Ddod, ddangos! Felly dyma'r bobl sy'n ddweud, eich bod yma'r rhaid i ni wedi gweld arall, yma'r llwyddoedd yma, ac mae gennym nhw'n rhaid i chi i chi'n gwybod o'r ffordd â'r rhain ymlaenau a'r ddweud cymaint sy'n cymaintio'r 1916 ymlaenau. Mae'r ddweud, mae'r wych yn ymdill yn ymddill o'r ddafod o'r centinoi. Y ddafod ar yr awdyn sydd cymaintio'r 100th anforsau ymdill chi'n ysgrifennu o'r awdyn between 1912 and 1923 which brought about momentous change in Ireland, marking the end of British colonialism and the birth of the Republic of Ireland. It was a traumatic birth which rose out of a very divisive events. The centenary spanned those listed on the slide here and include the First World War, the 1916 Easter Rising and culminate with the War of Independence and the Civil War. Now this is a hugely important thing for Ireland. There is no precedent for this decade long period of commemoration and historical collections are absolutely central to these commemorations. Archives have never been more prominent in the public consciousness and for a while there you couldn't open a newspaper or turn on the TV without seeing a historical document or an archivist there to interpret it. In my book that is all a very good thing. The decade provides an unprecedented opportunity for the country as a whole to engage with its own history. However, commemoration does mean different things to different people. And many of the events that happened a hundred years ago have an enduring legacy in modern Irish politics today, both north of the border in Northern Ireland and south of the border in the Republic of Ireland. And indeed in Ireland's relationship with Britain and we can see this in the news at the moment with Brexit and the border negotiations. And I think that a sensitive understanding of the nuances of history should be compulsory when it comes to political diplomacy. So how these events are commemorated still has the power to cause division and conflict within communities. Rather than a shared history, the decade of centenaries could be viewed as a sequence of events, each one with different stakeholders and different political undertones. Now the sensitivities of commemoration have been at the forefront of the considerations of the main organizing bodies both north and south. Now the text on the left here illustrates the initial statement of the Irish government's advisory group on centenary commemorations which acted as a sort of mission statement for all the events from 2012 to 2016. This recognises the emotions involved in any act of commemoration as well as a phrase that Geoffrey Crossock made mention of earlier today, the multiple readings and multiple identities which form part of the Irish historical experience. The slide on the right is taken from the decade of anniversary's toolkit produced by the Northern Ireland Community Relations Council as a guide to those planning commemorative events in the north. They developed four principles for commemoration which included recognition of the implications and consequences of what happened and an understanding that different perceptions and interpretations exist. Now the commemorations of the centenaries on that list that I showed you so far have offered the opportunity to re-examine the past and also to shed light on aspects of a shared national story which may have been omitted or perhaps forgotten. This has been demonstrated through the traditional view of the First World War, formerly considered to be more or less solely that of an Anglo-Irish tradition. Through the centenary commemorations of the war there has been a rediscovery of different stories and nuances and there is a sense that history has been reassessed and reclaimed by all sorts of different communities around Ireland. Once again the act of going back to the archives, back to the primary sources is absolutely central in this. So the 1916 Easter Rising has occupied a central place in the decade of commemoration so far because of its positioning as a crucial turning point for modern Irish politics. The Rising was an armed insurrection against British rule in Ireland which lasted for six days in April 1916 culminating in the execution of 15 of the leaders in May 1916 by British forces. This started a shift in public opinion and more importantly international opinion which paved the way for Irish independence. Commemoration of this particular event can very much be fused contemporary politics and this was evident during the commemorations of the 50th anniversary in 1966 which proved to be a very somber militaristic uncritical experience which went on to have very divisive after effects. This is something that community leaders had been very mindful of in the run up to last year's centenary and as a country there was a desire to avoid any comparison with the 1966 commemorations. Needless to say the different sectors involved in commemorative activities approached the centenary with some trepidation but the commemorations of 2016 proved to be a far more creative, inclusive and pluralistic experience. There has been a bewildering range of responses to the centenary from heritage agencies and all across the arts basically you name the medium, 1916 has been commemorated through it. The level of engagement from the general public was absolutely unprecedented and could be seen physically in the numbers turning out for events and lectures and virtually through the proliferation of community and family websites and just general online discussion. There was a particular focus on the Easter weekend with state-led formal tributes to those who died accompanied by what can only be called a city-wide street festival which was part carnival, part history symposium, part family day out and that was organised by the National Broadcaster RTE. The estimate is that one million people took part in that weekend in commemorative activities and to put that into perspective the population of the whole of Ireland at this point is estimated to be about four and a half million people. So what did the library of Trinity College Dublin contribute? Well, we recognised that we are perhaps most famous for the Book of Kells and our collection of medieval manuscripts and we wouldn't be known even inside Ireland as a repository for 1916 archival material. This is perhaps due to the role of Trinity College itself during the rising as an outpost of British Empire where troops were garrisoned and these are British troops in the middle of Front Square and Trinity College during 1916. The centenary provided us with an opportunity to reexamine some of the complexities and to banish some preconceptions. So strategically what we wanted to do was to alert researchers to the potential of the collections to increase their visibility online to act as a catalyst for research on 1916. We also wanted to contribute to the fleshing out of those multiple identities mentioned earlier and also to tell the stories of previously unknown protagonists from different political viewpoints. We also wanted to interact more with our audience and connect with 1916 researchers through social media. So we did this through a number of different initiatives, most notably the Changed Utterly blog project and Twitter account. We also had a physical exhibition in the library's beautiful long room and an online exhibition that we developed and used Google Cultural Institute. So this was the Changed Utterly blog project. It was run by our departments, the research collection departments within the library and it was the library's flagship response to the centenary. It was a year-long blog project with its own website and Twitter account with posts on a weekly basis and its name changed utterly after the most famous line in the Yates poem, Easter 1916. In designing the project, we took our cue from the 1916 collections themselves, which are lots of small, disparate ones that would lend themselves very easily to an episodic project like a blog. And one of the most important aspects for us was that we started a year ahead of the anniversary in April 2015 to use that time to draw attention to the library's resources to act as a springboard for other people's research rather than to simply tell a linear story. The posts were written by library staff, academics and other experts. Lots of our academic bloggers were tapped on the shoulder whilst they were working in the library reading rooms and they perhaps had books coming out that we could promote on Twitter. So it worked as a real reciprocal relationship. Each post links to the library's catalogue entries and to the digital collection site where applicable, bringing viewers through to other library resources. The content is made up of diaries, letters, pamphlets, photographs and objects from all sides of the political divide. And in exploring this material you can see the stories of ordinary people as they struggle to comprehend what is happening to them. And in others you get a sense of the real complicated social and political situation in Ireland at this moment. And this is illustrated quite incredibly by the library's own copy of the Proclamation which is Ireland's Declaration of Independence. It was taken from the walls of the GPO, the rebel headquarters during the rising. And during conservation 11 First World War recruiting posters were found to be attached to the back of it. And this articulates the two opposing political viewpoints coexisting at this time. One group recruiting for the British Empire, the other rejecting empire and proclaiming a republic. Now one of the strengths of the library are the accounts written by women caught up in the rising. And some of these have only been highlighted for the very first time through the Change Duttley project. And there were many other diverse posts from Trinity's role in the rising to that of other institutions such as Dublin Zoo. And that post proved actually quite popular with primary schools as a way into the material for their students. As well as posts on poetry, armoured vehicles, rugby and stained glass. The content is incredibly rich and varied and I think this is why people really engage with it. It was telling lots of unexpected stories. It kind of didn't know what was going to come up week after week. Sometimes we didn't know what was going to come up week after week. So that leads us on to our attempts to measure the impact. Now we've tried to take a look at the immediate impact felt during the project. So from April 2015 to April 2016. And we also have noticed since the project sort of finished, so to speak, a change in the sorts of engagement that we've received in the 18 months since we finished blogging and tweeting. And I'll also briefly mention our approach into the project for the future. And I have to say that I wish we'd built in a few plans on how to do this when we started. We kind of didn't. Firstly, some quantitative statistics in terms of views of the blog and Twitter followers indicate that we finished the year of blog posts with 65,000 views of the blog. And the subsequent year saw 10,000 further visits with the spike on the anniversary. These are of course just the number of clicks and they don't convey any opinion on the quality of what we produced or how it helped other people. Twitter enabled us to tap into research communities and spark discussion about our blog content and provided another access point into the material. Our followers were mainly academic, local history and official 1916 sites. But we also had a great deal of engagement with teachers and with general researchers at large. It's also been very useful for calls for information. Our followers identified the location of a photograph of an armored car, for example. And we also found out how many armored car enthusiasts there are out there. By April 2016 we had 2,200 followers on Twitter with a further 300 arriving the following year even after the blogging had finished. So when trying to organise, sorry this is a