bags.  Stay back with the photo. Guess what, I gotta walk down the street, too. Because it's one way street. So we gotta grab all these joints and get busy. Damn, look at that, look at this. Look at this. Look how many bags this is, G. They out of pocket. They out of pocket. 17 bags. Let's go. Order number eight. We drove down the street to get this cake. It was right around the corner, baby. Right down the street. Drop like it's hot. Stay back with the photos. Let's go. Order number nine. We on the ground tonight in the time. Y'all gonna continue to see us shine. Drop like it's hot. Doorbell, step back, photo. Y'all already know those? Let's go. Hello. I have a present for you. You were waiting for it. There you go. You have a great one. Order one, one. We're here to get the job done. And get these funds, son. And get on the run. Drop like it's hot. Doorbell, step back, photo. Let's go. This. Put it at the back, though.  Guess what? Order one, two. If you want to win, do what the winners do. That's all you gotta do, baby. Photo smack. We're gone. Order one tray, keep on grinding, even if they hate. Even if they hate, keep grinding. So you keep on shining. Keep on climbing to the top. Doorbell, step back, photo smack. We gone. Order number 14. If they sleep on you, let them keep snoring. That boy be rhyming. That boy be rhyming. Photo smack. Let's go. Order number 15, Star Living, not just existing. We gotta find this name so we can do our thing. Right there, baby. Ball, time to ball. I'm here with your Madison and Rain order. We in the game, coach. You know what I'm talking about? I already know exactly where to take it. Go over here right quick, right quick ski, so we can get busy. Drop like it's hot. Step back on with that photo, smack. Let's go. Order number 16, guess what we here to do? Drop this order so we can get cream, not in between. They got these right on this red, this red bench. Drop like it's hot. Step back on with the photo, smack. When that door be up. Let's go. Order 17, I don't know if y'all know, but we in the recession, babe. I gotta put this G14 classified code in. Y'all can't see what's going on. Hey, y'all remember this door? This the one that broke. In the game, coach. Step back on with the photo, smack. Let's go. Order number eight, in order to be great, you're gonna make some mistakes along the way, babe.  But keep on grinding, keep on shining. You want some more? Step back, photo, smack. Let's go. Order 19, go out there and get that green by any means necessary. You know what I'm talking about? We got three bags again. Doorbell, step back, photo, smack. Let's go.  We're here to get dough. Sometimes they be hating, because I don't know. I don't know why they be hating. We're gonna keep on skating. You know what I'm talking about? Being great and getting this catching. Drop like it's hot. Doorbell, step back, photo, smack. We gone. Order 21 savage. I want my money green like cabbage. I want my money green like cabbage. Doorbell, step back, photo. Alright, you know what I'm talking about? Smack, let's go. Order 22, I say I like my money green, but I love it when it's blue. Drop like it's hot. Step back with the photo, smack. Let's go. Order 23 like Jordan. Sometimes when they hate, you gotta ignore them. Sometimes you gotta ignore the haters, baby. And keep on getting that paper. You know what I'm talking about? I'll add them later. Step back with the photo, smack. Let's go. Guess what I did? You guys would not believe what I did. I tweaked. I dropped off the wrong bag to this house. And we had to go out the way eight minutes, come all the way back. Hopefully it's going back outside. Come up a little, let me know if it's going to be out here. Dang, they got it. Nah, I rang the doorbell in a way. I tweaked. I can't believe I did this. I'm a goofball, Jamal. Hopefully home. Cause if they not home, then I don't know what I'm gonna do. Hello, I left the wrong bag. You know what, I just emailed about it. Okay. Sorry about that. No, no problem. You have a good one. Okay, thank you. Yes, man. Order 24, we're here to get it popping like Chris Cole. What are you talking about there? Oh, they didn't usually get it. Hit him with the photo. When that doorbell, let's go. Order 25, guess what? Since I tweaked, I had an extra 20 minute drive. I tweaked. I can't believe I did that goofball stuff. I'm mad about that.  It's all right though. Hit him with the photo. Smack, let's go. Order 26, we got three more than we did. After we deal with these three, we gone, baby. We're getting it home, baby. Doorbell, step back. Photo, smack, let's go. Order 27, guess what? Messing up, taught me a lesson. The lesson is make sure you check the addresses. Learn from our mistakes. Just be great and don't hate. Hit him with the photo. Smack, let's go. Order 28, first they load and they hate. And guess what? It's cause they fake. Jump like a solid step back. Photo, smack, let's go. Order 29, guess what? I'll see you guys next time. Smack. And just like that, it is in the video. I hope you guys enjoyed the content. Guess what, baby? We drove 36 miles and it took us three hours and 15 minutes to get in it. It really would have took us two hours and 55 minutes but y'all saw me mess up, baby. Hey, it covers the territory, baby. But I hope you guys enjoyed the video. Again, if you're not subscribed, go and smash the subscribe button. Give me a thumbs up and do not skip those ads. And like it tomorrow if you subscribe, it's just bad. Why are we not just bad? The story is story three. It's a big bad season. We'll keep on getting more reason. I'll see you guys on next one. We going, we going before you are ignoring the DDK and I'm on my way.


