What's the word, y'all? For me, the MVP race is down to two different players and those players are Nicole Jokic and Luca Doncic. Now, I wanna explain, I am not predicting who will win. This is just like if I had a ballot, which I don't, this is the way my ballot would be as of right now. And since we're getting closer and closer to the end of the NBA season, I'm thinking about my ballot quite a bit. I know my DPOL, I know my Rickey of the Year, I know my Six Man of the Year. I'm still trying to decide between clutch player of the year and a coach of the year. But for the most part, I know who's winning every award on my personal ballot except for my MVP. Now, all season long, it has been a basically, well, not all season long, because until Joel Abigat injured and he's coming back, until Joel Abigat injured, it was him number one. But once Joel Abigat injured, it became a four man race between those two dudes I mentioned a couple of seconds ago and Giannis and Shay Gilgis Alexander. Now, I do wanna say the way I kinda do my awards is not necessarily what Giannis did wrong or what Shay did wrong, but instead it's about those other two guys kinda propelling themselves. I think right now Giannis is having one of the most dominant seasons ever and it's definitely gonna be weird to see him not end up probably top two on most ballots. It might be the best third place finish or best fourth place finish ever, but I think that this recent stretch for those other two guys kinda propelled them at one and two, for me. In the month of March, this is what Luca D'Anges did. He averaged 32 points per game, 10 assists, 10 rebounds, 47% from the field, 38% from three in his team. Won all of these games, all of them. And you wanna know the scary part about this. This month of March was basically his worst statistical month of the season. His 32, 10 and 10 on 47, 38 is maybe the worst month since the turn of the calendar season and that is scary stuff. Now, this month for Jokic is 27, 8 and 12 and this is probably his best statistical month of the season so far. Now, obviously, count of stats will say, hey, Luca D'Anges has the upper hand, right? I just talked about 32, 10 and 10 for this month and overall it's like 34, 34, 9.8 and 9.1. Like, statistically, Luca D'Anges has that advantage. But obviously, it's not all about the count of stats, right? And has it ever really been? It's never really been all about the count of stats. For the majority of my NBA fandom, the MVP award for most cases had been best player, best team. It was that way when I was growing up and over the last five, six, seven years it kind of has shifted to no longer being that Russell Westberg won an MVP as a lower seed. When I say lower, I mean just not one of the top two seeds Nicole Jokic did a few years ago. We've seen it kind of evolve from best player, best team and I kind of like this way a little bit more. But sometimes, sometimes I tend to overthink these things. Like, I'm trying to be candid with you. Again, my vote doesn't mean anything. I don't impact the voters whatsoever but sometimes I feel like I do overthink things kind of a lot, right? And I firmly do believe that Nicole Jokic is the best player in the world, in the world. I watched him play last night against the Cleveland Cavaliers and again this is just one game out of 82 but it's just embodied what it means to be the best player in the world. There was no Jamal Murray in this game. You know Jamal Murray and Nicole Jokic have like the best two man game in basketball right now, right? There was no, he was not in this one and last two games they played against Phoenix without Jamal Murray and they played against Minnesota without Jamal Murray and you could definitely tell that they needed Jamal. Jamal's an awesome caliber player and again they have the best two man game in basketball. Well against the Cavaliers, that kind of shifted with a two man game ended up being with KCP, ended up being with Peyton Watson, ended up being with Aaron Gordon. Like he found it. He didn't need Jamal in that specific scenario and he ended up in this game with 26, 16 and 18. And I promise you, let me see if I can find the clips. I promise you, of the 16 assists there were maybe five of them and I was like, this is a pass that only Nicole Jokic can make. And there was a few of them that didn't end up in assists. But let me see if I can find some of them. Now this touch pass is not a specific Nicole Jokic singer. I actually seen Braun do something like this a couple nights ago. But again, just the awareness to know that, hey, one little tap over to Christian Braun, or Brown, I'm sorry, is the thing. And this one's just beautiful. I mean, this is not a rare occasion I guess for the Denver Nuggets for something like this to happen. But it's just so flawlessly ran that I can't explain it. Just knowing that AG is gonna be there to make this play is just beautiful basketball. And his very next assist might be the most impressive one of them all. As you can see, it's him empty side with Christian Thompson. You see the cut in Christian Brown is gonna draw a lot of attention, so much attention that George's Nyang is looking at the cutter and Peyton Watson is sitting there on the baseline creeping, creeping, perfect pass to find him. Because again, even though it feels as though that George Nyang should be in this playoff preventing his pass, he's worried about the cutter so much that Peyton Watson is right here in the dunker for easy lay. And it's just scary how fast the top tier players in the league can process the game. In this moment of time, they decide to send two and Nikola Jokic off this screen, right? And in a matter of 1.2 milliseconds, he feels the second man coming and then it opens up Peyton Watson. And again, this is not something that's super unique to the NBA, but this is not something they were doing for the entire game. It's not like he just saw this three possessions ago and now he's processing that this is gonna be open. This is one of the few times they did this. And it just like that in an instant second, boom, Peyton Watson there for easy lay. Like that is the stuff that, again, makes me say that he is the best player in the world. And I know that this award is not best player in the world award, but him being the best player in the world simultaneously makes him one of the two most valuable players in the league. But then also after that game wrapped up, I tuned in to watch the Dallas Mavericks play against the Houston Rockets. The Rockets were on a double-digit game win streak. I think it was a 10 or 11 at this point. And this is a huge game for both teams. If the Mavericks win this one, they propel themselves over the six seed and they were gonna be the five seed, which maxes them up against the Clippers in a series. Again, they're still moving left and right every single game. And then again, the Houston Rockets are trying to take the 10 spot from the Warriors. So it's a huge game. And you know how I just talked about how Jokic picked apart a team mostly what is passing again at 26 points.  Luca picked this team. I can't even, let me see if I can find some clips. Now at this point in the game, Luca already has 10 points. He's got 10 of their 24. And things really shifted when Jaclyn Dale got on this court.  