bit wordy, when trying to organise the qualitative result for the impact felt by the end of the project, I've tried to separate it out into visibility, student engagement, academic engagement and external engagement. And I'll just pick out a few prominent examples. Firstly we started to notice a huge increase in image orders for items featured in the blog. Many of these were used in other institutions exhibitions such as the flagship exhibition at the general post office or popped up on hoardings around town like this one that I just took on my phone that popped up outside Ulster Bank on O'Connell Street. Another unexpected outcome has been the increase in levels of donations of new archival material, like this autograph album which came from the Frongurch internment camp in Wales where many of the revels were taken afterwards. And the blog post on this was actually written by the relative of this internee here. As well as an upsurge in researchers of 1916 material in the reading rooms, we also assisted student groups with project work like the digital humanities students who were working on a visualisation of one of the eyewitness diaries. And we also worked with the student philosophical society on their centenary week exhibition. We collaborated a great deal, one of the great things about this project is we made other connections within the institution and we collaborated a great deal with the public affairs and communications department and built relationships there. So I don't know why we were taken by surprise by the media attention but we certainly were. We featured in the Irish Times and on the RT news and there was also an exclusive on our copy of the proclamation in the Guardian. We also formed a collaboration with the Irish examiner newspaper who turned six of our blog posts into articles in their newspaper. We were also finalised for a blog awards Ireland and also an e-government award for promoting Ireland overseas and we were described as one of the more innovative responses to the rising on RT's history show. So thinking about impact after the project, when scoping out the project initially we spoke to Trinity's history department and they advised us based on their own experience of the commemorations of the 1798 rebellion in 1998. They said that there's a huge surge of interest from scholars for the anniversary period and then it's generally left alone as a subject for study for the next ten years. So we were expecting the same and in fact what we've noticed is whilst students engaged in pure historical research that has kind of tailed off. We have found ourselves speaking to students working on more reflective and analytical research on the nature of commemoration and how institutions have gone about this. I've been in contact with such students from Ireland but also beyond St Andrews, Glasgow, Paris and most recently earlier this month someone in Detroit. We are also aware of the publication of works drawing on our collections that's starting to come through and we're also one of the joys. We're really delighted to be involved in the piece of onsite theatre called Meeting Ghosts in College Park which drew a number of its characters from personalities that had been featured in Change Duttley blog posts. Another huge benefit actually is that our work has been noticed by college authorities and the research community. We've been asked to speak about Change Duttley at conferences and also to contribute to reports and booklets. We appeared prominently in our Provost's annual review last year and I was asked to contribute an article on the project for the Dean of Research's report for this year and that to my knowledge is the first time that the library has been profiled as a producer of research. The success of the project has also given us great confidence in producing other digital projects and great leverage when encouraging others to collaborate with us. This was the case on this summer's Jonathan Swift online exhibition which was led by us but in partnership with other Dublin cultural institutions. We've also been approached by academic schools wanting to model student capstone projects on the Change Duttley blog and that's been enormously gratifying as well. Sustainability is always a problem with a digital project and so we wanted to see if we could set Change Duttley up as a portal for the 1916 collections that could be used into the future. This Easter we gave it a revamp. It still has the 52 blog posts but it's now enhanced with audio-visual material either produced by ourselves but mostly from Trinity academic colleagues. So it now contains podcasts to the Humanities Research Centre, the Trinity Longroom Hub, held a symposium on the proclamation. It holds those podcasts and also it has a YouTube lecture course produced by history lecturer Patrick Gagan on the background to the rising. So it's a really rich research resource now and it's supported by Trinity's research IT department to help to try and assure its technical survival into the future. It's also been captured by the web archiving projects that were run by the British Library and the Bodleian and also the National Library of Ireland earlier, well late last year. Now heading into the commemorations at the beginning of last year the whole country worried that it might be divisive and nationalistic but in fact it enabled communities to come together and was enormously enriching. So much so that the government have recognised the societal and political importance of heritage and culture in fostering this feeling of togetherness and exploration of the past for the future. This reached a very welcome crescendo at the end of 2016 with the announcement of a state sponsored framework to build on the legacy of the 1916 commemorations called Creative Ireland. This programme aims to continue the sense of community pride and appreciation for culture which emerged during the centenary. There is of course concern amongst arts and heritage agencies that government rhetoric is backed up with appropriate investment and this wasn't completely fulfilled in the budget announcement of 2018 but it is a five year programme so we watch with eagerness. So now we look to the future and the more potentially contentious commemorations such as that of the Irish Civil War. The advisory group statement for commemorations from 2018 to 2023 was released last month recognising we should also be conscious that on this island we have a common history but not a common memory of these shaping events. And that the aim of commemorations should be to broaden sympathies without having to abandon loyalties and in particular to recognise the value of ideals and sacrifices including their cost. As for cultural institutions I foresee more collaboration as we all got to know one another and built a great sense of camaraderie throughout the 1916 commemorations and whatever we do we will take our cue from the material itself and let the archives speak for themselves. And I'd just like to take this opportunity to thank my co-lead Shane Moore and all our fabulous colleagues and contributors and thank you all for listening.


In this video we provide the solution to question number 15 for practice exam number three for math 1050 We're given a rational function f of x equals x squared times x minus six all over x plus four Times x minus three squared We have to graph this function, but it also asks us to list some informations like what's the x intercepts of the graph That's the first thing it mentions the x intercepts are going to come from the Rational functions numerator what makes the numerator go to zero and then it's in reduced simplified form There's no common factors here, so it's already simplified. We're going to get intercepts at zero when this guy goes to zero and Six when this factor goes to zero we needed we need to indicate their multiplicities here so notice that zero has an even multiplicity and Six has an odd multiplicity. This tells us that at zero we will touch the x-axis, but not cross at six We will cross the x-axis To find the y intercept There's only the one that happens that when x equals zero so we plug zero into the formula there You're going to end up with zero squared times negative six All over four times negative three squared. This just turns out to be zero So let's label what we discovered so far so zero zero was an x intercept and a y intercept and then we had x equals six Going forward here. So one two three four five six Like so. All right. No big deal there Next Let's consider the asymptotes. So let's look for the vertical asymptotes. These are going to be the things That make the denominator go to zero again when you're in reduced form If you had something that cancelled out that could give you a remove point there isn't one in this example though So we have an x plus four we have an x minus three squared So we have vertical asymptotes at x equals negative four and at positive three again It asks us to list the multiplicities negative four has an odd multiplicity of one three has a even multiplicity of two So at negative four we are going to cross infinity, but at positive three we are going to touch infinity So I want to add these to my graph here So we have a vertical asymptote at negative four add that in there And I'll just add a little label here. This is x equals negative four and then we also have one at positive three Do your best to keep these things straight Usually like to do a dash line to emphasize. This is a vertical asymptote This is a place we want to avoid x equals three in this situation And then the last thing it wants to also consider it wants all asymptotes So is that our bleak asymptote is there whores us until what's the in behavior of this thing? So notice that the in behavior is going the in behavior here We're just going to look at the leading terms. So you have this x squared Times x over x times x squared again That is a week nor everything except for the leading terms This looks like x cubed over x cube, which really just comes becomes a one This is going to be our horizontal asymptote And so let's add that horizontal asymptote to the graph now There's no scale on the y-axis just on the x-axis. That's okay. You see the little tick marks right here We didn't do that on the y-axis and that's because the scale of the y-axis doesn't really matter so much for these pictures It doesn't have to be a perfect one a perfect scale vertically speaking Mostly it's we were trying to pin tension. How does it how does the function behave near its intercepts and near its asymptotes? That's what we're looking for do label these things of course especially since there's no scale in play here So this is y equals one. Okay, and so with that now in play. We're ready to start drawing our picture What's going on here? So we have for example our intercept here at six So we have x equals six right here. What's gonna happen? Well, if we're at this intercept, we have to somewhat we have to we have to either go up or we have to go down Right, those are only two options. Well, do we go up towards the asymptote the horizontal asymptote? Are we go down towards whatever? Well, if you go down, you're gonna have to at some point turn around Right, and there's no x-intercept so you can't cross or you have to go off the screen and come back to the side But there's no vertical asymptote. So those options are not available to us So it seems that we have to go up from x equals six there So go up and bend towards your Horizontal asymptote right there We could we probably should be checking is it do we cross or horizontal asymptote? We do something like this. Is there ever a point where the function equals one? You can check that out real quickly There is no such location This one it doesn't cross it's horizontal asymptote that doesn't mean that's always the case but in this one We don't Since at six we cross the x-axis. We have to come back down from the other side, right? Now we can't go back up again because well, we don't have an x-intercept. We have to go off towards infinity Meaning we have to go closer and closer to our vertical Ascent of there at neg or a positive three now at positive three Remember we are going to touch infinity. So we're gonna come back from the same side We started with draw a little arrows right there Then we're gonna come towards x equals zero x equals zero we touch the x-axis So we're gonna come back down like so again. There's no x-intercepts to go up from so therefore We're gonna have to go down towards our Vertical asymptote at negative four we cross infinity so we come up from the other side Like so and again, there's no more exit. There's no more x-intercepts or any more asymptotes So we're gonna have to start approaching our horizontal asymptote giving us something like this All right, so let me draw that last part again a little bit better a little bit more smooth But in general a picture is gonna look something like the following. It's not perfect It's a little crude. You see my finger wiggling a little bit as I'm drawing this not a big deal Even though it's crude. This is exactly the type of picture. We want we should label things of course So we have the intercept zero zero. We have the intercept zero six zero We already labeled the asymptotes x equals negative four x equals three and y equals one And so that was everything it asked us to do we listed the multiplicities of the x-intercepts and the red asymptotes So we're good. This then finishes question number 15