We are going to begin the study of second order partial differential equations. In this lecture we are going to study about special curves associated to second order partial differential equation. So the outline for today is we start with some illustrative examples exactly like how we started off our study of first order partial differential equations where in lecture 2.1 we looked at 3 Cauchy problems which exhibited all the 3 possibilities for the number of solutions. Unique solution when the data is datum curve is of certain type and when it is of another type it was either 0 solutions or infinitely many solutions. Then we will make an attempt to solve a Cauchy problem for a second order quasi linear PDE. So second order quasi linear PDE in 2 independent variables the most general such equation is of this form AUXX plus 2BUXY plus CUYY plus D equal to 0 where ABCD are functions of 5 variables. So they are defined on open subset omega 5 of R5. So in the above equation we refer to that as 2 QL second order quasi linear equation we suppress the dependence of A, B, C, D on X, Y, U, X, U, Y. Otherwise the equation will be very long when it is understood there is no need to repeat it. So that is why the dependence is suppressed here. So A stands for A of X, Y, U, X, U, Y etc. Now second order linear PDE we will consider this also in this chapter. In fact the second part of this chapter will be as exclusively dealing with second order linear PDE and a general such general form of such equation is here. So we refer to that as 2L second order linear partial differential equation. So let us look at some illustrative examples. Let us start with this second order partial differential equation UXX equal to 0. I have to mention it is in 2 independent variables because it is not clear from the equation. So it is general solution we can write down. This is actually a ODE with respect to X. Therefore when you integrate first time UX of XY will be constant with respect to X. So it will be some arbitrary function of Y that is K1Y. Integrate once more you get UXY equal to K1YX plus K2Y. See solutions of UXX equal to 0 or straight lines if it is a ODE straight line will have A they look like AX plus B. Now because this there is a Y variable involved A and B will be functions of Y also. Now what are K1 and K2? K2 of Y if I want to find out I need to finish this term kill this term. So I put X equal to 0 that means U of 0 Y this term is gone what remains is K2Y. So K2Y is nothing but U of 0 Y and what is K1Y that looks obvious you are differentiate with respect to X and UX of XY in fact is K1Y but I am taking X equal to 0 to be uniform with this because later on we are going to look at prescribing conditions to solve UXX equal to 0 conditions like our Cauchy data that we have seen in first order PDE. Therefore let it be U0Y and UX0Y then what we get is K1Y and K2Y. So therefore this analysis shows that you have complete freedom to prescribe U of 0 Y and UX of 0 Y if you want to solve UXX equal to 0. Now let us look go to the second example equation is the same we are not changing the equation UXX equal to 0 therefore the general solution continues to be the same. Now suppose we want to prescribe U of X0 arbitrarily what should be that can we do it is that allowed by the equation because this equation immediately the solution is coming like this right. So therefore asking this question is same as asking whether equation allows us. So what is UX0 when I put UX0 Y equal to 0 I get K1 0 times X plus K2 0. So it is not allowed by the equation you cannot have arbitrary functions UX0 must look like AX plus B for some real numbers A and B. Let us prescribe UX0 equal to AX plus B prescribing UX0 equal to AX plus B means that K1 0 is A K2 0 is B we just saw that on the previous slide. So since UX0 equal to AX plus B the derivative of U with respect to X is determined on the X axis UXX0 will turn out to be A. So there is no way that we can prescribe UX on the line X0 it is not possible. Then we ask the question can we prescribe UI in that case UI of X comma 0 let us find out. So UI of X0 from this general solution will look like this now derivative with respect to Y that means K1 dash Y X plus K2 dash Y when Y equal to 0 it is K1 dash 0 times X plus K2 dash 0. This means even UI of X comma 0 must be like CX plus D it must be linear function for some constant CD in R. So therefore to conclude we have the following two scenarios when UX0 and UIX0 are prescribed what are they this is just to recall that the general solution is this for UXX equal to 0 what are the two scenarios prescribe UX0 equal to AX plus B and UIX0 equal to CX plus D. Now prescribing UX0 equal to AX plus B fixes the values of K1 and K2 at the points at the point 0 X equal to 0 K1 0 is A and K2 0 is B. Now the other condition UI of X0 equal to CX plus D that fixes K1 dash 0 equal to C K2 dash 0 equal to D. Therefore there are infinitely many solutions to the Cauchy problem because these two conditions namely UX0 equal to AX plus B and UIX0 equal to CX plus D does not determine both K1 and K2 the functions what are all the things which are determined by this condition are simply the values of the function and the derivative at the point 0 for both K1 and K2. So you have so many functions which satisfy this criteria that is