And I'm gonna show you back to back to back to back. This is the first play, right? Yeah, this is the first play he gets to switch that he wants. He's got 13 points at this moment. And it's like, may I have this dance off glass? And he looks at his teammates and like, yeah, that was fun, I did that. Next time down, it's again, the switch that they wanted. Jaclyn Dale on an island, step back three. He cannot miss right now. Uh-oh, what we got? Another switch was Luca and Jaclyn Dale. Fayow, another three. Second quarter, late in the second quarter. Guess who we got? Jaclyn Dale. That's a center he just went over by the way. And he said he was too small. Now I've watched this with the Rockets broadcasts in mind, listening to it. And the whole game, they were just talking about, man, this guy is just reminiscent of what James Harden used to do here. And obviously you see that James Harden was a guy that was pretty heiso heavy. One of the best iso players ever. And he got switches that he wanted and he dominated those switches. And I 100% feel them when they say that. The scary thing about this is that Luca Dodge is what, 25 years old now? He's 25 and his game is ever evolving as we can see. Now he's one of the better three point shooters in the league. Now he always been a good three point shooter, high volume, difficult shots. But now those difficult shots are falling more. And he ended up with 47 points in three quarters. In three quarters. I mean, he didn't score in the fourth. I don't wanna make it seem like he didn't play in the fourth because he did play I think six or so minutes, but he didn't score in the fourth. So he had 47, 12 and seven in three quarters. And then he also was doing stuff like this, which is just the amount of pressure on Dante X to hit that shot is crazy. Now the worst part about NBA discourse when it comes to awards and so on and so forth is that it gets extremely, extremely toxic. Y'all know that's the antithesis of who I am and what I support with the enjoy basketball brand. But again, I do wanna say again that like a guy like Giannis or a guy like Shaggy was on his end, there's no slight on their season. But again, I think these guys have kind of propelled themselves over. So I would never to try to decide between two players of MVP say like, hey, this guy did this wrong. But instead it's about empowering the people that are doing amazing stuff. So let me know what you think about the MVP race. Do you still have Shay or Giannis as your one or two? Like I honestly still believe that those are viable options. But for me at this point, it is Luca and it is Jokic. And number three, I don't know, it might be Giannis. I'd probably put Giannis three and Shay four. But again, it's evolving. Let me know what you think.


I am the Sniper of the Intimidating Straw Hat Pirate Crew. To thank you for trusting me, I'd rather risk my life here. My name is Usopp. Remember it. From now on, I'm going to become your legendary hero. Hello and welcome to One Piece 101, the series that breaks down every one and everything in the One Piece world. Today, we will be examining the bravery-impaired sharp shooter of the Straw Hats, Usopp. Usopp is a tanned and exceptionally long-nosed boy who was first introduced in the series during the Syrup Village arc. Other than his nose, Usopp's most defining feature is his constant lying. And we're not talking little lies either. We're talking huge, over-the-top world, breaking exaggerations that are often used for comic effect in the series. This habit comes directly from his childhood after his father, Yasopp, left the family home in Syrup Village to become a pirate. Sometime later, Usopp's mother fell fatally ill and Usopp would run through the village yelling that pirates are coming in the hopes of strengthening her spirits by making her think that Yasopp had returned home. After his mother passed away, Usopp continued to tell this lie on a daily basis to cope with the pain of having lost both of his parents in one way or another. And while this did become an annoyance to the rest of the villagers, they tolerated his antics and even began to use the daily false pirate alert as somewhat of an alarm clock. However, Usopp's lies would find a home with a young girl living in the village named Kaya. Like Usopp's mother, Kaya was a very sickly girl. And Usopp would tell her made-up stories about his, quote-unquote, adventures to cheer her up. Usopp found further allies in three young children and even began a pirate crew with them, rather creatively named the Usopp Pirates. Eventually, the beginnings of the Straw Hat crew landed at Syrup Village in search for ship, with Usopp almost immediately bonding with Luffy due to the fact that he had actually met his father, Yasopp, who was now a part of the Red Hair Pirates. Shortly after, a ploy by the Black Cat pirates to kill Kaya and sack the village was discovered, and although Usopp warned the town about the impending attack, they paid him no mind because, well, that was essentially the same thing he said every day, just slightly more detailed. Even the captain of the Black Cat Pirates allowed Usopp to flee because he knew that nobody would believe him. And so Usopp, alongside Luffy, Zora and Nami, fought off the Black Cat Pirates on their own, managing to defeat Kuro, Jango, and the other weirdos. In the end, Usopp requested that the attack be kept a secret so as not to distress the village. However, Kaya and Mary, who had become part of the conflict, knew about their brave actions and rewarded the crew by gifting them the Going Mary. Usopp then decided that it was time he set out to sea in order to pursue his dream of becoming a brave warrior of the sea. And after a socially awkward exchange, Usopp ended up officially joining the Straw Hat Pirates. Throughout Usopp's journey with the crew, he has gradually, and I mean very gradually, made strides towards achieving his dream. However, these moments have been exceptionally important and generally involve Usopp standing up to an individual infinitely stronger than him. Sometimes these bursts of courage and ingenuity can be as a direct result of Usopp wanting to further his own personal growth, but he has also been known to put his life on the line for the sake of others. With that said, Usopp is often portrayed as one of the more comedic characters in the series, which can be seen in particular when we delve into his arsenal of attacks, which includes Usopp's spell, Usopp rubber band of Doom, and Usopp's noise. Usopp also brings a lot of miscellaneous skill to the Straw Hat crew, particularly in the fields of artistry and engineering. In fact, Usopp was the one who actually made the iconic Straw Hat pirate Jolly Roger. In addition to that, he was the creator of Nami's Climate Act, and prior to Frankie joining the crew, Usopp even served as a shipwright. However, Usopp's greatest skill most certainly lies in the area of marksmanship. As a result, Usopp's weapon of choice is various forms of a slingshot with a wide variety of strange ammunition that when combined with his superhuman eyesight and ability to calculate factors, such as trajectories and wind influence almost instantaneously, make Usopp an incredibly dangerous person to face in long-range combat. Conversely, Usopp has next to no close-range combat ability. However, when forced into such situations, he has been known to equip himself with dial technology, although he does not possess the strength necessary to use the impact dial in particular without severe physical ramifications. With that in mind, Usopp does possess an incredible amount of endurance for what is essentially