Hi, you guys. Lindsay here. Welcome back to my channel, Inside to Him. So if you've been following me for a while, you know that I'm still on the struggle bus when it comes to sewing pants. I just cannot get the fit right on most any pants pattern. The more fitted the pants pattern is, the worse it is for me. It takes me so much time to end up with something that's worse than what I can find in stores. So for now, I'm still buying my jeans, but I am buying them from Goodwill. So and today I want to share with you my best tip for buying pants second hand and making sure that they fit for my body shape.  If you have more of an apple shape, it would be better for you to fit to find pants that fit in your waist and then altering them to fit your butt and thighs. Taking apart a jeans waistband and altering it at the center back or side seams is really a pain and it takes forever. I have come up with a quick and dirty method that takes less than an hour and is easy to adjust as you gain or lose weight. Let me show you how I do it. Okay, so first things first on this, you need to unpick with your seam ripper, starting at the side seam and then unpick, I don't know what is that, like four or five inches on each side of your jeans. So we've got the waistband is attached on the front and then when you flip it to the inside, the facing of the jeans is what I have unpicked, allowing me to access the tunnel of the waistband. And I want to point out that in order to do that, I had to remove this name plate that was on here because it was sewn through all the layers. I am going to try and get that back on there. I just don't know with my home sewing machine if I'm going to be able to sew through all of this leather, but we will see how that goes. I'm pretty sure if I can't do it at home, I could take it to like a cobbler, like a shoe person. They have industrial leather type machines and they might be able to get it back on, or maybe even like a regular alteration shop. So we'll see what I can do here with what I've got, but there are other resources. Okay, so once you've done that, you need to determine how much you need to take out from your waistband. So I did that on my own already. I pinched it closed like so along the center back while I was wearing the pants and I was pinching it just outside of these belt carriers. So I'm going to, it's kind of an estimate in the beginning and you'll see how we will perfect that measurement here shortly. But you measure that and I'm getting like right around one and a half inches. So double that because we've got this doubled over. So that is going to be three inches. Our waistband, I'm going to measure half of the waistband from the side seam here all the way to the center back here. And I'm getting a measurement of what is that, 10 inches. So double that because we've got double a waistband and that is going to give us a 20 inch back waistband measurement. And I need to remove three inches of that. So my elastic is going to be cut to approximately 17 inches.  And we're going to add half an inch of seam allowance for either side. So I've got a measurement of 18 inches here. So now you're going to take your elastic. I've got this really cool elastic threader. I will have a link in the description box for where you can get one of these. It's awesome for projects like this way, way better than the safety pin because it's so long. And we have a center back seam here in our waistband that we are going to be trying to push through. So it's a lot easier with, you know, a pointy hard plastic long tool like this. But anyways, you just feed your elastic through the back. You can tell how powerful the grip is on this threader. It is not letting go, which is one of the other things I really love about it.  And I'm going to secure this with my seam allowance sort of where I want it to end with a safety pin. We need it to be a lot stronger than a straight pin because we're going to be tugging on this a little bit more. So just make sure you catch your elastic in the safety pin. So yeah, I've got it going through all the layers and the elastic. And on the other side, you remove this guy and do the same thing. Find your side seam. There we go. Find your side seam, tuck the elastic up into the waistband and secure it with a safety pin again. Like so. Now we've got to put them back on and assess how they fit. More often than not, I end up having to take it in even more. But it's easier to do that on your body than to keep like guessing at the measurements. So let's do that. Okay, so now