Hello and welcome to the Grand Line Review, your source for everything One Piece, and today we are here to do the impossible and rank every character who has ever held the title of Emperor in the series. Now to some of you that term may be better known as Yonko, which literally means four emperors, however over the course of the series there have been five characters who have held the esteem of being one of said four emperors, as well as a super sixth character who has even been labeled as the fifth emperor. But these are going to be the biggest of mums and the whitest of beards, and with that in mind before we begin it's time to play a round of Big Beard, a very simple minigame the rules of which are as follows. I am going to present you with a character and then we are going to adorn them with either a Big Beard or a Little Beard. You simply need to guess whether or not the beard will be big or little, and should you guess incorrectly then your punishment will be to subscribe to the Grand Line Review as well as forever be ostracized from the facial hair community. And if you do guess correctly then you will be awarded the title of beard master. But today our character is going to be Boa Hancock. Will she be given a Big Beard or a Little Beard? Make your choice now and we shall reveal the answer in three, two, one, and it's a Little Beard. I suppose that's because Boa Hancock prefers that kind of sleek, well maintained style. So if you guess incorrectly then you know what to do and please do say hi in the comments below if you are a new member of the Grand Fleet. Welcome. But back to emperors exactly how will we be ranking these stellar existences? Well the same way we generally go about these videos which is by me asking you to choose your favorite. And so I put up a poll in my community section which had just over 9,000 respondents who cast a total of almost 14,000 votes. And the discrepancy for these numbers is because multiple votes were allowed so if you simply could not choose between two, three, four, perhaps five or even all six then you could just go ahead and do your thing. Although selecting all six would have had a net impact of absolutely nothing so I hope you didn't do that and if you did your vote was pointless. But let's go ahead and discover just how popular our emperors are commencing with the six and last place which will be Charlotte Lynn Lynn who managed to acquire a fairly meager 869 votes. And look I'm not necessarily surprised that Big Mom ended up in last place because my anecdotal experiences would suggest that she is by far the most unpopular of the emperors and even quite unpopular as a one-piece character in general. There is a fair bit of hate for the Big Mom however with almost 14,000 votes delegated I still didn't quite expect her to do this profoundly terribly because for some perspective on that 869 votes works out to about 6.25% of the overall votes which is not great is it. And as for why this is Big Mom is definitely a polarizing figure there's a fairly large faction of the online one-piece fan base that get incredibly frustrated when Big Mom gets used for any sort of comic purpose because they view it as demeaning or undervaluing her status as an emperor. The idea being that one of the four most powerful pirates in the world should be serious business at all times. Or as Big Mom or even Big Meme as she is sometimes known is a bit of a joke. It's a thought I disagree with quite a bit because I actually really enjoy the variety that Big Mom brings to this group. I do like how she's pretty much the most powerful human being on the planet but isn't quite mentally capable of using that power to its full potential. There's a very nice balance of character there for me. But Big Mom also has a bit of a tough time because the only two arcs that have really heavily featured her are also quite polarizing being Hulkic Island and Ilwano. Both mega arcs that can at times feel very slow moving and there may be a lot of inherent association of the character to those experiences. And I'm also going to go out on limb here and say stuff like the amnesia incident on Ilwano probably did not help the Big Mom case here. Once again it was one of those things that I really enjoyed because we got to return to and explore the more innocent child side of Big Mom but with that said if asked I would be highly unlikely to label her as my favorite emperor so there's that. Also this poll isn't no way representative of how many people like or dislike Big Mom. People en masse just seem to prefer every other emperor a bit more. In some cases a lot more. Although that's not so much the case for our number five finisher Marshall Deteach also known as the man with the beard that is black. And he got and also quite disappointing 980 votes which isn't terribly far from Big Mom's total just over a hundred votes away really. And again it's not so much a case of being surprised at the positioning but rather at the sheer lack of votes for this man.  To be fair though Blackbeard has a pretty purposely grotesque design so that's always going to be a barrier to like ship for him and he really does represent the very traditional image of a pirate a filthy missing teeth alcoholic gun toting party maniac which the more I think about it the more fun it sounds. Still there is a lot of intrigue surrounding him he actually presents one of the biggest mystery existences in the series and every time more information is given about Blackbeard it only brings up further questions. Plus as someone set up to be the final antagonist of the series you'd figure that popularity was a thing that each may have some access to. Interestingly enough though even if we aren't talking about the emperor specifically Blackbeard has never really been a popular character in one piece and the highest he has ever ranked in an officially conducted poll was 49th place which was during the sixth poll where he was quite notably beaten by Farafra in 48th place that's right Farafra. Although to be fair to Blackbeard there was a very strong meme campaign in Japan trying to push Farafra to the top but still Blackbeard is very much sitting on the boundaries here. Edging rather dangerously between the very concepts of popularity and unpopularity. Blackbeard does also have a major disadvantage here though being that he is one of the most underexplored of the emperors so there's just not a lot to work with. However at the same time we had plenty to work with with the very well explored Big Mom and that didn't really end up helping her so who's to say how that would impact things. So only time will tell if Blackbeard will ever rise in the ranks however this snapshot is not favorable for him. So moving into fourth place we have at the time of this recording our current major antagonist Mr. Kaido who managed to secure 1,363 votes which is still not super amazing but a definite step up from our prior two contenders. And Kaido is interesting because quite frankly I can tell that he's a fairly unpopular existence in the fanbase because every time I make a video focusing on Kaido it does some pretty sadly average numbers. Whereas making videos focusing on any of the other emperors including Blackbeard can be the equivalent of YouTube Gold. Still I think that Kaido is definitely rising in popularity slowly, very slowly mind you but steadily. And I think that if I presented this poll a year ago Blackbeard probably would have defeated him. But recent events on Wano has given this emperor some much needed focus because for a lot of his time in the series Kaido was just sort of the blank slate emperor. He wasn't the inspirational figure of Shanks nor the ultimate generational presence of Whitebeard, not set up to be the final antagonist like Blackbeard and certainly not the zany character of Big Mom. Kaido was just sort of there being Kaido, whatever that is. The most underdeveloped of the four emperors and that was certainly reflected in his official popularity poll results where he has only been ranked once and he landed in 97th place. So all of a sudden Blackbeard's 49th place finish looks pretty all right. Notably this official poll took place in 2017 prior to the Wano arc so I think that pretty much says it all. The determining factor will likely be when we see some kind of Kaido specific flashback but even then there's no guarantee of a rise in popularity because once again the Big Mom example proves that having a fairly well fleshed out character does not equate to people automatically loving said character. Unfortunately for her and potentially unfortunately for Kaido. But heading into our top contenders now we landed the bronze medal which will be awarded to a certain monkey D. Luffy who has invoked the gear third equivalent of vote securement lashing out with a gigantic 2941 ticks of approval. And here is where the comment shall begin. Luffy isn't a real emperor, fake news Morgan's Luffy is an unofficial emperor you can't have a fifth yonko etc I've heard them all especially that last one actually because Luffy was never declared one of the yonko he was called the fifth emperor. So the yonko arguers just have a fundamental misunderstanding of one piece and potentially even life itself but regardless of whether you think he deserves to be here or not that's where he is and we do need to recognize it. But because I was curious I decided to put up a poll to determine where the fan base currently sits on the issue and 110,000 votes later which is a lot. It's looking like the overwhelming majority of you agree with Luffy's emperosity and I'd like to thank everyone who voted in that poll I was honestly expecting about 10,000 so 110,000 is just pretty well. Now granted it also might be unfair to put our main protagonist on a poll like this but after seeing his position I really don't think that it is. Most voters seem to have taken this question quite seriously as selecting their favorite emperor rather than character otherwise I have no doubt that Luffy would have won which means that we are evaluating him at least somewhat on the criteria of existing as an emperor. That or the reason why Luffy didn't reach first place is because the voters refusing to recognize him as a candidate although with the benefit of knowing on with the results I can tell you that 22% of emperor deniers would not have made the difference here. In any case it's hard to say anything more about Luffy that