a regular human being. A perfect example of this would be during the Water 7 arc where Usopp was able to take multiple attacks from Luffy before finally succumbing to his power. Speaking of this moment is actually very important in regards to an examination of Usopp's character, as it showed for the first time just how far Usopp was willing to go in order to protect his core beliefs. Upon arriving at Water 7, the going Mary gifted to them by Kaia was deemed unrepairable, and Luffy made the decision to say goodbye to them Mary and buy a new ship. Usopp saw this decision as quite cruel, accusing Luffy of abandoning a comrade just because they had become useless and even proposed that Luffy may one day do the same to him. In a fit of rage, Usopp then left the straw hat pirates and challenged Luffy to a duel to decide who was going to keep the going Mary. Usopp fought valiantly and did surprisingly well against Luffy, however he was inevitably defeated. To a shame to face the crew after this event, Usopp developed a new persona in the guise of a superhero named Soga King. Soga King acts very much like the regular Usopp except that he has a much deeper voice and addresses everyone with the suffix Koon. This alter ego allowed Usopp to fight alongside the straw hats as it served as a perfect disguise to the more intellectually challenged members of the crew and essentially meaning Luffy. Although Usopp did eventually tearfully apologise for his actions and was welcomed back into the crew with open arms. But Usopp's personal development would not stop there. When the straw hats were separated by Bartholomew Kuma, Usopp was sent flying to an island called Greenstone which is part of the Boine Archipelago. Despite being home to some aggressive giant bugs and man-eating plants Greenstone also contained a location known as the Forest of Gluttony to which Usopp may have indulged a bit too much in. But it would be on the Boine Archipelago that Usopp would be trained by Heracles for two years in the art of growing and utilising a unique series of plants known as Popgreens. And when Usopp rejoined the crew after the time skip he displayed an incredible amount of proficiency in using these plants for a wide variety of situations as well as became noticeably more buff in general. However even after acquiring these new skills Usopp still struggles with bouts of extreme cowardice although he has displayed a tremendous level of bravery compared to his pre-times himself. In particular during the Dressrosa arc when he decided to take on treble and sugar alone after Niko Robin and the Tontata tribe had been incapacitated. And while this didn't go very well for Usopp at all at first his actions did lead to a series of events that culminated in the defeat of one of the seven wallards of the sea as well as him being given the epithet God Usopp.  And so Usopp continues to steadily grow not entirely free of his coward-like tendencies no we're near it actually but this man is slowly but surely taking strives towards his goal of becoming a brave warrior of the sea. Some more fun facts about Usopp. Usopp is one of the few straw hats who currently has access to any form of haki which was discovered when he awakened an ability to use observation haki during the Dressrosa arc. Prior to Usopp having a bounty of his own his alter ego Saga King was recognized by the world government who issued a 30 million berry bounty on the superhero's head. Following the straw hats visit to skypeer the citizens of the sky island created a theme park dedicated to Usopp named rubber band land complete with a statue depicting him as a muscular warrior a further statue dedicated to Usopp exists on Dressrosa but this one depicts him in a more accurate fashion.  And finally a truly useless fact when questioned by a fan whether Usopp or Kaku had a longer nose Oda answered that it was indeed Usopp due entirely to its rounded tip and that pretty much does it for Usopp. If you enjoyed this video then please feel free to like, favorite or subscribe and if you are in any way inclined to support this independent channel then also please feel free to check out my Patreon discord server or Twitter the links to which are in the very handy description below and finally please do comment with who, what or where you'd like to see featured in the next One Piece 101.


Hello, Myself, MS Bhasargaon, Assistant Professor, Department of Humanities and Sciences, Walshan Institute of Technology, Solapur. Learning Outcome At the end of this session, students will be able to express the given function in four-year series in the interval minus pi to pi. Now, first of all, we will see the Euler's formulae. The four-year series expansion of the function f of x in the interval minus pi to pi is given by f of x is equal to a0 plus summation of n equal to 1 to infinity a n cos nx plus b n sin nx, where a0 is equal to 1 by 2 pi, integration from minus pi to pi f of x dx, and a n equal to 1 by pi, integration from minus pi to pi f of x cos nx dx, and b n equal to 1 by pi, integration from minus pi to pi f of x sin nx dx. Now, these Euler's formulae, we are using to calculate the three constants a0, a9, bn. Now, we will see the example, find the four-year series of f of x equal to x minus x square in the interval minus pi to pi. Let f of x equal to a0 plus summation of n equal to 1 to infinity a n cos nx plus b n sin nx, let us call this as equation number 1, where a0 is equal to 1 by 2 pi, integration from minus pi to pi f of x dx, which is equal to 1 by 2 pi, here f of x is x minus x square dx. Now, we can integrate with respect to x, which is equal to 1 by 2 pi, integration of x is x square by 2 minus, integration of x square is x cube by 3 with a limit minus pi to pi, which is equal to 1 by 2 pi, first we will put the upper limit, that is putting x equal to pi, we get pi square by 2 minus pi cube by 3 minus of lower limit, that is putting x equal to minus pi, we get minus pi whole square by 2 minus of minus pi cube by 3. By simplifying we get, which is equal to 1 by 2 pi, pi square by 2 minus pi cube by 3 minus pi square by 2 minus pi cube by 3, which is equal to we get minus pi square by 3. Now, we calculate a n, a n is equal to 1 by pi integration from minus pi to pi f of x cos n x which is equal to 1 by pi integration from minus pi to pi here f of x is x minus x square into cos n x dx. Now, here to integrate x minus x square into cos n x we can use generalized rule of integration by part that we already seen which is equal to 1 by pi first we will take u as a x minus x square as it is x minus x square and integration of cos n x is sin n x by n minus that is sin of the rule into bracket derivative of x minus x square that is 1 minus 2 x into into bracket integration of sin n x is minus cos n x by n square plus derivative of 1 minus 2 x is that is minus 2 into bracket integration of minus cos n x by n square is minus sin n x by n cube with a limit minus pi to pi. Now, first of all we will put the upper limit if you put x equal to pi here what we have sin of n pi is 0 therefore, the first term will be 0. Now, this minus into minus plus putting x equal to pi you get 1 minus 2 pi cos of n pi by n square this minus minus plus again we have sin of n pi is 0 minus of lower limit that is we have to put x equal to minus pi again here sin of minus n pi is 0 therefore, first term is 0 and in second term by putting x equal to minus pi you get plus 1 plus 2 pi cos of minus n pi by n square