I've got them back on. And you want to make sure too that you're wearing the shoes that you plan on, a shoe that's similar anyways, because that's going to affect how they fall from your hips. So you can see we've got the waistband and it's still a little bit too loose. So reach around, grab one of the safety pins and remove it. But obviously make sure you hold on to your elastic like so. And then just give it a giant tug until it pulls itself up to you all the way around. So you can see I pulled out a good like six inches probably. And just make sure that it's it feels good and that it's like I said, hugging you kind of all the way around. And that you're still comfortable, you know, you don't want it to be like, I don't know, holding you in so bad. So okay, so once you've got that in place, put your safety pin back in so that you can take these off. And we can officially secure our elastic. Okay, so once you take them off again, you can see we've got one side secured over here.  And when you pull it tight, you can see that how the elastic is kind of working in there. Okay, so you need to handle this kind of one side at a time. Remove your safety pin. And you know that this is in line with the side seam. So take it over to your machine and sew it into the facing. So you are going to be able to see these stitches from the outside. It's going to be a single row of stitches right here kind of in line with your side seam. So it's kind of obvious, kind of not, I'm going to use a navy thread. So it blends a little bit more. I'm just using a straight pin to secure this now, since I'm not putting it on and wearing it and stuff. And then you open this up, and you should be able to stitch right along your side seam, which is this area here right through here, stitch back and forth a couple of times to hold it in place. And then once you're done with that, you simply flip your facing back down, and then top stitch that closed. And it'll be easier to do whenever you don't have the pins in and stuff. Then come to the other side and do the exact same thing. So I'll show you what I've got when I'm done doing all of that sewing. All right, the jeans are all done. You can tell that along the front of the waist band, they still look nice and flat and just like your regular jeans. And then when you flip it to the back, you can see you've got like a bit of a gathering situation going on. Now when I put these on, they do stretch out a little bit more. So the gathering isn't so obvious. And if you remember from what these looked like before, I had, I mean, three inches to take in, which is a lot. Not all jeans that I buy need three inches. A lot of times it's like one or two. And so the fewer, the less you need to take it in, the less gathering this will end up looking. And I might try and get this guy back on again too. Although with all of this, yeah, I don't know. I'm gonna try. I'll let you know how it goes. I did it. I got it. I got it back on there. And considering I am sewing through two layers of denim interfacing and now that elastic, I think I did pretty good. So I had gold topstitching thread for the waistband. And so I used that for the outside. And then I had just regular polyester thread on the bottom. And there's a little bit of, I don't know, elastic, I think, popping through. But that's okay. I'm really happy with this. So let me put these on and show you what they look like. And there you have it, the quickest, simplest way to alter a denim waistband. You can see the before and after side by side. I love how these jeans fit. And when I wear my top or sweater untunked, you can't even see the waistband. So no one knows it's gathered in the back. And if I gain or lose weight, which I tend to do, I can unpick one side of the waistband and tighten or loosen the elastic as needed. If your weight fluctuates a lot like me, tuck in a couple extra inches of elastic into the waistband so you have something to work with in the future. Here's another pair I recently did. These are more fitted through the thigh. And the waistband isn't as big to start as the first pair. But you can see the elastic trick still works. So if you have a hard time finding jeans or pants that fit in the stores, give this little trick a try. But that's going to do it for me today. Thank you so much for watching. If you enjoyed this video, please give it a thumbs up. And if you're not subscribed already, please do so. I have a lot of tips and tricks like this coming your way. But I'll see you all very soon. Thanks again for watching. Bye.