hasn't already been said in 700 of my other one piece videos as well as those of every other one piece youtuber on the platform so I think I'll just leave things here as we move on to our top two candidates and with that our second place is none other than Edward Newgate with a very well-deserved 3,213 votes almost a quarter of the overall total and really this is because Whitebeard was not only an emperor of the sea he was pretty much the emperor. He is the bar that we measure all other emperors to because Whitebeard was the first to be exhaustively explored in terms of individual power greater military might and overall character which of course occurred during the paramount war and in a really weird way you could say that Whitebeard is the character responsible for the more underwhelming impressions of Big Mom, Kaido and Blackbeard because how is any one of them meant to compete with him? Whitebeard was just so well-rounded with a superbly compelling history and personality and this gets to the point where even to this day Whitebeard's legacy is still active and felt throughout the world his actions in life and in death are continuing to help shape modern events as we know them and speaking of death I would also flag that as a pretty huge contributing factor for Whitebeard's popularity. I recently made a video on the most rewatched scenes in the entire series and Whitebeard's death was stated by an astounding amount of people as one of their go-to scenes so I guess one of the greatest and most accurate things you could say about Whitebeard is that he has had a colossal impact on the hearts and minds of One Piece fans which, Luffy aside, I'm not sure we could so conclusively state about anyone else featured on this ranking. Well, until we reach our number one position that is and you've probably figured it out by now but it is of course one red-haired Shanks. Utterly dominating this poll with 4535 votes we have the emperor we first met all the way back in chapter one of the series and despite it looking quite obvious in retrospect Shanks ranking in first place was a bit of a shock to me because he is surprisingly enough probably by far the least export of the emperors. We know very little about his capabilities, his forces, territory and all that fun stuff but at the same time that mystery is probably one of Shanks's greatest assets in a poll like this because previously I stated the characters like Big Mom, Kaido and Blackbeard had no hope of living up to the standards that Whitebeard set but because we know so little about Shanks there is still the potential for him to not only meet but exceed those standards and without the curse of knowledge the imagination is free to perceive Shanks exactly as one sees fit which means that the entirety of the One Piece fanbase likely has a universally great opinion of Shanks's emperor because he is currently able to fill all of their ideal individual criteria simply through omission and of course I do need to mention the obvious which is that the readers and watchers have a clear emotional connection to Shanks which was always going to help just as it did with Whitebeard but the relationship with Shanks runs to an even deeper core because he really is the reason why Luffy embarked on this journey in the first place so he really has this disgustingly perfect combination going on he's still mysterious enough to inhabit the role of an emperor not risk disappointing anyone but he's also heavily bonded to the audience resulting in a clear undisputed number one emperor of the sea but what do you guys think please do leave your thoughts in the comments below or even join my discord server and if you'd like to see more videos like this then please do go and check out some of my other content or even subscribe to the channel for more glorious One Piece business uploaded straight into your youtube feeds but for now this has been the ground line review and I'll see you next time


The heat is a big story this weekend. I heard a mild spring is taking a scorching turn tonight We have first with temperatures reaching the century mark in many areas in California this week Officials are saying it's a wake-up call for what to expect this summer So we're taking advantage of what's happening naturally to also remind them that summer is coming and with summer in California We have some pretty extreme heat and now with advances in technology and forecasting The National Weather Service has developed a new online tool that makes beating the heat easier. It's called heat risk It puts the heat into a climatological context It kind of tells you if this is normal heat for this time of year if it's something that might be more susceptible to pets and sensitive groups Or if this is going to be something that we need to kind of sound the alarm for for everybody that everyone needs to be changing their behavior We talked with a scientist who developed this tool and Michelle Meade gave us an inside look showing how all that complex weather data is Simplified in just a few clicks. You can see here. We have the forecast out to seven days So you could click through each seven day to see or each day to see how the heat risk is going to be changing Or you can zoom in to where you're interested in We'll do downtown Sacramento And the seven day forecast is going to pop up and it shows you what heat potential is each day of the week on that one Point location They call this the experimental heat risk tool and are hoping people will begin to use it when planning their outdoor activities this summer As long as folks know the steps they need to take now When we do get into the heart of the heat season when those temperatures can become You know hazardous to everybody they'll have the tools in their toolkit to make sure they're prepared and safe And we won't have those health and human impacts that can happen for me


OK, hej. So, I start by recalling you the last lemma of Thursday. So, we have that, we prove that if we have a measurable integral function and such that we have the integral of f over a, where a is any measurable set contained in e is non-negative, then we show that we prove that f, we can infer something about the sign of f. So, we have that f is larger than equal to zero almost everywhere in e. OK, now, so we want somehow to relax this requirement. So, here we are asking that this, we have to be sure that this holds for any measurable set. So, we take as test set any measurable set. OK, now we want to relax this a bit. And we consider, as lemma 2, f, an integral function, this time we define it on an interval i, interval, and such that the same requirement on the sign on the integral is verified, so f is not equal to zero, but this time u is enough that we have to check for any u open, OK, for any u with u open, and then, OK, then the conclusion is the same, then we have that f is larger than equal to zero almost everywhere in a, in i. OK, so somehow here we relax the requirement, because it's enough to check only for u open, for u open. OK, so, so somehow this is also a consequence of the, an application of the Lebesgue Convergence Theorem. OK, of course we want to use this result, OK. So, we recall that if we have a measurable set, then we can approximate it in a quite nice way, OK. So, we know that then the resist set g, we always call it g, belonging to the class g delta, which are countable intersection of open set, OK. Such that we know that a is contained in g, and the measure of g minus a is zero. OK, so let's, we have that a is equal to this minus g minus a of f, this is zero, OK, because the measure is zero, so we have that this two is i. OK, so by definition we have that of g, we have that we can represent g as the countable intersection of open set, so u and open, and here with no loss of generality we can think that this has a decreasing sequence of, of set, so with loss of generality we can assume that this is a decreasing sequence of open set, so u1 contains u2 and so on, because here, no sorry, u3, un, because here the idea is that for instance if it is not, if it is not fulfilled, then you replace u2 with u1 intersection u2, and so on u3 he would replace u3 with intersection of u1, u2, sorry, u3. OK, now we consider the characteristic function of each open set, un, so we have, we have that characteristic function of un that converts to the characteristic function of g, so in fact we can distinguish two cases if, so if x is in g, then it is in, we have that cg of x is equal to 1, and we have that c un of x is equal to 1, because g is contained in u1. OK, so if x, point wise, in fact, no, no, here is written in fact, here I mean almost everywhere, no, the point wise convergence of n, this is an explanation of why this is true. OK, so if x does not belong to g, then this of course is equal to zero, and if it is not to g, it means that there exists an index, and there exists an index x not, such that for any n larger than n not, we have that x does not belong to un, OK. So basically you have that from key of un of x is equal to zero for any n, OK, larger equal than n not. OK, OK, now, and we have that this sequence is a decreasing sequence, OK, because the set are decreasing. So now we want to multiply for f, this, and we get that this converges, of course, to g, OK, almost everywhere, OK. OK, of course we want to use the back convergence theorem, so we also noticed that, we started by the hypothesis that f is an integrable function, un, so this is bounded by f, and so by the denominator convergence theorem, we have that we can pass to the limit under the sign of integral, so we have that f of key of un converges to, OK, this is, of course, f of un, and, OK, so we have that this is, OK, these are open, OK, so by hypothesis we know that this is non-negative, and so we have that basically, OK, this is, we know that g of f is larger or equal to zero, but the beginning we proved that this was equal to the integral over the test, the measurable set a, to introduce, OK, so from the lemma before, from lemma one, we can deduce that f is larger or equal to zero in i, because, of course, a is an arbitrary measurable set, OK, now we want to relax more this hypothesis, so maybe we can test the positivity of the integral only on a special open set, which is namely on, on open intervals, OK, so now comes the lemma, so again we have f integrable, and we have, and we assume the sign of f, the sign of the integral of f over j is larger or equal to zero for any interval, for any j containing the i, where j is an open interval, OK, and then, OK, we have the same conclusion, we have that f is larger or equal to zero, almost everywhere in i, OK, this time to prove lemma three we want to use lemma two, and again here the strategy is analogous of the one that we use here in the sense that we want to approximate an open set to use lemma two by an open interval, OK, so we saw, we know that, we know that if you start by, let's call if u is open, then u can be represent, as a countable union of this joint open set, so i and one to infinity of, called i n, i n, and this joint open interval, this is, we prove this one of the first lectures, OK, so now we want to say something on the sign of this integral over u, OK, now we use of course the countable additivity property, so we have this is equal to i n of, yes of f, and this we, OK, we know that this is positive, each of them is positive, and so by lemma two we can say that this is true for any arbitrary open set, so we can, we have that since u is any arbitrary open set, then we can infer that f is indeed larger or equal to zero, OK, I must everywhere in i, OK, OK. And now instead this theorem, you consider again a function, an integrable function over an interval, define an interval with values in a, it should be, must be integrable, assume that i is an open interval, and assume that the integral of this product, now I told you it is phi, phi is always zero, for any, this is true, for any