and sin of minus n pi is 0. Now, which is equal to 1 by pi 1 minus 2 pi and we know cos of n pi is minus 1 raise to n divided by n square minus 1 plus 2 pi cos of minus n pi is cos n pi therefore, you can write the value of this as a minus 1 raise to n upon n square which is equal to 1 by pi by simplifying these two terms we get minus 4 pi into minus 1 raise to n by n square which is equal to minus 4 into minus 1 raise to n upon n square this what a n. Now, we will calculate b n b n is equal to 1 by pi minus integration from minus pi to pi f of x sin n x dx which is equal to 1 by pi integration from minus pi to pi x minus x square that is f of x sin n x dx again using here the generalized rule of integration by part first we will keep x minus x square that is u as it is integration of sin n x is minus cos n x by n minus that is the sign of the rule now derivative of x minus x square is 1 minus 1 minus 2 x integration of minus cos n x by n is minus sin n x by n square plus derivative of 1 minus 2 x is minus 2 into integration of minus sin n x by n square is plus cos n x by n cube. Now, putting the upper limit first putting x equal to pi we get pi minus pi square into bracket minus cos n pi by n and here sin of n pi 0 therefore, second term will be 0 minus twice cos of n pi by n cube minus now we are putting the lower limit that is x equal to minus pi minus pi minus pi square into bracket minus cos of minus n pi by n and because of sin n pi you get here 0 minus 2 into bracket cos of minus n pi by n cube. Now, by simplifying we get which is equal to 1 by pi minus pi into minus 1 raise to n by n plus pi square into minus 1 raise to n by n minus twice minus 1 raise to n by n cube minus pi into minus 1 raise to n by n minus pi square into minus 1 raise to n by n plus twice minus 1 raise to n by n cube. Now, simplifying this we get which is equal to 1 by pi minus 2 pi into minus 1 raise to n by n which is equal to pi pi we get cancel minus 2 into minus 1 raise to n by n. Substituting all the values of a naught a n and b n in equation number 1 we get f of x is equal to a naught that is minus pi square by 3 minus 4 into summation of n equal to 1 to infinity minus 1 raise to n upon n square cos n x minus twice summation of n equal to 1 to infinity minus 1 raise to n by n sin n x. Now pause the video for a while and find the Fourier constant a naught for the function f of x equal to e raise to a x in the interval minus pi to pi. I hope you have completed here you can calculate a naught by using the formula 1 by 2 pi integration from minus pi to pi f of x dx is equal to 1 by 2 pi integration from minus pi to pi e raise to a x dx which is equal to 1 by 2 pi integration of e raise to a x is e raise to a x upon a with the limit minus pi to pi which is equal to 1 by 2 pi a putting upper limit we get e raise to a pi minus e raise to minus a pi and as you know e raise to a pi minus e raise to minus a pi divided by 2 as a sin hyperbolic a pi divided by pi a. Now we will see the second example find the Fourier series of f of x is equal to minus x minus pi and minus pi is less than x is less than 0 equal to x plus pi when 0 is less than x is less than pi. Now here let f of x is equal to a naught plus summation of n equal to 1 to infinity a n cos n x plus b n sin n x let us call this as a equation number 1. We have a naught is equal to 1 by 2 pi integration from minus pi to pi f of x dx which is equal to 1 by 2 pi. Now first we will take the limit minus pi to 0 in that f of x is minus x minus pi dx plus now we will take the limit 0 to pi in that f of x is x plus pi dx now which is equal to 1 by 2 pi integrating now with respect to x minus x square by 2 minus pi x with the limit of minus pi to 0 plus integrating x plus pi with respect to x we get x square by 2 plus pi x with the limit 0 to pi. Now here by putting upper limit we get 0 minus of lower limit that is putting x equal to minus pi we get minus of pi square by 2 plus pi square plus and now here putting x equal to pi upper limit that is pi square by 2 plus pi square and for lower limiting putting x equal to 0 we get 0 which is equal to by simplifying this we get pi by 2. Now an equal to 1 by pi integration from minus pi to pi f of x cos x dx which is equal to 1 by pi first we will take the limit minus pi to 0 in that f of x is minus x minus pi into cos x dx plus integration from 0 to pi x plus pi cos x dx which is equal to 1 by pi now integrating with respect to x keeping minus x minus pi as it is and integration of cos n x is sin n x by n. Now derivative of minus x minus pi is minus 1 and integration of sin n x by n is minus cos n x by n square with the limit minus pi to 0 plus now keeping x plus pi as it is and integration of cos n x is sin n x by n minus derivative of x pi x plus pi is 1 and integration of sin x by n is minus cos n x by n square with a limit 0 to pi, which is equal to 1 by pi. Now, we will put the upper limit as you know sin 0 is 0 therefore, first term is 0 then minus into minus into minus we get the sin minus and cos 0 is 1 that you get minus 1 by n square. Now minus again putting x equal to minus pi as a lower limit as you know sin of minus n pi is 0 then minus cos of n pi by n square plus for the second integral putting x equal to pi again sin of n pi is 0 minus minus plus cos of n pi by n square minus of lower limit that is putting x equal to 0 first term will be 0 because sin 0 is 0 then cos 0 is 1 that you get 0 plus 1 by n square which is equal to 1 by pi minus 1 by n square and minus 1 by n square minus 2 by n square and this minus minus plus 2 cos n pi by n square which is equal to 2 by pi n square and cos of n pi is minus 1 raise to n minus 1. Now, we will calculate b n b n equal to 1 by pi integration from minus pi to pi f of x sin n x dx which is equal to 1 by pi integration from minus pi to 0 minus x minus pi into sin n x dx plus integration from 0 to pi x plus pi sin n x dx which is equal to 1 by pi again integrating by part we get we will keep minus x minus pi as it is and integration of sin n x is minus cos n x by n minus derivative of minus x minus pi is minus 1 and integration of minus cos n x by n is minus sin n x by n square with the limit minus pi to 0 plus keeping x plus pi as it is integration of sin n x is minus cos n x by n minus derivative of x plus pi is 1 and integration of minus cos n x by n is minus sin x by n square with the limit 0 to pi which is equal to now putting the upper limit x equal to 0 here what you get pi by n minus sin 0 is 0 minus lower limit putting x equal to minus pi here we get minus pi minus pi plus pi we will get 0 minus of lower limit putting and next second term is also sin of minus sin pi that is 0 plus now putting x is equal to pi pi plus pi 2 pi that is minus 2 pi and cos of n pi is minus 1 is to n divided by n and sin of n pi is 0 now putting the lower limit 0 that is minus putting x equal to 0 here we get minus pi by n because we know cos 0 is 1 and plus sin 0 is 0 which is equal to 1 by pi into bracket by simplifying we get 2 by n minus 2 pi minus 1 is to n by n which is equal to 2 by n into bracket 1 minus of minus 1 is to n now substituting the values of a naught a n and b n in equation number 1 we get f of x is equal to pi by 2 plus 2 by pi summation of n equal to 1 to infinity minus 1 is to n minus 1 by n square cos n x plus twice summation of n equal to 1 to infinity 1 minus minus 1 is to n by n sin n x references higher engineering mathematics by Dr. B. S. Gravel. Thank you.