statistics and excel poisson distribution excel function and graph get ready taking a deep breath holding it in for 10 seconds looking forward to a smooth soothing excel here first a word from our sponsor yeah actually we're sponsoring ourselves on this one because apparently the merchandisers they don't want to be seen with us but but that's okay whatever because our merchandise is is better than their stupid stuff anyways like our crunchy numbers is my cardio product line now i'm not saying that subscribing to this channel crunching numbers with us will make you thin fit and healthy or anything however it does seem like it worked for her just saying so subscribe hit the bell thing and buy some merchandise so you can make the world a better place by sharing your accounting instruction exercise routine if you would like a commercial free experience consider subscribing to our website at accountinginstruction.com or accountinginstruction.thinkific.com we are in excel if you don't have access to this workbook that's okay because we basically built this from a blank worksheet starting in a prior presentation so you could go back there or you can open a blank worksheet from here and you should be okay as well if you have access to this workbook three tabs down below example practice blank example in essence answer key practice tab having pre-formatted cells so you could get to the heart of the practice problem blank tab is where we started with a blank worksheet and are continuing with it here let's take a look at the example tab to get an idea of where we will be going we've been talking about the poisson distribution and we've been looking at different families of curves in a prior presentation looking at a uniform distribution now the poisson distribution which could be useful to help give predictive power about certain data sets if certain conditions are met we talked about some of those conditions in a prior presentation we looked at the formula which you don't want to be too intimidated by because if we know the conditions that are met we can use what we're going to work on now which is the poisson dot dist function will then plot a graph and then in future presentations we'll use another kind of random generator tool that's a little bit more complex than we've seen in the prior presentations to generate some data that we can plot as well let's go to the blank tab and so like I said we have this information from the prior presentation but if you don't have it that's okay you can start from just a blank worksheet so we're going to then just say our condition to be able to use the poisson dot dist is to have the mean which we might have access to from prior data so oftentimes a poisson distribution might be useful when you're looking at like say line situations and you're trying to determine how many people might show up in a given interval or what is the likelihood of so many people showing up within a certain interval and so what we need then is the mean which is often represented with our sigma or our lambda with the poisson distribution it's often the lambda so i'm going to be then entering the lambda i'm going to go to the home tab we're going to go to the i'm sorry insert tab going into the symbols i'm going to make a symbol and the lambda is right here because it's recently used if you don't have it recently used you can look under the greek and coptic to find your lambda let's insert it and so there it is i'm going to say close this and then i'm going to enter and then go back into it and i'm going to put the equals next to it and then the cell to the right i'm going to say that that is 10 so we know what the mean is represented by lambda and that's going to be 10 i'm also going to say the x number of rows number of rows in our table uh i'm going to make it conditional we'll start off with like 29 rows and it will use a little bit of a formula and that'll help us to basically adjust our graph when we put our graph in place so i'm going to say this is going to be x and actually let's put it to the right here i'm going to put it to the right here i'm going to say this is going to be x and this is going to be the uh result of our formula uh which will be result here okay which i might maybe i should put f of x let's say f of x here okay all right let's center these i'm going to center these up top home tab alignment group i'm going to center it and then we'll go into the font group i'm going to put this and make it black and white because it's a header that's what we typically do with the headers i'll select these two and just make them blue and bordered which is what we often do for our other data input areas home tab font group bordered bucket drop down if you don't have that blue more colors standard there's the blue that i typically use you can use whatever color you want i'm going to make the end a little bit skinnier so we have a little bit skinnier of an end holding control scrolling in a little bit so we'll zoom in a bit so zoom in a bit more and so okay so then the numbers i just want this to be numbered from a zero down to 29 now there's actually a formula for that which could be useful when we're then going to make a graph out of something because then it could adjust the the the amounts that are picked up in the graph so let me show you what i mean here instead of just doing this which is what we will typically do in the past which would be just say zero one two and then using excel to memorize the sequence if i want this sequence to change i can use a function and then pick up this number to change the sequence so let's show you that we're going to say this