phi, which is c infinity with compact support in i, OK, phi is c infinity with, OK, this must be true for any phi, OK, and then what can we say about phi, f in compact support, compact support, you know what it is? Ah, OK, OK, phi, compact support means that the set where f of x is different from zero, the closure of this set is strictly contained in i, the, the, no, no, no, you just, you consider the set where phi of x is different from zero, take the closure, so it's, and you have that, this must be contained in, strictly contained in i, OK, this means compact support, OK, so what we want to infer is that a point-wise property again, that f is equal to zero almost everywhere in i, OK, so this is a very fundamental theorem, especially it arrives in distribution theory, probably we will see later. OK, so let's prove it. OK, so we consider an opening interval, j, an opening interval, such that j is contained in i, OK, just, for instance, j is equal to a, b, just to fix the idea, and then, no, just believe me, we can construct, we can construct a sequence phi n, in, so, we satisfy this property, such that we have that, yeah, non-negative, and they converge increasingly to key a, b, j is integral from zero to b, from a to b. J, j is n, b, j is integral from a to b? Yeah, it's an interval, it's an opening interval. I don't know, it's called a to b, not from zero. Ah, no, but a to b is, and this is the same, OK, just to fix the idea, OK. OK, so by hypothesis, we have that, we have that, the integral over i of f times phi n is equal to zero. OK, and moreover, again, we want to apply the dominated convergence theorem. OK, so we have that, this converts pointwise, b, n. Yeah, which is increasing. Yes, pointwise, I mean, maybe it is not necessary, just consider this, this is enough to what we have to do, but in addition, it's also good. OK, so f times phi n is less or equal than f, just to be, and she's integral, and then, I give the next convergence theorem. OK, we have that, of course, that n converts to f, this is equal to zero. Ah, OK, let's, so this converts to f, to the integral of f over i. OK, now you can think, so basically you have that, so you have that, this is zero, OK? And then, think of this as the fact that this means two things, but you can see this as these two facts satisfies it tangentially. This means that f is, must be larger or equal to zero, almost everywhere in i. This is j, OK? f is less or equal to zero, almost everywhere in i, so basically what you get is that, indeed, f is equal to zero, almost everywhere in i. OK, now we'll try to, somehow to outline how we can construct this such a function phi n. Is it clear or not?  And you can think of it, as these two inequalities satisfy, OK? This is the equivalent to say that the integral of f over i is equal to zero is equivalent to say that these two are satisfied, OK? Yeah, yeah, yeah, you played the last lemma, no? This is, no, we said it with the larger or equal, the last year. Yeah, we said it like this, like this part, the integral over j of f is larger equal to zero, then f is larger equal to zero. And then, of course, you can prove that it's the same for the less or equal. OK, so how we can construct such a function phi n. So consider just to fix the idea i to be an interval p of n points of a, b. And then we have that there exists a sequence phi n of function in c infinity, c has infinity with complex support in phi n on i, in sej that they are in between zero and one, OK? And they converge. We have that, what we used before, they converge increasingly to the characteristic function of a, b most revenue. OK, so the proof is a constructed one, so we have to proceed by step. So, first of all, let me introduce the following function, f, defined in a piecewise way. So f is defined from r to zero, one as follows. OK, it is zero for x, if x is less or equal to zero. Otherwise, is equal to e to the minus one over x, if x is positive, very strictly positive.  It's continuous on r, because if you let x tend to zero from the right at the limit is zero. OK, so what about the derivative of this function? So what about the derivative? So we have that f prime of x is equal to zero, of course, if x is less or equal to zero, otherwise, is equal to one over x square, so do the computation times one over, OK? Again, you can prove that f prime is also continuous. Square, yes, x square, x square, f prime is continuous, also in zero. OK, you can prove this either by considering the incremental quotient, or just by observing that the two derivative exist in the two pieces, and they coincide in zero. OK, now, if we go on, so the second derivative and so on, what do we obtain? So you would have that second of x is equal to zero, x is, and you get, or here you get minus, no, plus x to the power minus one over x, x positive. And basically here, if you look, the idea is that if you go on with the derivative, here you obtain a polynomial of one over x, OK? But of course, we have to prove that this is true, and we will do this by induction. So again, here, second is continuous, and so on. So we claim that the k derivative k of x is equal to zero, if x is less or equal to zero, otherwise, it is a polynomial pk of one over x times e minus one, one over x, so x is positive, pk is in the variable one over x, OK? The step k equal to zero is true, for the step k equal to zero and the claim is true. We have that, so we prove, we want induction, OK? So we have that if is equal to zero, then OK. And so we assume that the claim is true for k, it's true for the index k, OK? So we have that if you do the computation, what you obtain? It's zero, it's x is less or equal to zero, otherwise, you get something of the type minus one over x square, pk prime one over x, e minus one over x plus pk one over x, x positive. So basically, if you collect e to the minus one over x, you get the t, so you have that pk plus one would be minus x square pk prime one over x plus pk one over x, OK? By analogous consideration, you can prove that this is continuous in zero. So indeed, we finally prove that we have that such an f is c infinity r, OK? And another mean related property is that such a function, the function f, provide an example of a function which is c infinity, but not analytic, OK? Because you have that the derivative of any order are zero, so the series converts to zero, near zero, but the function is not zero, OK? OK. And now we want to construct using this f, this function phi n, which converts point wise to the characteristic function of an interval, OK? OK, just to visualize the things, this function is zero here, and here it's very flat, because the derivatives of any order are zero, so it's something like this, as as an asymptote in correspondence of one, OK? So at level one, you have the dot, OK? So how we can modify, we can handle this function f. We define a function g to r in such a way. G is defined by means of f, like this, f of x times f of 1 minus x. OK, just to observe that if x is non-positive, you have that f of x is zero, and then g of x is zero, and if x is larger than 1, f of 1 minus x is also zero, then, again, g of x is zero, OK? So just to, so the graph of g looks like something like, something like this, OK? OK, now we want to use g to proceed, and we want to construct this function, so we want to construct function g, capital G, which is c infinity r, and which satisfies the following properties, that g of x is equal to zero for any x less or equal to zero, and it must be in between zero and one, if x is in between zero and one, and must be equal to one for x larger than one. So a way to, so to provide such a function capital G by means of this small g is the following, and we define g of x as the quotient between the integral, between zero and x of g of t, dv t divided by the integral between zero and one of g t. Because you have that, if x is less or equal to zero, then g is zero, OK, and also g is zero. If you have that x is larger than one, then you can look at this, the numerator has zero, plus one x, and this is zero, and so g is one. In this case, you have that also g is in between zero and one, so, OK, if you have x between zero and one, then g is positive, and so this quotient is between zero and one. OK, now we define a contraction of this g, so how it looks like, this g looks like this, you have, OK, this is zero, so here is zero, and then you have a transition from zero to one in this interval zero, one, OK, this is the level one. So now we make a contraction of this bg in order to make this transition from zero to one somehow faster, OK, so we define g epsilon of x, g epsilon of x, has g of x divided by epsilon, so it's a contraction, and so how it looks like, the graph, you have zero, this times transition happens between zero and epsilon, so you have something like this, and of course g epsilon is still c infinity, OK. The first graph is for capital G. Sorry? The first graph. Yeah, the first graph is for capital G, and this is for g epsilon, OK. See, the only thing that changes this part, this transition. OK, and then basically what we want, so we have to define this function phi n, which approximate the characteristic function of a, b in a c, of course they have to be c infinity. OK, so we are interested on somehow on this transition, on this part of the graph, OK. So we will use this g epsilon, how it is defined in this part from zero and epsilon to approximate the characteristic function, so OK, you will have basically the level one, and here I can construct this function phi n, I just show you, with drawing, in by using this transition between a plus epsilon, for instance, and a minus b, b minus epsilon, OK. And I define this function phi n in between a and a plus epsilon has g epsilon is defined here, OK. So it would be like this, and here it's like, you can argue, you can reflect the graph, and this, the dashed line represent the graph of the phi n, OK. So basically what I found is that the phi n are c infinity function, they have compact support. OK, actually you can, no, to have compact support, but actually I had to, so let me consider, OK, let me move a here, and here is a plus epsilon over 2, OK, so a is here, and here is b minus epsilon over 2, and b is here, OK. So there must be some room to put then phi n equal to 0, OK. So they have compact support. From here, epsilon is 1 over 2, I took a plus epsilon over 2, I mean, OK, a plus epsilon over 2 plays the role of 0 here, OK, while a plus epsilon plays the role of epsilon, OK. I define it by the drawing, I mean, just you understand that, I mean, the delicate point is how to define a transition from 0 to 1 in a c infinity way, OK, this was somehow the difficult part. So then you can, OK, you can do the details, but the meaning was this, OK, OK. Yeah, and also let me observe that the construction of such phi n, I mean, also, I mean, with the construction of this phi n, we can also improve a former result that we prove, just when we introduce the first result in major theory. And so more precisely, we can mistake this theorem. So basically, we prove at some point that we can approximate a measurable function by means of a continuous function, if you remember, in some, OK, in some proper way. And then we can say more, we can say that we can approximate a measurable function by means of c infinity function with compact support, OK, with the, if you do the, sorry, so if you have let f be a measurable function function, so you must have something, you have to require that it must be, it must take simple values, but we have to assume that the measure of the set where f is plus minus infinity is 0. OK, then we can approximate such a measurable function in this improved way. So then, given any epsilon positive, we have this time that there exists a sequence, OK, there is some h, h, which is c infinity with compact support, OK, in a b, such that the measure of the set of the, OK, the measure of the set of the x in a b, where we