when God's revealing things, right? He's revealing it for that time as well. For example, there's a prophecy. Deja is like the Antichrist. He's gonna travel the earth on a flying donkey, but this donkey has a wingspan of an airplane. So he's explaining it to people who they have no concept of an airplane, but he's essentially describing an airplane. It's the same thing for almost all these prophecies. Like you have to put it at the time of the people. If they misunderstood it previously and we understand it more correctly now, how do we know that we're not misunderstanding it now and we'll understand it more correctly in the future? That would be up for everyone to decide, right? They have to look into religion. I understand everybody's gonna have to decide for themselves. There's things that are more likely. Is it more likely that someone wrote something down and later on people interpreted it to be consistent with new facts or someone revealed information to someone that they didn't understand at the time and then gained understanding of later? Which of those two scenarios is more likely and how do you tell? I mean, we'd have to investigate that.


My name is Larry Powell and I teach at the Business School of Merit Circle. It's one of the companies that I think is my trustee. It's the one that I trust most of the time. I used to work there. I worked as a leading drug worker for Washington, for New York, and like me, I checked out my friends who are at home, and I said, well, how can I do this? And I'm going to give you examples of how this works, but I don't know. So, the title is Impressive, by the sounds of the proactive transmission. So, it was called, it worked for so many, so please don't blame me too much. So, speaking of which, And basically, this work is about contests that literally happen in the NRI, the business sales trade. And I think we have a focus on the contests that are held for the health of the gains that are long-term as it can be. Long-term use and long-term work. So we know what the manager has to do later. We will just sample these other experiments. So the two terms of the experiment are the reason those treatments are terrible, the reason they're not. Because it's probably not going to go too well. It's a great number of reasons we have. And these expressions are basically simple. We want to understand how people will be saved in those kind of desperate situations. So we characterize it. And believe me, I also think it's just likely that not many parents go home and study that game. So if you see that I'm safe as well, I think that there's a little dynamic outside the game. And then there's the conditions to come. It's to seek out. It's the way the people are doing it. It's here, right? Oh, we need the conditions. So it is. If you want to think about sort of not making a reasonable, maybe more specific or kind of situation style, like a very close to the model, that would be, say, a competition between the managers of these two branches on the chain for a sample. So the two branches that I'm going to go to, because two are going to the rest of them, they're all the same parts. They're all the same size. So I don't think even Q1 is going to end them, but Q1 is going to be Q2. So that's the kind of our Q1 case. I don't know what it's called. I don't think else being people. So what is the difference? Well, let's go a little deeper about it. By the way, how are you going to work? It's very complex. It's very simple. Well, these are two branches sitting here, right? But eventually one of those managers is going to be appointed as a district manager. So for a branch manager to be district manager. And the assumption is that both the company and the company would have the performance of these two restaurants and companies become private companies. So if a company makes it private company, both the platforms would be the district manager. As I'm going to talk about that, that's the question. Well, the location, of course, we do the same work. It's a market city in San Francisco, and if you look at the western part of the city, it's the same work between the two restaurants. So a few terms that I'm going to use. Positivity, non-positivity, the domestic context of both of these terms. Positivity, you know, it's the work of the business which is B is political A, and C is political B, and C is also political A. It's a trans-political. So correlation is non-positivity. Again, B is political A, C is political B, but A trans-political, of course. So it's some kind of cycle. Now, context. The realistic context, for example, the realistic context would be the auction, the standard auction, English auction, Dutch auction. Coral is the fairest people that can use the auction. So any answer, I'm the fairest people, I'm the only one. Most of the realistic context is the example of the drug context. Also, there are two papers that we've had some why, but each of them is each of the equalities. It's not the realistic one. For example, our beauty plan, say, we plug in the X, X gets more than just X or X possible. And so far, I'd love to work on it. What idea? And how do you call it? How? That's the one, so if it's more than X, it's simply humanization. Humanization, of course. The thing that's known is, also, if you're in a conference, if you're on the screen, sometimes they say, what part do you say you haven't been given any kind of call? All right. All right. All right. All right. How about here? How about over here? That's basically it. All right, so it's almost a long, transitive term.  Now, when we talk about non-transitive teams, the relations are defined between strategies that belong to different places. So, in this case, say, we have this rock and when we say, well, we have this rock, it's considered, if you play rock, it's really quite a big deal I won't show you. But then, seasons we play a lot, generally, we play from a couple of seasons. And occasionally along also in seasons. So that's an example for us to tier and us to list all of this basically all of the chance that And so the whole pre-optional is what you have. You can obtain a number of cells, one to two hundred cells, which is a polar. And the polar is a one polar model. So to us, we both see, those are very small speeds, compared to the angle sketch. Well, it's the first speed, because the object goes first, both edges. So it means to be, because of that, for all of these cells, the polar versus the polar, that goes in the best direction. Now, one of the most interesting things, an example is non-transitive types. It was kind of a complication of our program as well as the, I don't know, I think it's all about, you know, one day, one day, but here is some more, one day. Ah, yes, right. So the pre-optional solution, if you think about choosing these different types, different types of stretches, and about, which is, I think, P, it's A, C, it's E, or you think it's D, A, and A, it's C. So in this case, the probability of being is 5 over 1. So what I would like to comment on at the bottom is, being is like, Bill Gates, I want to see what Bill Gates is the person, so I'm going to explain this. Bill Gates once went to a research institute in Boston. He went to a research institute where he took where he was getting. So he gave, you know, three, five, six, so well, two, two, whatever you want, but I choose where I'm going, and one of them is, I don't know, the price of the contest is about, but Bill Gates expected that he does, and most certainly you choose first. So the idea is that you know, more of the material of the first mover, whether the first mover chooses, in this case, the second mover has an advantage for almost five or nine, but I believe it's available. Just like paper or scissors, only three options on the list of contents. And the one we are going to, we are starting here, in this paper, is it this letter? Can you do