equals sequence sequence this is the formula so there it is and then i'm going to say the number of rows is going to be 29 plus one you have to have the plus one because there's going to be a zero in there as well so we started zero up to 29 which means there's 30 rows but 29 plus the zero and then the columns is going to be nothing in the columns because i have one column so i could put a zero there or just two uh commas to get to the next point and then we're going to be starting at not one but zero which i'll just hard code or type in so i'm going to say okay close that up and enter and then it gives us our little spill down here down to 29 now again the reason that's nice in this case is because then i can change this to 30 and it'll then populate down to 30 i can i can i can decrease this to five and it'll give us uh whatever result we want which is great let's bring it back 29 now we'll use the poisson function so i'm going to put my cursor in p2 and to put that in place we're going to say equals and then poissons we want poisson dot dist double clicking on that the arguments we need are the x value which is going to be in this column so in this case zero and then the mean represented by lambda 10 in this case and then we'll talk about whether it be cumulative or not in a second so we're going to pick up x so i'm just going to pick up that zero going to say comma and then and by the way we will go back into this in a second and represent x with an array and that could be useful when we're graphing it but for now we're going to pick up that o2 and then comma the mean is going to be this 10 now that 10 i don't want it to move when i copy it down therefore i'm going to select f4 and the keyboard dollar sign before the m and the one and then we're going to say comma now it could be cumulative or not so if it's cumulative this will make more sense possibly when we look at the graph and we'll look at more examples shortly but cumulative means it's going to be all the results up to a certain point whereas if it's not cumulative you're just going to get the results for one particular point so you can type in true if you want it to be cumulative or false if the probability mass function we want it to be false so i'm going to just type in a zero you can also type in a zero or a one zero for false and one for true closing it up and then enter and so there we have this i'm going to then then i could double click right here to copy it down to how many cells we have you could make it a percent home tab number percent because that shows you a few more places out and then add a couple decimals so this is so this is what we have as our results and so what this is basically saying we'll get into more examples in future presentations is if you're in for example a line type of situation and you're trying to say that x represents a time frame say minutes what's the probability that two people show up within the two minute time frame something like that and the probability being point two three percent right and so when you're looking at the cumulative proper probability then it would be adding up the probabilities of zero one two three or four we'll talk about more examples of that in future presentations for now just note that one of the problems if i'm not using like an array is is notice this if i go down to 29 you get zeros after that point now if i wanted to bring this back up to like five notice that because i copied this side down then it doesn't really it doesn't shorten this column when i shortened this column whereas if i use an array it will shorten it so for example if i put this back to 29 and let's redo this formula so i'm going to delete this whole thing and i'm going to do it again slightly different using our array so i'm going to say this equals hoisin.dist tab and this time instead of selecting that just that cell i'm going to put my cursor there hold control shift down arrow that takes me to the bottom if i want to get back to the top i can hold control backspace takes me back to the top now i see an array in our formula is what we're doing here and then i'm going to say comma the mean is still this cell 10 i want to make that absolute still so i'm going to say f4 on the keyboard and then comma and this is still going to be zero or false closing it up and enter so now you can you can see it populates with an array same results but one of the benefits of doing it this way might be a little faster sometimes the arrays cause problems when you put tables in place so there's pros and cons to them but one pro is one benefit is if i put a five here notice it changes the whole size of my of of both columns which is kind of nice so let's put it back to 29 and and so then let's go ahead and graph this thing so i want to graph this i'm going to close this up a bit let's make this a little smaller so i'm going to select my data like so and then i'm going to go up top and say insert charts and i want to make basically just a bar and so we'll put this in place i'm going to have to adjust the data that it's picking up here so i will then go into it here chart design i want to select the data here's the data that it's picking up sometimes it's easiest just to delete these two and re enter it so i'm just going to delete both of them uh and say just remove both add a new data set and i'm going to say the series name that we want represented on the y is going to be the f of x as the name and i'm going to pick up the data putting my cursor on uh p uh to holding down control shift down which takes me to the bottom of the data set control backspace takes me back up okay and so there we have this and