why you have infinitely many solutions. Now the second scenario is at least one of them you have not prescribed as a linear function what will happen no solutions no solution to the Cauchy problem. So if you compare both the examples PDE is the same in both the examples namely UXX equal to 0 when Cauchy data is given on Y axis solution is unique when Cauchy data is given on X axis two possibilities exist infinitely many solutions are no solutions. So recall the three Cauchy problem that we considered in lecture 2.1 for first order PDEs where we had similar observations the curves gamma 2 which give rise to 0 or infinite number of solutions turned out to be a special curves for the PDE and these curves were called base characteristic curves later on. Now the natural questions are are they a special curves for every second order PDE how many will be there how to find them. Next few lectures are devoted to finding answers to these questions. So solving Cauchy problems for second order PDEs we cover some preliminaries. So we are going to pose a Cauchy problem and then implement a classical strategy to solve that before posing Cauchy problem we need to introduce a few terminology we are going to do that. So let gamma 2 denote a planar curve described parametrically by gamma 2 X equal to FS Y equal to GS SNI where I is an interval in R and FGR C1 functions. Further assume that gamma 2 is a regular curve what does that mean for every SNI F dash G dash is not equal to 0 0 and geometrically speaking gamma 2 possesses a well defined tangent at each of its points. We have come across the notion of a regular curve even in the context of first order PDEs. Let N of FS GS denote the unit normal to gamma 2 at the point FS GS in gamma 2. FS GS is a point on gamma 2 and N denotes the unit normal to gamma 2 at that point. Defined by this because unit normal is not unique it will be there will be two choices for example if this is your gamma 2 this is the tangential direction and what you have here is the normal direction. So plus or minus of each other we do not care which one we are taking for this problem it should be given by this we are giving the formula here N FS GS equal to this. Notice this is well defined if the denominator is not 0 that is precisely the assumption of the regular curve. So we are now ready to state the Cauchy problem given two functions we will see where they appear H and chi. Cauchy problem for the second order Quasillinear equation consists of finding a C2 function that is twice continuously differential function such that U is a solution to the PDE which is a second order Quasillinear equation 2ql and U satisfies the two conditions U of FS GS equal to HS that means U is prescribed on points of gamma 2 as H of S and dou U by dou N which is called the normal derivative of U is also prescribed as chi S at every point of gamma 2. So suppose this is your gamma 2 and you take a point here at this point this point is like FS GS this is how points look on gamma 2 you are prescribing what should be the value of U at this point and you are also prescribing dou U by dou N at this point the normal derivative why not any other derivative that question we will discuss at the end of this lecture why not any other derivative. So actually if you see the normal direction is like that right or maybe any of the direction let us take this side you can actually define the directional derivative in any direction that can be prescribed. So what can be prescribed is dou U by dou V that can be prescribed that is also fine what all you should not prescribe is the in the direction of the tangent you should not prescribe if you recall if you consider U x 0 equal to some function H x U x is already determined okay. So therefore you cannot prescribe this with freedom and what is U x x 0 it is a directional derivative of U in the direction 1 comma 0 that is the direction of the x axis. So that is what is called a tangential derivative okay and at every point the direction of the tangent and the normal they will be linearly independent. So you can prescribe 2 derivatives but one derivative tangential derivative is already determined if you have prescribed the function U therefore there is U this you cannot prescribe therefore any other directional derivative we can prescribe where V is independent of the direction 1 0 but to be very clear we are prescribing on a normal because tangent and a direction which is immediately connected to a tangential direction is a direction perpendicular to that which is a normal direction. So that is why we prescribe dou U by dou N that is a secret you can prescribe any other derivative also and we require as usual the condition should be met for S belonging to a sub interval of phi which means we are looking at local with respect to data kind of solution. So geometrically speaking if you define a space curve gamma in R 3 by putting z equal to H s we get gamma 2 will be the projection of gamma 2 xy plane and the surface z equal to u x y defined by solution or the Cauchy problem will contain a part of this gamma okay. Now we discuss a classical strategy to solve the Cauchy problem. The goal is to find solution to Cauchy problem near points of gamma 2 because functions will be defined in a neighborhood of gamma 2 the same thing was true for the first