that? So the question is so far, so the news and the problem, the news and the problem is a classic single period program on Google. It was introduced by Edwards in 1988, and Edwards, who was trying to figure out who is working with a lot of cash, but back home, he really wrote for customers, for customers' loans. So some of the customers can show up on any one day, and if they can pass it on or if they are getting a problem, they may miss customers. So, for example, for a customer, you will call too much cash, or you are not going to get an increase of cash if it's too much. So there is a cost of going too much cash. And then, so in the context, let's say, even normally, because of the basic problem with it, there is a newsboy, so he draws a lot of money, with some administrative pros and cons, he's probably a good person with you. Besides, how many newspapers do you get from the publisher? When everyone wants to start, how many people are still speaking, how many newspapers do you get? I want to sell them, which is quite as cheap because of how many people should go and buy some sweets. And of course, it's called as the price of the newsboy, the price to the publisher, and the demand is random. So it's a long time for people to show up. It may be random, we don't know. So questions come in English. How much you buy from the publisher? You know, it's much more expensive. Well, the solution, you can, because in many cases, you can't calculate the expected profit. Most online, immediately, it's actually very, very, very, very, easy to use, just to look at other conditions directly, and say, he is the same, but both are very, very, very expensive. Now, the one thing, of course, in this case, it's a little bit complicated. So they find the margin, think of the value, and the variable, and that is that, well, suppose we already had two units of stock, for whichever reason, and we decide, what is the order of buy from the publisher? So in this case, it's 5. And the way we figure out what we, in this case, we can only sell that as the margin unit once we sell the original utilities here. So that means that the chance that we are going to sell that unit is that we have probability that we have found with, of Q. And that gives you the margin value. That's how it works. So in this case, if we sell the original utilities, they are going to sell them with one key. And the margin cost that we can only sell them to is just very relatively quick. So, for example, we use in the study or in the experiment, so we just standardize the same with the probability that it's not very different. It's one, and then the other. One more means is saving money. So it's going to be quite efficient. It should characterize the units which is the power. So that's what we've got. Any questions? Yeah, and this problem, so why does the problem be caused, it travels pretty much, whether it's not on the last one, or planning problems in the next project. Let's see. An example is going people who sell flowers, but it has characteristics of a single period of it. So if tomorrow the product is going to expire, I don't think it's going to perish. If you tell a flower, it's going to last a long time. It's going to expire. In this case, in this case we kind of implicitly assume that there's a competition because it's a low price. So it's a right. So it's very successful. So no. There are a lot of kinds of competition which we use in the study. We find the price and find where you can put it also. It's very good quality. So we will explain it in spaces for better context. Either it makes it implicit or not. But the thing about all those models is that the the quality or the way users are going to be they basically still do not function. Or they sometimes don't function. So if I didn't talk about my market for too much demand then there will be demand to get it all over 40 years. So that's how it works. So let's start at the line because it's going to be crazy. So if you think about this kind of scenario for large scale, for example, we'll be a huge cracker. So what they do for many categories of products will be so you decide to start making a certain model. They actually talk about the symposium of your own material for that particular model for 40 years and it works. They make only one decision. I have no idea what demand is going to be but the first one I started to know was the creation for sure. So they made that decision and that second one said about making that decision. Now, this model is a simple model. Of course, I have no idea what this model was. So what was the indication of why it wasn't built earlier. So, to increase the people they built that model and there's a lot of people just to see how people are going to do because that's actually a very important practical problem. It's a problem to understand how people are actually going to be decisions in this area. So they've got the information about how people are going to be involved in that model and of course they will work and not make important decisions. So the higher the efficiency the higher each product is going to be 8 to 5% of the maximum work and all in surprisingly about 10% of participants will be able to make optimal decisions. Now, the participants will be based against what I mentioned and very only the second year it will be and they started this problem to be justified. There are a few people who are currently who are actually speaking for who are able to to ground the industry but then we play in the spirit of what we want to do while we select a design. So the problem is not really it's not for the model. We've got picture if you look at the enemy or the eyes or the qualities while they work well it's like energy highly and on this it's very high quality so on this GR you can judge from the quality but the point is that there are three conditions for this model. One condition in this model is low cost so in this case the quality is 7 to 5 just like for example when the other condition is 3 to 5, 6.4 so the model will be 2 to 5 and the same condition for the model 2 to 5, 5 to 4 so the model will be exactly the same. That's very good. Now the high quality model on this we just can say so this is a good problem because the high quality is very high quality for between both the model and the model that's the component of the ratio it's robust but if you look at the model and the actual of the model that you go to so this is the new demand that it's just like so they basically all have the same characteristics of the model yes they were experienced a long time ago but we can't date this but for the same time what makes this experiment more interesting for this problem is that the other is coercion no no coercion it's plain to think about how we are going to do it so we have a lot of questions what turns out that model is actually not the same so there is something else but there are other things that is good for example for example quality explanations and then health is insufficient so people have to continue they only feel they need they just don't have quality they don't have quality they don't have quality so they stop something then the launch is you know it's very obvious for us that the quality is limited it's not going to depend on demand or use of agents it's just it's also very fainted even quite a few subjects but 30% of people they will act as though the demand is going to be the same they don't spend the demand they don't pay so what's our demand was in the previous period as much to do is to become actually here so when we are working we are doing this but well speaking of hacking and censorship adjustment in fact 30% of people they feel kind of weak but never change but since 30% of people say it's more than the demand or the demand although there are other people who can't it is not there is not much more to do so basically it's just simple repetitions