then this data set i want to make sure that i have my own numbers in there instead of their numbers right so i'm going to edit this data set and say i want you to be picking up from this area uh control shift down control backspace and then enter so now it goes up to 29 because that's how many numbers we have there and okay so there we have it and then i'm going to say boom this is good and then uh let's just call this let's just name this uh poi let's just call it up top poi son poi son i hope that's spelled right again okay and then and then down here we have the x so we could then say okay let's add the uh we could add the axis titles and say that this is going to be uh equal to f of x and this is equal to uh x and then so so there we have it and you can see it looks kind of like a bell curd but it's basically going to be skewed a little to the right now once we have this then we can adjust this this one number the mean and that's what's great about the poi son distribution that's what you that's what you need to know in order to you know populate the poi son and so i can then say well what if this was uh what if this was free or something like that you could see then it's you could see the skew a little bit more uh in detail here now note that what's interesting is this still goes out to 29 because i told it to go out to 29 here even though we have a lot of information that we might not need on that tail end even though it's possible like if you're talking about a situation where where you're thinking about how many people might show up uh in a given interval of time it's possible for this number to go up to infinity but obviously there's some kind of practical limit right so i could then change this and say well it looks like i can basically cut this off at maybe like 14 and so that's going to adjust our graph down a bit so if i look at our graph notice it's still picking up the information down to here but if i then adjust this up i'm going to say boom and now our graph has been adjusted in terms of the x column which is fairly easy to do so let's go out further let's say this went out to like 70 and then i'm gonna and then so so now it populated uh down to here the spill uh looks good but i need to populate i need to adjust the formatting of all these cells let's just paint brush all of them and then make this one black and white and centered so now now it's going all the way down to 70 and you can see that actually populated automatically in the graph to pick it up to 70 so when i shrink it it might not do it exactly uh automatically but you get the idea and then if i bring this up to if i if i brought it down to like one or something like that you could see that kind of highlights the skew more if i bring it up to five you can bring it it looks more kind of bell shaped but it's always going to be skewed a bit so that looks pretty bell shaped but there's generally going to be a right skew on it which you could see more dramatically when the mean is smaller here so there we have it so there's our there's our our general table with it so now uh next time what we'll do is we'll use a a random generation tool that still has a random element to it but it's going to be designed for randomness related to a poisson uh distribution and we'll and we'll just so we'll generate that which will simulate a situation that basically meets these conditions still has randomness to it but the idea is that it simulates randomness where these conditions are met and that random information that we get then should then if we plot it be similar to this outcome in a similar way as the uniform distribution you know the straight line distribution would be similar to something like rolling the dice that we talked about uh uh last time and that's the idea because now if we can see that relationship then we can possibly use a poisson distribution to give us ideas about what to be expected in the future even though it's not perfect there's still randomness to it but it might give us that approximation where they could give us better decision making capacity


What's up everybody? I'm EvilRabbit and if this car looks familiar to you, it's because you've been around the channel for a very long time It was my competition car in Forza Motorsports 7. We are back here on Forza Motorsports 7 We're gonna see if I can still drift in Forza 7. So let's get to it Let's have a little fun. Let's get sideways and see if I can still drift in Forza 7 All right, so like I said, we are back on Forza 7 did hook the Xbox one back up for the moment So we are back here with the full Fendatech rig and everything going on But now we have two cameras with the new PC being able to see pedal cam and stuff here on Forza 7 So we're just gonna we're just gonna go to a free play session and see if we can actually still drift here I don't know how the settings are or anything like that Let's see about changing up tracks If I can still remember how to How to change tracks in this game All right now that we remember how to change tracks in this game We're gonna see I'm trying to think where the funnest track would be to take this car out right now And just see if we can still drift here in Forza 7 Trying to think of the funnest track. I could think about that we could go do for you know a lower horsepower car And I'm thinking we're probably gonna go to Maple Valley is a pretty good track to go to right off the Brit We don't want it wet. Okay. Okay. Dry. Dry. Dry. All right Let's go. So like I said, I'm not sure what my wheel settings are right now I'm not sure how the wheel settings are set up because haven't been on the Xbox one in Well, since I got the PC so bring it back for the motorsports seven with the anticipation of the new fours are coming out But I know a lot of you guys have been around with the Forza content So decided to bring you know for the motorsports seven drifting back and to really see if I can Still drift here on seven being on a set of course or for a very long time. I don't know why I'm trying to use my mouse So we're just gonna go out there and well see what happens All right. Well Definitely already different feeling right now. All the wheels feel so much different than a set. Oh, the wheel feel is definitely Totally different. We're gonna dial back the audio limits.  