order partial differential equations also. So take a point on gamma 2 p0 in terms of x and y you may call x0 y0 in terms of the parameter running on the gamma 2 fs0 gs0. Determine derivatives of all orders of a possible solution at p0. Determine derivatives of all orders propose a Taylor series around the point p0 using the information on derivatives at p0. What information do we need to propose a Taylor series of a function all the derivatives at a particular point in this case p0 that you have determined in the this step determine all the derivatives. So Taylor series can be proposed and hoping that the series converges and it would be a solution to the Cauchy problem that is the hope one needs to prove that. This is the essential idea behind the proof of Cauchy-Kowalski theorem for details you may consult the book of partial differential equations by Fritz John you will find the details there. So in other words somebody comes to you and tells you that boss I know that this Cauchy problem has a solution which can be expressed in Taylor series format. It has a Taylor series expansion in other words he is telling you that solution is real analytic. Now your job is only to find that to find that what all you need to do is find all partial derivatives of the function at the point p0. If you can determine them uniquely then you can't hold off all the derivatives and propose that series Taylor series. And since somebody told you that he has a Taylor series expansion you hope that this will be solution to that. But to implement this strategy we would require the data in the problem namely the A, B, C, D to be smooth to be as many times differentiable as we want. Similarly F, G and H which are prescribed functions or maybe H and Chi yeah F, G are determined defined by gamma 2 gamma 2 is defined by F, G and then we are given the Cauchy data in terms of H and Chi. So all of this of course we need to assume are C infinity functions then only we can implement this strategy. We limit ourselves to just computation of all derivatives we are going to inquire into the possibilities of computing all derivatives at a point p0. Can we do it or not? Whether somebody stops us from doing that if so who is that we will identify. Of course we have no a priori knowledge of the solution the person who told there may be a solution he is not given as the formula so that I can compute derivative using the function. No it is not that you are given a function and then find its Taylor series it is not the case. You are thinking that there is a solution which has a Taylor series expansion and you are trying to find out if such is the case what are the derivatives and what is available to you is only the Cauchy data and the PDE these are the only two things that you can use nothing else. So we do not discuss the convergence aspects of the formal Taylor series which needs to be proposed after computing the partial derivatives of all orders. So for implementing this strategy we need to assume that all the functions involved in the Cauchy problem namely a, b, c, d, f, g, h, chi are C infinity that means they have derivatives of all orders. Now we are going to drop the subscript 0 in S0 and we write S with understanding that it is fixed but otherwise arbitrary in I. Now let us see the computation of first order derivatives using only the Cauchy data and the PDE and for brevity in notations let us introduce ux at a point fsgs on gamma 2 as ps similarly uy at a point fsgs as qs. So these functions are defined on gamma 2 we are just introducing we do not know ux uy yet we need to determine ux uy. So using these notations the normal derivative condition takes this form minus pg prime plus qf prime by root f tax square plus zeta square equal to Cauchy and dou u by dou n is gradient u dot n gradient u is ux and uy dot n is minus g prime f prime divided by root f dash square plus g dash square that is why we get this equation solution must satisfy hs equal to u of fsgs u of fsgs also right. So let us differentiate this equation because we want to get an equation for ps and qs.  So we have got one more equation. So we had one equation on the previous slide and one more equation on this slide both of them are linear with respect to ps qs. So let us recall both the equations in one place look we want to determine ps and qs this is one linear equation featuring ps qs f prime g prime h prime are known here also g prime f prime and chi are known so this is also a known linear equation in p and q. The coefficient matrix is invertible what is the coefficient matrix the first equation is f prime of s g prime of s second one is minus g prime of s divided by square root of f prime square plus g prime square and here it is f prime by square root of f prime square plus g prime square into ps qs this is a system equal to h prime into chi not chi into h prime and chi. Now what is the determinant of this it is f prime square plus g prime square divided by square root of f prime square g prime s square therefore determinant is equal to square root of f prime square plus g prime square and that is not equal to 0 due to the regularity of the curve so therefore there is exactly one solution for ps and qs so we can find ps and qs uniquely. So both the first order partial derivatives have been determined at all points of