have that f, OK, f of x minus this h of x is larger, larger than epsilon, is less than epsilon. OK, if you remember, we proved this theorem and we proved that there exists an h, which is only continuous, do you remember? And at some point, if you remember, we had this problem arise that we had to approximate with a continuous function, the characteristic function of an interval, OK? And we do this, so we did it in a quite, if you want, rough way. We just took the function g like this, we call it g, I think. Now we can improve, because now if you repeat the same argument and you replace the function g, the continuous function g, with this function phi n, OK, so the graph will be slightly smoother, OK? Then you can prove this theorem, OK? OK, now I will introduce the notion, just to conclude the part concerning measure theory, the notion of convergence in measure. So we give the following definition. So we have fn and f, where fn is now measurable. OK, then we say that fn converts to f in measure. The sequence fn converts to f in measure. If you have that for any epsilon positive, the limit as n tends to plus infinity of, sorry, n, OK. The limit of the measure of this set, so we have that where fn of x minus f of x is larger than epsilon, OK, this limit must go to 0. And so in analogy, you have the definition of a Cauchy sequence in measure. Give another definition. The sequence measurable function is a Cauchy sequence in measure. OK, if you have that for any epsilon positive, there exists an index and bar, such that for any n, then larger than this n bar, the measure of the set of the x in e where fn of x minus fn of x is larger than epsilon, is less than epsilon. OK, then we prove a theorem. We state a consequence of the convergence in measure, OK. OK, so be a sequence of measurable real valued function, which is a Cauchy sequence in measure, such that, OK. Then what we can say about the point was convergence of this set fn, we can infer that then there exists a subsequence of fn, there is such a function, a measurable function, f subsequence, call it fnk of the original sequence fn, such that we have that for any epsilon positive, there exists a measurable set a with small measure, less than epsilon, such that we have that outside this set, this set of small measure, the convergence is uniform. OK, on e minus a, OK. Of course, e is the domain of the fn and of f. By hypothesis, we know that fn is a Cauchy sequence in measure, so hypothesis, we have that, if we fix, we have that, OK. For any, instead of taking an epsilon, we take a k, positive integer, and we have that, there exists a corresponding nk, such that for any n, for any m and n, larger than n is mk, we can have that measure of the set of the points in e, where fm of x minus fn of x is less, make a particular choice of this epsilon k is less than 2 minus k. OK, again, now we do an extra hypothesis, but this with no loss of generality. We can assume that the dependence of nk is increasing with respect to k, because otherwise you can replace, for instance, nk plus 1 by the maximum of n1 and the form of 1, OK. OK, now we define the set ek as the set of the point of x in e, such that we have that fn of kx minus fn of k plus 1x is larger or equal than 2 minus k, OK. OK, and then we consider the union of this set, starting from an index m, for instance, so you take fm to be the union for k, which goes from m to infinity of this ek. OK, now we consider a point, which is outside fm x, which is in e minus fn. OK, how it behaves, so, and we have that and let 2 index jk, such that, for instance, they are larger than n. OK, then by the triangle inequality, we have that fn k of x minus fn j. Let x, let j, there are 2 index, no, j and k, which belongs to n, and they are larger than our index n, OK. So they are there here. OK, now we use the triangle inequality and we have, we can bound this as taken index l from k to j, for j minus 1, fn l of x minus fn l plus 1 of x. And since we are outside fm, we have that this is less or equal in the sum of 2 minus l, which is, OK, which is 2 minus k plus 1, 1 minus 2 k minus j minus 1. OK, from this we have that, we get that, that if you take the limit as k and j, this 2 index tends to infinity, yeah. This is the summation, I took the index l to k to j minus 1, because I supposed j is larger than k, 2 minus l, 2, yes. OK, so the supremum over x, which belongs to e minus fm of fn k of x minus fn j of x, this goes to 0, OK, we call it star. OK, so we have that r is a complete space, so fn k is a kushi sequence, so there exists a limit in r, so this f, such that we have that fn k, so we set this fx as the limit, has nk tends to plus infinity of fn of x. OK, so f is defined on each e minus fn, so f of x is defined union of v minus fn over n, OK, and this is e minus the intersection of fn. OK, so we have that, OK, by star what we, by this, we know that fn k converts to f uniformly on e minus fm, and this for any m, OK. And then of course we have, we will have to say something about the measure of this fm, if it is small or not. And of course we have also that if we took f, has taken the intersection of this fm, this is just by definition, this is the intersection of a union k, which goes from infinity of this e k. OK, so just let's estimate the measure of fn, so we want to infer that this is small in some sense, OK. This is less or equal, OK, it's equal to the measure of e k, k, which goes from m to infinity, OK, then of course this is less or equal to the sum of the measure of e k. OK, but by hypothesis the measure of this e k is less or equal than 2 minus k. So equal 2 minus k, OK, this can be that. So for any m, so basically this of course tends to 0, as m tends to 0. And so what we prove is that the measure of f is equal 0. So actually we prove more, so we prove that. So we add that fn, OK, converts to f almost everywhere e in e, because it converts pointwise in e minus f, OK. It converts pointwise and f as measure 0. So we prove that if we have a Cauchy sequence in measure, then it converts uniformly outside set of small measure, and it converts almost everywhere in e, OK. So we prove two facts, actually. OK, and now in this last 15 minutes maybe we can do some left exercise. So these are exercise on the monofon convergence theorem. OK, so just start from a general fact. So you have, for instance, a function f from 0 to 1, take 1, include it in 0 to plus infinity, which is continuous, for instance. So basically by the monofon convergence theorem we can compute. This integral has the limit as epsilon actually over a sequence, epsilon goes to 0, of epsilon of n f of x, OK, so fine. OK, so, for instance, we can specify the choice of f. So if you have, for instance, f of x is equal to 1 over x alpha with alpha positive, OK, so what can we say? We can divide three cases. So we have the cases when, for instance, epsilon over 1 over x alpha in the x, you have this is, the primitives is x minus alpha minus 1 minus alpha plus, OK, plus 1, so this is over x epsilon x equal to 1, and then this is 1 over minus alpha plus 1 minus alpha plus 1, so this goes to plus infinity when minus alpha plus 1 is negative. So as we have that the integral between 0 and 1 of 1 x alpha is equal to plus infinity for alpha larger than 1. And then when alpha is 1, I mean, these are course things that you already know, but just to see them as an application of the monofon convergence theorem, this is the logarithm of x epsilon over 1, and this minus the logarithm of epsilon plus 2 plus infinity, this is for alpha 1, OK, c, this is a, this is b, OK, so a and b, the integral is equal to plus infinity if alpha is larger than 1, in case c, which is the finite one, you have that 0, 1, 1 over, for the same reason that you have this, this is equal to 1 over 1 minus 1, so it's finite. OK, then we can slightly change, for instance, the denominator alpha x, so what you have is that, for instance, if we, in the exercise, might be to consider this integral of sin x alpha in the x, OK, here, alpha is always, OK, so we have that in this interval sin x is less or equal than x, then you have that sin x to the alpha is larger or equal than 1 over x to the alpha, so we try to use this result, and so we have that the integral of 1 sin x alpha in dx is equal to plus infinity alpha larger than 1, but on the other hand we have also that, we also know this is of k, a standard fact that when x tends to 0, the sin of x divided by x is the limit of this question goes to 1, OK, so we have that, this means that there exist some delta positive as such that, for instance, 1 over 2x is less or equal than sin x, x in between 0 and delta, and then you have that the sin x alpha is less or equal than 2 alpha x to the alpha, we can split this interval 0 delta has sin x to the alpha plus delta 1 sin x to the alpha, OK, this is finite, it's finite in a problem because we are outside and call this a, we can integrate between 0 and delta of sin of x, the alpha is less or equal than the integral between 0 and delta of 1, OK, if you want 2 alpha x alpha, but we already completed this, less or equal than 2 alpha 0 delta, OK, no, OK, less or equal, OK, then this, which is finite alpha less than 1, OK, so now maybe you can do, you can study by yourself a strain, a slight variation of this, and you can use somehow the same argument, OK, what you can, you can, for instance, you can try to solve is the following, so over the interval 0, 1 over 2, you can try to solve this x alpha and log x beta in dx with alpha positive and delta also positive, so you can try to somehow to discuss the convergence of the integral with respect to these 2 exponents alpha and beta, OK, so probably when alpha is equal to 1 beta plays a role, otherwise it's more easier, OK, so for today we can stop here.


I would have voted no. Thank you, Ms Harper. I'll make sure that's recorded. The result of the vote on amendment number 12079.1 in the name of Paul O'Kane is yes, 18, no, 94. There were three abstentions. The amendment is therefore not agreed. The next question is that motion 12079 in the name of Shirley-Anne Somerville on delivering record social security investment in Scotland to tackle the cost of living crisis and inequality be agreed. Are we all agreed? Parliament's not agreed. There'll be a division and members who cast their votes now. The vote is closed. The result of the vote on motion 12079 in the name of Shirley-Anne Somerville is yes, 62, no, 51. There were no abstentions. The motion is therefore agreed. The final question is that motion 12087 in the name of George Adam on behalf of the Parliamentary Bureau on committee meeting time be agreed. Are we all agreed? That is agreed. That concludes decision time. We have a brief pause before we move on to members' business. Members leaving the chamber should do so as quickly and as quietly as possible.


Hello everybody, welcome back to another Adobe Premiere Pro tutorial. This one, I'm going to show you how to freeze frame in about 45 to 60 seconds. Here I got some footage here of a puppy dog doing this thing. Oh, there's the button here and there they go. Puppy puppy puppy puppy play.  This is a shot where the puppy looks in the mirror and sees himself or herself. All right, cool. How do we freeze frame simple? First off, click on export frame here in this toolbar. Left click on that and then you're going to see here you get the option export frame. More importantly, you can import that image. I'm selecting a JPEG into the project. So make sure that's selected. Click on OK. Now in your project panel, you're going to see that picture of the puppy. I'm going to drag and drop that puppy onto the video track too. And then I'm going to say make it. I don't know how long you want this shot to be. Let's go to seven. Yeah, that's whatever. That's good enough. Now here we go. Watch this freeze. Back to business that is literally all there is to it to making a freeze frame. Thanks for watching. Be back soon.