Oh, man. So different But I feel like we could still drift here in seven as I say that we go right to the dirt Oh, I feel like I need to Definitely adjust some wheel feedback settings and everything because this is definitely a bit different. I remember Trying to get super smooth on this game and definitely struggling with it for a while Like the fanatic is very twitchy right now. And I'm not sure why I'm gonna have to go back and fine-tune some settings But we can still kind of drift here in seven Nowhere as smooth as I used to be for say like competition, but oh gosh Yeah, we need to tweak some settings here. So let's uh, oh wrong button. Let's go and see about Some wheel settings. Oh, no, I wanted to go options. I wanted to go controller and Now we're gonna go to advanced. I'm unsure Where I really need to change all this stuff But I think we're gonna lower a bunch of stuff and just see where This leads us to And go back I may have to go back and take a look at some of my Original settings I had okay. Now. That's very light. So We need to ah wrong button again. I think we need to add some of that mechanical feel Maybe this will give us a little bit more of a road feel Or a self-steer in a sense. It's a little better. Let's see Wheel just feels super light to me right now. Oh boy compared to a set of Corsa and I think I'm gonna have to go back and really check So my old settings and stuff like that to see if we can't maybe get that to feel back how I want it to But then again, like I said, we are so used to running on a set of Corsa That everything is different I'm gonna kick this one back up Kick this really high. I'm gonna just kick this back up, dude. We'll go the opposite direction than what we just did See if that feels any better.  Let's see if we can't if that feels better. Oh now we got some self-steer That feels a lot better actually Just flat footing this FC. I should have broke Got how much the sand stops you right away in this car See if we can get this I think I can still get it. I think I still got oh I still got it here It's definitely so different than what I'm used to on the PC side with a set of Corsa But we're good. We got it for sure. I believe we're getting back in the fields and we go So could we still drift in fours of seven? Yes. Yes, we can Oh and so we do that Hard hit into the wall. So let's why don't I keep hitting my keyboard? Keep forgetting that I am on Xbox not on PC. I'm trying to use my keyboard and stuff. So we're gonna switch up cars. Maybe switch up tracks And uh, see what we can do.  That would be the Corvette That I became so comfortable in when I was preparing for esda and stuff So little uh A little back to basics here on the channel for today My train of thought on that Car didn't grip how I expected it to A little clutch kick though I feel like if I wanted to get back into throwing down some sessions online in fours of seven I will definitely have to get some practice behind the wheel again Brakes do not work like my cars in a set of Corsa at all Gonna have to maybe tune these cars a little differently now Yeah, brakes do not work how they do In a setto gonna have to fine tune the brake settings and stuff I want to continue to drive because I've basically changed my driving style switching from The xbox to the pc and going onto the set of Corsa So if you guys want to keep seeing fours and more sports seven content and all You know some back to the xbox stuff as well as a set of Corsa and the pc side Let me know down in the comment section down below I hope you guys enjoyed this throwback to the original og Game that I used to play all the time and as always I thank you guys for watching. I'm evil rabbit. I'll see you guys on the track


Coming to you live from my home office in Makiki, and from downtown Honolulu, from the studios of ThinkTech Hawaii. Today, our topic of discussion will be on health on the front line. It's vital to maintain your physical and mental health today more than ever. What I would like you to take away from today's discussion is the idea that living a life which is healthy both physically and mentally allows you to really live your life. Today we are very honored to welcome a dear friend Moina Cameron, who in 2007, after serving for 22 years in the British Army, became the first, mind you, the first woman to be appointed as a human water or beef eater at the Tower of London. The beef eaters are the guardians of the tower and they guard the palace for Her Majesty Queen Elizabeth II. Aloha Moina and welcome to Hawaii. Aloha. Please share a little bit about yourself so our audience can get to hear your beautiful accent and get to know you. Well, first of all, thank you very much for inviting me. This is really exciting. Just to remind everybody though that it is it's one o'clock in the morning here in London. Anyway, so yes, my name is Moira. I was born and brought up in the west coast of Scotland. And it's an area that is very quiet, very rural. There's not