gamma 2 with this in fact we are interested in determining at some point of gamma 2 that we have fixed since the point is arbitrary we are saying at any point in gamma 2 using only the Cauchy data the PDE did not play any role so a remark computation of first order derivatives at points of gamma 2 required the knowledge of u and its normal derivative on gamma 2 only these not surprising the Cauchy data contains information on directional derivatives of u in two independent directions what are they they are tangential through u of fs gs equal to hs normal through the normal derivative which is given explicitly. Since information on tangential derivative is inbuilt in this condition u of fs gs equal to hs one needed to prescribe derivative in any other direction which is non tangential this is what we discussed in the beginning of this lecture. For definiteness we have used normal direction that is all the partial derivatives or directional derivatives in two coordinate directions for a differentiable function knowledge of any two directional derivatives of course directions must be linearly independent that is enough to determine any other directional derivative as this map is a linear functional on r2 v going to dv u that is a directional derivative of u in the direction of v at the point p so p is fixed then the mapping v going to dv u at p is a linear functional which is fully determined once its values on a basis is known at any point on gamma 2 the tangential and normal directions are always linearly independent. Now let us look at computation of second order derivatives here we need to use a PDE we have determined the first order derivatives at all points of gamma 2 that is ux of fs gs and ui of fs gs we call them ps and qs on differentiating with respect to s we get p prime of s equal to uxx into f prime s plus ux y into g prime s similarly we get this expression for q prime s note here that the LHS p prime s and q prime s is known because p and s are known functions this is known f prime and g prime are anyway known so what are unknowns here uxx, uxy and uy y note that we are not making any distinction between uxi, uxy and uyx y because we are planned to compute all the derivatives and then propose a formal power series expansions for the solution therefore we are assuming that solution is smooth and for smooth functions the mixed partial derivatives do not depend on the order in which you take the derivatives so thus in conclusion 3 and 4 represent 2 equations 2 linear equations in the 3 unknowns therefore it will be nice to have one more equation so that we can hope to determine the unknown quantities namely the second order partial derivatives of u along the curve gamma 2. So the equations 3 and 4 feature 3 unknowns which are uxx, uxy, uy y it would be nice to have another equation satisfied by these unknowns so that we can hope to determine them the PDE the second order quasi linear PDE gives us a third equation because PDE is an expression for some combination of second order partial derivatives thus we have a uxx plus 2b uxy plus cuyy equal to minus d I have written in this form I have taken d to the other side because I wanted to write finally a system of linear equations for the unknown quantities where zeta s is fs gs hs ps qs. So the equations 3, 4, 5 may be written as the linear system in this linear system notice this is these are known quantities the matrix on the right hand side is known function of s therefore we can determine uniquely these quantities provided this determinant is non-zero. What is the determinant of this matrix let us denote it by delta of s because it keeps coming throughout this lecture it has this expression once you expand this determinant it turns out it is this is it is this. Observe that the system of linear equations on the last slide determine all the second order partial derivatives if delta s is not equal to 0 at points of gamma 2 wherever it is non-zero you can determine the second order derivatives at that point. So from now onwards assume that the above condition is satisfied. Now let us look go to the computation of third and higher order derivatives how the second order partial derivatives are computed if you look at pde determines a combination of second order derivatives of u along gamma 2 fine second thing is knowledge of first order derivatives of u on gamma 2 yields another two relations when we differentiate that with respect to s. So that is how we got the three equations and we could get all the three second order partial derivatives. Now if you want to repeat the above process what you need is a pde which gives a combination of third order derivatives how will you get that differentiate the pde if you differentiate the pde with respect to x or y you will get a new pde which has the third order derivatives in it fine and now second order derivatives we know on gamma 2 therefore if you differentiate that that will give you some more relations which involve third order derivatives of u. pde satisfied by third order partial derivatives we want to find so let us differentiate the given quasi linear equation with respect to x. So by product rule it turns out to be this notice the third order partial derivatives are appearing here and their coefficients are a to b and c which are exactly the coefficients