of the same problem of the only thing that might change is maybe you learn a little bit about hacking but each time is just like a new part so minimizing and if you think about demand chasing if you suppose what exactly will be passed in modernization then from Irish or Irish right because the new demand changes in Irish but that explains that's how the demand chase so similarly there are people who try to minimize that's possible so what is this large world demand is small it's not all that much and so much on but similarly to demand chasing in Irish and the consequences in modernization people really don't appreciate the way it is the problem generally will begin with performance and instead of solving this problem of the demand they effectively how they act and solve the problem of the single point of demand say 25 so that kind of squeezes out a certain development so they are very close to the probability of how well they can repeat all this well the problem is that the top of it is still fine actually most of the time it exists only in any kind to understand more as the actual decisions that should be made and all this is most of what we are doing on this research we are only trying to make decisions that fall in each center but we realize that most of the time decisions are possible sometimes you could do different studies so the typical results look like in terms of the decisions so all the time there is only one right and the decisions are in this case where people play the habit periods to play some learning school or I think so interestingly also that people join all kinds of studies so we realize that about 60% of the people will own the article quantity you know 25 to 70% should be ordered the year in the month that's all and so we didn't get to be more than yet so some people believe that 50% of the people is in prostitution some people believe that it can't be substantial numbers like 25% of boys 25% of all people 25% of the people that is in prostitution so all kinds of things and and at the top of that we are going to study the reputation of people in this world so I already understanding what people do in all of the reputation what is the task of the the whole which is going to make sense but in this world we are looking at the first two companies now just like those store managers they they want to maximize their product but also they compete for the size of the size of the because the boys should remember this very well the promise where does the promise come from which happens with the probability that they are high and it's powerful but that's kind of normal with prostitution before we have it's kind of broken that's what we talk about so far I will start with two of the companies that compete because of the interactions that come from boys so I much on some of my customers who are new in a way in this case the product is completely simple and perfectly isolated and the only way we have come here is through companies no that depends on the organizations so here we are looking at the organizations so we spend a lot of time just playing for one year and depending on who is going to be involved also if you are dealing with this what you do is actually a dedicated function and then when you take the speculation it becomes very close that's really been more of a this game the key point is that this is the one that is simple just thinking about it if you are more of the organizationized here then to my opponents it would be pleasure to order by organ storage you know I wait for the order 75 and my opponent would order 74 0.99 and they would win it would be a pleasure to order myself so moving just a little bit closer to the center is actually a job a little bit difficult there is a discontinuity also if you are I order 75 or something like that then it would be 500 in those cases but if you just a little bit less if you go outside because of this slightly less because of this discontinuity there is a non-continuity relation so each quantity is going to stand apart until a certain point because if you go all the way to the ATM and you are respectful because so small will be a person with a certain price and that margin you know she pays a lot of money so it's not that obviously it's costly no no if you get a lot of money it's got no idea from the customer perspective it's not that big it's only in compliance so what do we do with the dynamic of the study game and so of course there could be two there is one equilibrium which is 4 to 5 but equilibrium can be something like 69 and 5 to 6 5 to 6 you know there and the other equilibrium is what you see here is that in those hours most we do in the second one of the study so what happens a lot it's impossible to get into the probability is already imposed it's going to be a curve or it's going to be a curve so then it's going to be a curve or it's going to be a curve or it's going to be a curve And what this is what are the decisions and how it turns out that's what the methodology feels to me there's two different how collaborative junior So, in this case, we are going to have, because of the intensity, we are going to have this equilibrium. And it's possible, in this case, it's going to be uniform. To do this, it's possible. So, the physical equation, the physical self, or the one. So, if you think it's actually uniform, you have to be careful to do this. You just want to make sure that it's uniform. Yeah. But it's actually a question of a soul that we can't solve. So, here, we have a score of two. So, we have score two different. In this case, we have the price. So, if you have a large impulse, you can have this kind of question. Well, for that, you can see that most of the time, most of the time, it's going to be very close to me. And again, because, as we go to the center, it's going to be the right size. And if you decrease the size of the price, it's going to be close to you. So, it's still a mistake. So, if you decrease the price, it's still a mistake. So, if you decrease the price, it's still a mistake. So, it's still a mistake. So, if you decrease the price, it's still a mistake. All right. So, that's what we did. Now, the most interesting part is the audience. It's a form of stone. So, it's just a mistake. And one side of it, try to see if people actually follow the slope of gravity. It's not a way that it has many points, or maybe it's just an origination. And the way we do this, you know, we test the direction of the area, basically. And we use, we just want to use the one by four, basically. One factor point, four levels. So, the manipulation we use to try to induce competitors is called the core range of our strains. So, if you zoom, you don't need to do anything. You don't need to do anything. All right. Up to this form is possible. Which completely is going to be introduced in the other parts. So, in three conditions, on three levels, we're going to start with four. We do not use any horizontal bars. And so, in the last condition, we use horizontal bars. And on slide two, we wanted to... It was a simple slide. It's exploited by the person who was testing the power of the air. But we wanted to test the power of the air. Some of us came up with that. They actually showed that the power works. Now, we wanted to see if that's the air that comes here can also help this part. So, what I'm going to do is... I'm going to explain how to do so. If you're trying, so, in the table, you're kind of trying to make a reference to, you know, transition. If I have two minutes, I don't call that. So, slide two, the renovation process, slide two comes from this. That's kind of competitive, as we are going to see it, actually, about your, you know, sort of, you know, our relationship, but also our health benefits, perspective benefits. And, you see, well, this is a different problem. But in some of the similar contest, there's a, you know, a lot of