Make sure But we back man. We back. We back. We back. There we go. We got people in here There we go now we got people Yo, oh my god, bro. I'm hurt bro. I'm hurt I Saw you know, I'm a slot bro. I'm wondering like what's going on Well, they fix the truck guys It's the truck guys again ready for the big race the big race Yeah, guys, I had to get the out of get the phone on life support man. This is a little IV wire I Now we got resuscitated Was about to be gone, but we had to go get the line to bring it back and now we're back and now we're back and now the smoke It's still free That gap sauce that motherfuckers again. Hey, just run. It's got embarrassed You want to see it go to you? Oh Shit my girl Yeah, we got Pimper TV.  We got Pimper TV right there before we got bad Bonnie V8 boy Dang Yo Yo, he's frying me right now yo, that's funny Ha ha ha robot everywhere I can't do that But I said maybe I can give you some tips bro Yeah, you tried to sit out of it with that one, bro You tried to sit out of it with that one Hey, that's it didn't work. I don't know what the girl the pussy juices wasn't strong enough.  That's just supposed to grow out. Oh my oh my god Oh my god, this really ain't for children Yo, yo, I think I think I think we're past it, bro We went there, bro The moment the moment the moment you said the moment you said you ain't something ain't work. Yeah, we went there Yeah, yeah, yeah, you can't come back, bro I really don't know This is just off the cuff They're preparing the track now That's a real strong fertilizer Y'all, bro, I'm doing I'm doing with y'all, bro. I'm doing with y'all, bro I said after 2,000 They don't say I want me one about 23 or something But then you know what I'm saying? I got acting like I'm in a prison movie. I'm doing Chair and leader.   I got the mask right here in my bag. Oh my bad. No. No. No. No. You got the life support We being safe and this ain't that This ain't that and plus Her lipstick is repainting corona repellent The mosquito repellent too How's the how's the weather bro almost really good the weather's nice. Yeah It's better than Florida I don't take any damn thing And then it got really cold towards the end of the end of the day this morning was cold as hell. Yeah, I was I literally just took my hoodie off not long ago. It was frosty as hell under when we got here Man, I'm sitting on this bro. I see everybody else doing it Bad body there are you looking like Selena, you know About say by the time you're done with her life, she's gonna have so many dms What's the challenge? Act like you didn't have to like that I told you and go check out my girl, man. Go check out.    It's just a different version of him when I ain't got the costume on but it's still me I don't turn this off It almost got turned off Yeah, it almost got turned off No, let me see this at the Polynesian the spicy joint Yeah, round of rail is there. Um, I believe I believe um, I don't think he's on the track yet. He's um, he's doing his thing Now I think he's chilling. I don't think so I mean he would have ran by now Yeah, I feel like he would have ran by now because now it's past eight now Yeah, he ain't running He definitely here Yo, here's um, yo, this sounds to some snaps. Yeah, it sounds to some snaps, yo I don't need some trap mode There's the trap mode right here. This is where, hey, this is where Omar bro calls her You know what I'm saying? This is the way Tomas Palente calls her up, you know what I'm saying? There, I finally got to be a novella niggaz out there and I heard Me I'm more That's the phone for all the thirst. That's the phone for all the thirst She said she said she needed a life to be clear so she leave all the thirst in there and then she got the real phone over there She got a phone that say D-Tix Y'all, I'm sending them damn joints for you Yes, you do. Yes, you do Yes, you do Money trying to be modest man That's a p-game step Hemilife 44 is out the track as well Scarface. I've seen this car. Um parking around here somewhere Let's see It's still preparing the truck y'all Let's do some walking around y'all Yeah, they still They still getting the um The track ready y'all Yeah, we got all the youtubers, um Joshua ronk race if you're gonna watch the other life Octane red what up what up what up? Oh damn, it's like mad packed over here y'all You gotta get in the bra and I'm cut man. I'm moving around right now. Yeah, that's a little broke Excuse me real quick, bro Oh Yeah, who should was on the floor like that bro my bad dog.   What's up? What's up? What's up? Oh Yeah, man, yeah, man, you can't be leaving your drinks on the floor Yeah, because why would I like what I'm walking I'm recording. I'm not looking at the ground like that I never saw that till I came here. I'll be to be honest with you. I've never seen that Yeah, I've never seen that before No, no, no, we got the leggings that look like jeans in florida Hey, left the two guys asking if you found some purple drink I Want to try that spicy ketchup everybody keep talking about oh, it's really good It's really good No, I gotta watch round seven because I feel like I'm gonna kick someone else's drink over Nice and sticky now Yo, hey yo pimping they say you got to hit up houston for the screw tape Bro, they kicked everybody off and I guess they let them back on again. I don't know Uh-oh, let's go the black bird. Here we go Is it I don't know. I think it is. I think it is. It is he vapors Yeah, I guess he vapors is racing dehumanology I mean not dehumanology. God damn. Yo, it's it's e-vapors versus dumb. That's my bad The guy said do you know you in the comments and I just ran with it. Yeah, rest up.  This is the real race right here. Yeah, it's not even vapors lefty. It's uh, yeah, it's someone else So, yeah, they're definitely locking some money behind this He is racing dehumanology octane red That's the yeah, I guess they must have locked this up before the race with dehumanology Uh-oh Yeah, we was over there earlier, but they told everybody to get off. Yeah, I'm good. I'm good right here Yeah, this is my people over there Don't mess this up. Get it together y'all Yo, does anybody know the name of the dude in the Porsche?  Yeah, that Porsche was fast He shouldn't be mad. He shouldn't be mad. Yeah Turn the quality up on my life No Some generic, bro, you know some generic Yeah, I had uh, I had some brisket nachos. It's pretty good. Oh, really? The whole scenario So I bet so come on here I serve myself I'm all right Right, right Hey, you heard him No, no, no, no, you missed it, bro. Say the part about you from New York. You know I Got cusses up Okay, they got good weed out I never see me talking full Spanish You play Maybe you were shortstop, maybe what is you doing? Hey, you know what I'm black What Hey, yo, they're talking about turning the quality up on my live, I don't know how to do that Can't do it No, no, no, I I chuck because I wasn't um, I wasn't public now in public. I got like The Wi-Fi connection so Yeah, so it's it's it's kind of cap right now.  They were just the white ones Straight up People be not here because everybody out here on the Wi-Fi You see we'll see what at these at an open event like this.   You stay on bet your bars go That's how you can tell the best you can do is just stay on and just rock with it Like the hotel you should say no more internet available.  I can't be talking to people like that. You really gotta be you gotta be focused. Yeah Oh, yeah, yeah, he's working right now The thing is is like Like like if I really want to do that, I'd have to have a second dude I understand truly because the drone like you can't be like I couldn't be talking about anything like this Because I have to be on the drone. Yeah, that's just for real You need to pull up in part Yeah, I honestly feel like honestly feel like the drone is good too if you have like a second man with you And you like yo bro just focus on that or you got a really dope location Yeah, yeah, but yeah the drone. Yeah, you can't multitask with the drone Yo, yo, what's happening now?  That's t rosé. Okay T rosé versus Okay Yo, what was the other dude's name? I didn't hear him really pronounce it t rosé and who that's t rose Right, right, right.  Yeah Oh, okay. That's not fair. Yeah, that ain't fair. He's sticking bro That ain't fair The actual race Or the exhibition or is it it's an exhibition race is this an exhibition? It's t rosé versus uh First somebody I don't know who that is QD Hey, his wife ain't cooking him no dinner, bro. It's done. He's sleeping on the couch. She's keeping the x box She's keeping the x box Yo, who's racing now? I don't even know your me too. She ain't okay. The woman's in the one Oh, that's fucked up man Yo, t-roll yo, let's say t rosé hang under that roses Oh, man, yeah, that's a lot looking crazy. Oh, yeah, that's a lot looking crazy Yeah, who's racing now? It's probably be the last Yeah, what's Here it is y'all here it is y'all Here it is y'all here it is y'all the race y'all been waiting for Here it is y'all Yo, like I could like I could show this the race is about to happen y'all Oh, it's a practice run. It's not Yo, cutie, I think this might be a practice run because I don't see demonology bad bunny.   I got about six pairs of mine. You gotta respect these. I appreciate that God Yo, what happened Yo, they said bad bunny gonna be at Russell mania today Yo, bro Yo, bro The college beyond snap, bro. Oh my god, bro Yeah, I think he's just doing a test run, but he is racing. I mean, uh, he is racing demonology, but I think this is just a test run right now Yeah, we're in this Oh We got our own little section here.  I'm not sure yet. I'm not sure yet. I'm not sure yet on that I Have you seen her in her space jam costume? Remember I was talking about that.  It just come out Savagery, bro There's a security guard police officer. I don't know.  You talk about security guys.  It's like on a roll Like that motherfucker was moving Like god damn, bro I know Maybe you look like paul blark out there He said bad money how you deal with it, girl Hey, if you stand up comedy clubs are hoping again.  Oh, I haven't heard about That's funny Oh Like this video like and share appreciate y'all man You're here to lay off, but I think you almost I think you're through with sliding Yeah, yeah, because he had to lay off and you heard it, right Yeah Yeah, yeah, yeah, because he definitely he laid off the gas for sure. Yeah Huh Yeah, but who's this I can't I can't see the people the people are in the way.  Look at him. Look at a bro right here I think there's something wrong with demonizing car or he's just letting dunk master cook He didn't even took a chance. Well, this car ain't even been up there once. Yeah, I don't know what's going on Okay, yo guys, this is exclusive content right here the track pack Drag pack y'all one of three That my butt click baited us.  