in the given equation and this is nothing special for differentiation with respect to just x it will also be the same when you differentiate this with respect to y there will be a third order derivatives a different third order derivatives but coefficients are a to b and c even if you differentiate it 10 times even then you will get if suppose you differentiate 10 times then you get an equation which is a 12th order derivatives but with same coefficients a to b and c this will not change. So after differentiating we get this equation now we need to explain slightly what this notation stands for notice here a or a b c d they are all functions of x y u of x y ux of x y u of x y so that we are differentiating with respect to x that is why we have written this kind of notation. So let us introduce what this notation is so it is dou phi by dou x for any function phi a or b or c or d what it stands for is this dou by dou x of phi of x y u of x y ux of x y u y of x y so from the PDE that we obtained after differentiating the given PDE with respect to x it follows that a u triple x plus 2 b u double x y plus c u x double y namely this this is a known function along gamma 2 or this is a known function of s provided the rest of the things are known functions of s notice here a b c d are known functions and the second order partial derivatives u x x u x y u y y have already been determined along gamma 2 therefore they are known functions and on the next slide we are going to show that dou by dou x phi for any of these functions a b c d is a known function of s and then it follows that these are all known functions of s and hence this quantity is a known function of s that means this particular combination of u triple x u double x y and u x double y is known function of s. Take any function phi in a b c d what is this this by chain rule is exactly this differentiate phi with respect to x at this point zeta s differentiate phi with respect to the third variable which we are calling z and then that is u so u derivative of u with respect to x that is why ux and phi this is p this is q so phi p and phi q and this is ux therefore it is u double x this is u y therefore it is u x y therefore this is a known function of s the conclusion is that u a u triple x plus 2 b u x x y plus c u x y y is a known function of s. What is remaining is u triple y that is not appearing here for that we need to work separately we will discuss that later. The following system of equations holds for the third order derivatives at the point fs gs in gamma 2 see we knew u x x u x y u y y as a function of s these are been already determined therefore we can differentiate them d by d s d by d s d by d s. So, earlier for the first order derivatives we called p q as u x and u y now we can call r s and t but it will introduce new notations I want to avoid that that is why I am retaining it as it is but by chain rule this quantity is given in terms of a combination of third order partial derivatives of u so this we can get this expression. So, here all third order partial derivatives are evaluated at this point zeta of s the functions on the LHS are known because we know all the second order partial derivatives of u along gamma 2 so they are all known functions of s and hence their derivatives and on the right hand side we know f prime g prime the only thing we do not know is the third order partial derivatives which we are trying to determine. So, the system of linear equations given by 7a, 7b and 6 6 is the equation that we obtained after differentiating the given equation with respect to x and 7a and 7b are the first two equations on this slide and the right hand sides are all known functions I am not writing explicitly what they are because that is not important for us what we need to know is these are known functions these are the unknown functions which we are trying to determine and this matrix is interesting because exactly the same matrix that appeared in the computation of second order partial derivatives and we have assumed that is invertible therefore we can determine all these three derivatives. Now, we had an option of writing equations u triple y also featuring so we can write four equations but for this reason I avoided that this is convenient for us u triple y we will do similarly how to find u triple y okay so thus u triple x u double x y u x y y are determined along gamma 2 what remains is to find u triple y how do you get that many ways one of them is do the same procedure instead of differentiating the PDE with respect to x differentiate with respect to y and consider the last two equations on that slide where we had the three equations the equations which came out of differentiating second order derivatives along gamma so exactly same computations you repeat otherwise differentiate the given PDE with respect to y and then in that only u triple y will be unknown rest of the third order derivatives have already been determined so therefore you can determine u triple y of course you would need that the coefficient multiplying u triple y is non-zero what that can be done so the procedure described above can be continued indefinitely and all higher order derivatives of u may be determined there is no need to impose any more assumptions on gamma 2 other than requiring delta s not equal