competition between us, who are in the EPA, and also welcome to the transit companies. They try to see how the use of social preferences may affect the efficiency of that supply chain. So, in this case, there's just a lot of competition. They're in relations with the EPA, very different countries, but the point was that they have a couple of people in which they did nothing. Then they had a status in the agency, in which they could say to the participants, oh, you will, this time, you may have one of your partners. And there was a relationship by condition in which they settled the meeting before the meeting with the prime people. Imagine that personally, you would take that role and just come to the person, share your plans, build up your future relationship. So, that's going to be the end of the exam. So, if you look at the status relationship, what we can see on this job, so, again, the future is in supply, there's a supply chain. If you look at the status relationship, it's actually supposed to be in the status condition, it's mostly in the relationship condition with the practice. So, the competition here is destructive. It's a stress point, but you can compare and patch with the behavioral practices if you might agree. So, that was our power meter, which was one of our projects I think it's about. So, in this case, again, we'll study one. This is an example of what we see in the case of the exam. And what we've done, remember when I was taking Ultracenter, I mentioned several models that can't be explained Ultracenter. So, those are professional factors that apply in this kind of situation. And another here, is that you control for a load and keep only the company. Remember, as you know, it's not very good for control for everything else. Therefore, what we've done, we just explained, it was a standard, it was a standard, it was a standard, it was a standard, it was a standard, it was a standard, it was a standard, it was a standard. So, that's why we were able to use PostgreSQL and CLE1 to see a postgreSQL line. Now, the key standard we've done was for an option for CLE1. First, we introduced a network signal where the projects are. So, that pair was not yet present in the lab that we were at. So, it wasn't in dept, so it took a bit of time as it was in the military was the federal government. So the same conditions were equitated. And you're saying that, well, in the same conditions, somewhere to the past, it's going to be rain. This is it. It has been over 85 years. That's what our subjects mean. And second, let's follow the control. So first of all, we use conflict is the reference point, it's language. And second, how we control the earth. Else, we put it along for the privacy to control. It's restricted, it shows what a conflict is. One more, one less than the reference point. And we want it to simply go over. Right. And the way we manipulate, so what we manipulate is a lot of the experience was the number of columns in the history of data. So after a certain number of periods, you would then produce the next column, which shows the other periods. Not saying anything about, you must not wear it. Not saying that you, well, you have to, in other two periods, either in the historical article, in which you need to make a list of the works and say, oh, there's a very hard one, you know who you are. And then finally, under the first regulation, you enter this number of columns in which you explicitly hold the value of columns. In that case, those are just columns. So that's what we've got. We're much more ready for the beginning. So that's an example of a logistic regression. And it shows how likely, how likely it is that people should make a column closer to the area. And already in the, so the intercept shows the place of the condition we will be seeing, just to show the order of the possible area. And that's where we're going to start. It's closer to here. It's more likely for people to choose one quantity closer to the area. Why is that? Well, as we do these calculations, we'll be size of the health crisis. But in the second part, it's a significant impact on the health crisis. We're still not changing. I guess if you have no data, you can hardly grasp it. And that's what the data looks like. We didn't have very many decisions made by the past, the past, the world, the future. The circle, the size of the circle, represents a number of iterations in relation. So let's go over here and see what we see here. And what we're going to see is the order of the possible area. The first is the shift towards the knee. So it's 25 for the possible area. But our average, so now we're going to have the average shift at the rest of the knee. It's quite a few opportunities, five, so it's going to be 25 points. As we move from knee to knee, one knee to knee, we have all that's become smaller. So people choose that relatively close to knee. Something else, of course, not so many people can see. Yes, indeed, most people do call this put aside, to show it, which is like, and some people show it that way. So they have called us away from knee to knee. In study two, let's see, let me get these, we begin in point, yes. They have passed on to university, they are in study two, and in second study, they could do other different things. So we also, but we just think that if, if it's too close to you, to be able to argue, to make a smaller argument, but...


I couldn't even record it. There's Google. I know it's big news. Did you see that guy? Google? Yeah, big news was bad. Oh really? Yeah, what's Google News? Like Google? Oh, Google News? Google News? Google News? Google News? Yeah, Google News. Yeah, I'm not there mainly. I'm not the only one. Is it now? Yeah, just one way back. Bye bye.


Another way to reduce red tape pretty obviously is to simplify the eligibility rules that created in the first place. In my opinion this is one of the best outreach strategies available. Both parties in the transaction benefit, there's the lower verification and other burdens on the customer side. On the, we've heard the other presentation presenters talk about this, the cost savings. On the agency side that the reduction in workload, the federal rule changes. It's going to repeat that in the past 10 years it's made this easier. The expanded categorical eligibility that was mentioned earlier, Maryland's adopted that. And we've dropped the asset limit for our TANF program as well. And it was, again, the data showed that it was this tiny sliver of clients that had assets that either led enough to close their cases or have them denied. And it just wasn't cost effective.


The inside Peter Young off gate two. Bjarne Pedersen, gate three in white, and Matty Anowski off the outside. Bjarne Pedersen has put a subject of a lot of speculation about his position with Birmingham. And I noticed earlier that when he gave an interview of Bjarne, he'd actually unpicked the Birmingham logo from his baseball caps. All the irregular chances are of Bjarne riding for Birmingham again. Slim. Bjarne Pedersen off the inside, and five racers from that position with Peter Young. And that was a bit of a collision there. Youngster Holt is now there. Watch out for Anowski. Yes! And the men straight for the second time are asking that Matty Anowski out in front, showing the reasons why great-handcourt breaks him so highly. Fast out in front, and he take giving turns. Absolutely right. Matty Anowski in the world under 21 champion. Highly made it, of course, right small. He came 20 stars in the Premier League. And he's a crash rider. With the lead, second place is Andrea Shelton. Third in weight is a Bjarne Pedersen with Peter Young. And the big dance steps at night. Once again, you know, he's in front. You can generate more speed with him. And then he'll pass. Not the greatest speedway, but certainly Matty Anowski won't worry about that. He's got some points on the board.