much really not much in the way of work. It's very tourist oriented and I really, I really needed to get away from the area. My mother, I was a bit of a wild child. And my mother, she very strongly supported me in this. She used a bit of reverse psychology when I said I wanted to join the army. And the reason I wanted to join is because my brother had been in the army. And every time he came back, he just, there was just something about him. He had the self confidence he had. He's bearing you stood properly and the everything was different about him. And he also had money money in his pocket, which was always something. That's a good persuasion. The stance, the standing up straight, the discipline. That's all, that's all wonderful. But just tell us a little bit more about the beautiful sites and what was it like growing up in Scotland? Scotland is, it's absolutely beautiful. There is no two ways about it. When it's sunny, there can probably be very few places in earth that will ever rival it, especially when I was brought up. I lived halfway up a hill. So we looked over a lot and across the other side of the hills. And it was just very, very beautiful. But when it was cold and it was winter, it was wet and miserable.  You had lovely summers.  You said you love so well. And as you were growing up, that's what you were surrounded by. Yeah, that's it. That's from that's from my bedroom window. Yeah. Wow, gorgeous. Just like Hawaii. Tell us a little bit more about your family. Is it just you and your brother? There's my brother. He's five years older than me.  But my, that's my brother and me there. Wow, look at that picture. I could reach the piano. I was playing it. Whether it was music or not was, I thought it was a symphony. My mother wasn't too keen on it. But then my father, he, he had another, he had another daughter, my sister, and she has got, she's got two boys. So yeah, so I've got two nephews. My family, my cousins and I were all quite close. Wow. And that's so important. Yeah, very much. Do you go back to Scotland to visit? I go back about normally twice a year, normally. Yeah, but they come down here as well because they love it. And my nephews were down last year. And they just, it was the first time in London. Really? Yes, first time in London. And it was the first time my sister had been to London. She's been down to the south before, but not actually London. I meet her. Yeah, you see her twice. Wow, that's amazing. It's fabulous. Wow. And so you had a beautiful childhood growing up in Scotland. So again, what made you decide to join the Army, Moira? Just because of my brother, really, and it was, and I knew that I had to get away. There wasn't going to be the work for me there. You know, I just, I needed to stretch my wings a little bit more. And because he had such a, you know, this bearing and he just, he just looked different. I just wanted a bit of it. So I thought, well, I can always try it. And if it doesn't work out, then it doesn't work out. So my mother gave me three years. In fact, she didn't give me three years. She said I wouldn't finish my training that I wouldn't be able to stand the discipline. Wow.  And so you needed not to just do it for yourself. I'm sure you wanted to tell mom, I can do this. You watch, I'll do it. And not only did you do it, but you serve for 22 years, I believe I did. Okay. And then you are up for a position. And that would be the beef eater position. And I just want you to tell us what made you want to become a beef eater slash human water. Well, it was actually a bit of a, it was just one of these things that just happened. An opportunity came along. That's me with my military uniform beside my foot, the chief and the jailer at the time. But yeah, it was, I'd retrained as a plumber and electrician. So I was going to go home to Scotland to work. And I just happened to be, it was about a year before I left the army. And I looked at a military magazine one day, and it had a picture of a beef eater on it. And I went, Oh, that looks quite interesting. Opened it up. And it said it wasn't just a job for the boys. Now, I'd already known that it was a former military position. You've got to do the 22 years, right? Have your long service and good conduct meddling. You've got to have your royal warrant, which just means we were sergeant majors. And, and it's that's the equivalent throughout the three services. And I thought, I'm going to phone these guys up. I've never been posted in London. They'd never served in London at all always wanted to. And I phoned up and I was on the phone for about half an hour with the HR manager. So, yeah, it was really interesting, very interesting. Wow. So now you see an ad that they're opening up the position of a human order, and women can apply as well. As long as you meet the criteria, and I believe it's 22 consecutive years of service to the British Army.  And then so you say, I can do this. And then you apply. And then what, how did you get notified? I mean, how did this, does this all work? Well, I was interviewed for the position in the, I think it was the early December late November the early December 2006. And I was at home on Christmas leave in Scotland, and I had a phone call, basically saying that if I was still interested that they wanted to offer me the position. So yeah, but because it was the first one, yes, the first female, it was, I had to keep it all quiet and secret.   