He's gonna bust out with the whole dam Yeah, no I think he about to just run Yeah Just never know what these events man That's why when I left I was like, why why why I'm gonna go get it real quick I got the drag pack Oh Oh boy. Yeah, true.  Yeah It might be some mobile official for all we know because they're not selling those cars yet Whoever it is. There's somebody Yo, dumbass is not racing. Yeah, that's somebody else right now. It's the drag pack verse, whatever green challenges that is Dump master. He did a test run though. If you want to go back and check that out. You can check that out Whoever that is. There's somebody I told y'all the drag pack, bro. That drag pad is serious, bro. I can't see Yeah, that drag pack is nasty, bro Yo, but yo, but yo, uh, don't sleep though. Um the drag pack you gotta learn how to drive it. It's not just Yeah, yeah, it's not it's not something where you're just getting you automatically fast Yeah, drag pack is serious. No, when he hit the bump with the gas, bro He got a reinforced transmission. He did fuel injection. He had to pour that all I heard.  Yeah Yeah, it was moving man Moving man exclusive content man the drag pack. Yeah drag pack. It's not even you can't even buy those yet Yeah, yeah, like that's super exclusive. Yeah That car comes out next year Yeah, that car comes out next year It's gonna be like a hundred and twenty thousand dollars. Yeah Hey the coup to come out this year, you know that, right? Oh, yeah. Yeah coup to come out. Yeah. Yeah. Yeah Okay, I'm curious to see how it's gonna be received. I'm curious to see how it's gonna be received Niggas is broke.   Shit.  Niggas Jordan's ain't coming out every month.  They're doing the rehashes on the sinkers Yeah, they're buying they're buying old inventory. I didn't think about that, bro.        Um, we were both learning but yeah, he took he took me on both, man I usually like to race from a day, but I didn't want to do that since I'm driving 16 hours We had the rally rails. He took he took me on both runs.  I don't know maybe maybe not because I don't want to say yeah, they have um It is um, you know, I don't know I had it on another live if you want to go and check that out It's the one after this one Wait before not after before before I'm in reverse.  I don't know. I don't want to tell you and I feel bad If you went and actually miss content How far is your car Run real quick and come right back, but I would just I wouldn't try to talk to nobody I want to try to you think she has enough time to go run to her car and come back I would just go get the cord come right back don't talk to nobody don't make eye contact with nobody We'll we'll say if you spot Yeah, we'll save your spot. But yeah, I would just make a beeline and come right back Yeah, that race has to happen. Yeah, so I think you may have time. Yeah Yeah Where can I find the live? Um damn, man. I wonder if I did it unlisted Um, I can't get off this live right now, but I'm a double check I think I left it unlisted though because I knew I went live three times the first live The first live it was good, but then the second time I went live I must have put it unlisted and I think that's the one where I caught the race that you're talking about But don't worry. It'll definitely be on though. That's my bag. Yo, I'm trying. I'm trying to catch all these races fast and I put it unlisted, man Hey, hey pimping left you wants to know who won fly by or ZTRD ZTRD Okay, see there you go Yeah, no, no, no, love T2 guys. Yeah Oh appreciate you, bro. Appreciate you. Appreciate you the love swaggy key.  So if all of a sudden you're in mid in mid stride and you hear it come right back That's real Oh, thanks Scarface Oh appreciate it man way mon. Hold on way mon Leans, I'm really bad at pronounces. I'm sorry Yo, so what's the hold up, man? We saying a prayer we come by iron. Yo, what's going on right now? Yeah, they talking about the bag right now.  He was going slow himself to the beat as you can see Yeah, something might happen to this car.   She does the first set of shoes. That was 200.  I want my uh my concords. They're motherfuckers look like Exactly, bro. Come do these shoes. Yo, these drugs have been pretty comfy though. I'm surprised with these They went down.  I'm like me got there and I see you look at my motherfucking mic Bro, there's fat leather when I got it. He didn't motherfucking his corduroy now Yo, no, hey, yo real quick man. They talking about the mask. Tell them that everybody every we we hanging out on our own little circle We got a little bit in the circle Yeah, my man got his mask off But you see we aren't together, but we're not bro, you know, it's not what y'all thinking He's not right up on me They can't live like that see we're from Chicago.  So, you know Yeah, so currently right now the update guys is uh, I guess they're talking about the bag Or like pennies said, um, something's going on with thump master's car So, uh, I don't want to go off the live because I don't want to miss it But yeah, man, I'm supposed to be racing some more, but we'll see. Yo, man Check out the merch though free smoke squad free smoke squad, man We got the merch. I got the last extra large on my shoulder right now Look look look at this right here the last extra large on my shoulder right now Someone need to come get it We've been giving these out for free Yeah, y'all man is racing the zio six of the burn. I believe he's racing the black book I'm gonna have to turn that up You really ran Yo, real rap that happened to me in high school, bro.   I don't know.  Hey, bro. Be safe about that, bro. You'll be safe, man You can see all his facial features everything nigga.  Y'all now y'all just being bogus now see Respect to y'all. We love you for being in but at least come with the good jokes though. You know what I mean? Yeah, come on, man. Come on, man. Come on, man.  Come on, man. Trust me. I want to be on I want I want to be on here I've always been on some type of way. Come on, man. I I just said I look like everybody but to be honest with you so I don't you know Oh, no We did today Kle ap cleveland and Dave PayPal Y'all My god, every man Yo what's wrong with people, bro? Hey there You're here Yo, come on belle Hey y'all bogus bro for respect though, that's a good one.  Just just jk bro. I love No, it is love bro. Hey, we want you to interact with us, but it's the shit that we this is why we do this love But y'all so you think you think you think they should talk about the bag or you think not not now I bet demonology like hold on nigga. I agreed to race this car now. You brought to get another one I think that's where they at now. Oh Oh Think about it. That was that always the plan exactly because you know demonology. He's an a bro one LS Why I'm good that one ain't the one I'm getting he don't know about me. You know I'm saying so now they like a bro You'll say they work this up now only thing good is he bring always three three cars I think when he go, but I think that's what they talking about right now And it's gonna take a lot another hour do whatever car he they agree on he would bring this shit out Running like you did at least twice then do it We don't know but now that he not running as he'll six Shit might Can I race my SRT eight tonight?  That's the demonology race, but yeah, you could have I mean They After this race, I don't know the race. Yo, what happens about to go down you got a better chance for the blur than the zero six That's what someone said It's not about which is the fastest car is about what demonology agrees to race That's why like when we show everybody got them at last swap supercharged They only raised 26 and up. You know I'm saying thought on the demon in regular race times Automatically already got an advantage. That's why he was winning the zero six He'll sacrifice a little horsepower and you know to keep his tires in slick so So, you know, it's giving take Everybody's seeing behind the scenes right now. Yo son You six is always breaking down. Yeah, it's been breaking down a couple of times man I'm gonna say, you know it bleaches and people buy it We got four to four people in here man Shout out shout out make sure y'all smash the like button y'all we got four to four people in here smash the like button y'all It's the black blur right there, right?  Yeah, there's a lot of power Yeah, so I guess yo, so just an update I guess I still agreeing on What's going on? Um, I believe that's what's happening. I could be wrong, but I sense that's what's going down right now Hear an engine I Yeah, true No, they haven't run we're trying to figure out now Oswaldo Yeah, I haven't answered now you go bro. We've been waiting man. They're supposed to raise what two hours ago, right?  Be out here John nation at a boxing match Ruben's stuttered oh my god Bro, so people don't know who that is bro. It's really stuttered, bro. Yeah, for real Oh demon on the track y'all there he is Jiu-jitsu Bossy about to teach a jiu-jitsu Yeah But I Know Sharon Sharon Sharon share this video, bro. I think this is it. I think this is it Thank y'all for patiently waiting. I think this is it. Oh, oh Yeah, man, make sure y'all like those 45 people in here like a command smash that like button Sharon.  I want to see those comments flooded like it Demonology, yo y'all saw y'all saw exclusive Yeah, yeah, yeah Yo Yo y'all saw man y'all saw man demonology for the win y'all demonology for the win Yeah Bro exclusive exclusive content Yeah, oh man there was right there man.   Yeah the car you want you seen it He pulled up that throttle quick. Yes, something happened So now you had to put another car I think he was supposed to make the CL-6 I Think but but I think something was wrong with it or something. I don't know Yeah Tell me we saw it bro. He came off that throttle. Oh, yeah, well, he felt something Well, one he felt something too. He kind of slid too. Yeah, he slid real bad He felt it You're right though. Dump master is going to say something about the CL-6 Appreciate bro for the live. Oh, yeah, no problem man CL-6 always breaking down now.  Yeah. Yeah suspension ain't built for that Yeah The car it's it's already the car is already out of character. Exactly. It's too heavy Yeah, man, we've read these comments though. Thanks for the live. Yeah, man.    I guess we can see what's going on. I guess we can see what's going on Yeah, yo, man My homeboy I'm a cat right here, man.  I'm trying to be the goat. We're here though.  He's in there. No, this is old rep on his skin band. Yeah, yeah, yeah Yeah, yeah, no, no, no, I was just saying yeah, because I So everybody's wrong with CJ.   I hope y'all can hear me.  We're on free smoke. Yo, make sure you go follow us on Free underscore smoke underscore official. Yo, I'm out of here man. I'm out of here. I'm tired Y'all wake up early in the morning and drive man. Peace