to 0 that is important of course you need all these functions to be infinitely differentiable now what are the curves gamma 2 for which delta s is identically equal to 0 this is a natural important question because such curves will prevent you from doing these computations okay we may not be able to determine the second order derivatives if delta s is identically equal to 0 or even if you are able to determine it is not unique so we do not say it is determined uniquely right so there is a trouble if delta s is identically equal to 0 what does that mean it just means this this is a equation equal to 0 for every s in i in example 2 PDE is uxx equal to 0 gamma 2 is x axis delta s is identically equal to 0 holds okay and we saw the trouble there either there is no solution or infinitely many solutions now how to find these curves how to determine these curves of course here the equation that we have is in terms of the parameter yes so now how do I find such curves in x-ray plane unfortunately it involves h s p s and q s okay because we have considered the quasi linear equations so therefore if you consider it is a linear equation it much easy okay assume that gamma 2 is a graph of a function for example x equal to xi of y then gamma 2 will be x equal to f s which is now xi s and y equal to g s which is s then this equation becomes this equation of course still zeta s is there so when the equation is linear then c zeta s does not depend on 5 quantities it only depends on the first 2 quantities which is x and y f s and g s so writing in terms of x and y we get this equation this is an ordinary differential equation first order but degree 2 there is a power 2 here so d xi by dy maybe one can compute using the formula of the solutions of quadratic equations and you are likely to get two equations likely to you may not get we will see that later in forthcoming lectures we will discuss about solutions of this equation. So let us summarize what we did a formal procedure to solve Cauchy problems breaks down if gamma 2 is a special curve one might ask why did we consider second order causality equation for this discussion simply because most general equations for which the method can be carried out are causality equations that is why we have done for causality equations for general non-linear equation we cannot do okay that a if you remember a depended only zeta s zeta s is already known the moment you compute the first order derivatives so whenever you differentiate as many number of times as you want the partial differential equation the highest order partial derivatives are always multiplied with a to b and c which are known functions and as a result the linear system that we may write from time to time will be the same whose determinant will always be delta s so that is the advantage since we could do we have done it for causality equations and the questions on special curves and their consequences will be important for the second order causality equations also we will come across them later interestingly we will run into special curves for PDE in a different context also which we will see in the next lecture in forthcoming lectures for linear PDEs we will try to find answers we will try to find answers to the following questions what are they do special curves exist for any second order linear equation if yes how to find them how many of them exist etc so in the next lecture we will take up another context where delta s makes an appearance delta s identically equal to 0 will become important thank you


So sales, so you got the customer contact list, typically an internal type of thing, the deposit detail, that's usually going to be an internal report. You're not usually going to be generating that to provide declines on a monthly basis. Estimate and progress billing, internal estimates by customer, internal helping us facilitate our communication with our customers. Income by customer summary, you could do that external, but that's going to be, but that's actually that could be an external report, but that's a good one as we saw possibly to make a graph out of, although we made the graph and the pie charts out of the sales, the sales by customer summary. So I think that type of information you might add as an added touch over and above your normal kind of bookkeeping reports that you might be providing to somebody to give the pie charts because their sales lines are usually the ones that they're most interested in. Right. And so, so then you got inventory and I'm not talking about like an audited financials, then we would have notes and whatnot to them. I'm just talking as a bookkeeper, giving your bundle of reports possibly on a monthly basis. So inventory valuation summary, probably, probably not something that's more of an internal report typically, not something you're going to be providing on a monthly basis all the time, payment methods, internal physical inventory worksheet, internal product service lists, internal typically sales by customer. And again, this is one that you might make like an external pie chart or graph from because that could be a good visual. Same with the sales by product summary. You could make that into a pie chart or a graph, which could be a nice external touch because sales is usually the big line item that people want more detail on.