And if you don't, then you don't get the title, right? That was the harder part than the service. Yeah, definitely. Wow. Wow, that's, that's amazing. That's amazing. And so now you say, yes, I would surely love to take the position. Oh, yes. Yes, definitely. Trying to keep very cool about it. You say, oh, yes, I'd love to. And but as soon as I came off the phone, I was bouncing all over the place. And it was, it was only my brother's dog in the house at the time and it thought it was going for a walk. So it started bouncing too. But yeah, it was phenomenal feeling. Yes, that is, that's a feat and accomplishment that I'm so, I'm so proud that I even get to know you and share this journey with you and I feel so excited. Like I'm even part of it because I kind of met you at that time a few years after but just, yeah, I just feel so I feel the excitement and the love. And then when you went to work as a, as a beef eater, what did you do and where did you work at?   And I normally it's about a couple of a couple of weeks, but because they wanted to control the press release, etc.  August is a very quiet month, as the media is concerned. So, yeah, so I was in September I finally went into uniform. And by that time I was really ready for it. That's that's it there. And that lovely chap that's facing us with a stripey tie in the glasses. He works for CNN International. He was a lovely chap. And so that was that one of your first media? No, because when I, when I'd been introduced, when it in the January of 2007, I'd also been introduced as unveiled to the press. So that was, I had no comprehension how big this thing was Wendy. No idea whatsoever. Really, I mean, the history of the towers like what 700 plus years of all know dominated right beef eaters. And now you come along after all that history. And that's historical news breaking. Yeah, definitely. And to be the first female beef eater in the Tower of London. That's all I got to say to people and they're like, wow, you know her. Oh, that's lovely. And so now now that you get unveiled and go through all the interviews and now you earn the rights of your uniform. Yes. Okay. So when you get put into the uniform now what what is your main job as a beef eater. We are still classed as the guardians of the tower. So we we guard the towers we look after the people look after our visitors that come every day. And we also because we live here at the tower as well, you know there's 46 families that live inside the tower grounds and the 37 beef eaters we all live here so it's a living palace so you have to make sure that people don't go into places that are meant to be going so it's so yeah. So we're still guardians. People you know audience some I don't know if you understand this tower is a palace it's it's massive. And you walk there and the most guests, they go through the gate they give their ticket and then they cross over a moat is it still called a moat still called the moat even though it's grass. It's a moat and it's like 45 feet or what's the distance of the mode 110 wide 110 wide you're crossing over going into the palace or the tower of London, and then there's a wall.  That's about 35 feet. Okay, so in the wall of the Tower of London is where all these amazing be eaters reside. And you're saying what because when you walk into the tower it's tower, it's cathedral, it's museums, it's everything as a tower would be so they have built their living quarters in the wall of the tower. And that's where Moita resides, and you're thinking oh my gosh, but let me tell you, it's very updated upscale. I mean, it's a beautiful. It's a beautiful home that Moina lives in, and calls her home for the last, what 13 years or 13 years now yes, years and just phenomenal because she she might have a day off and go out into town. All the guests are leaving, all the visitors are leaving. She's coming home from a day out in the town. And it's so cool because that's her home. People are leaving and she's going home. I just think that's phenomenal. Yeah, it's quite cool. Of course you know the big talking point is the bathroom window. Oh, I talk about it all the time. The bathroom window is about you open a little louver door don't you and you look eight feet to an arrow slit. And then that's it. That's the bathroom window. So that is the big talking point everybody has to look at that when they everybody has to go to that bathroom, because when you open the door is a regular toilet but behind the toilet. There's a door, a cabinet. And like she says, is that I believe where the archers would have sat in time of battle. Well, they wouldn't be that small it would be it would be a ledge. It's only because it's been built up. The wall would be that thick but it would be more of a little like kind of a tunnel down to it. And so people use that now for storage or because it's so cool and the stone keeps everything so cool. What would they also store in there. Is it good for keeping wine cold. I didn't know I could say that. So I said toilet paper, but yeah, it's amazing wine storage cooler what wine cellar right built in to the wall of the power it's beyond creative of what they've done there. And it's a beautiful home, which I visited a few times. Yeah, going there.  Reason for you to be work I mean to be late at work. But I know you on an average day how many visitors come to visit the Tower of London. Excuse me we normally average annually about 3 million. Yeah, so our busiest weekend is Easter. And that's our think our record then was 16,001 in one day. So it's, you know, it's crazy, it is crazy. But you lovely because the people especially if it's good weather because people just want to be there and you get good giggles and chats and you're telling kids stories and it's fabulous. Photo opportunities everywhere you turn I mean just taking a picture with Moira the first B Peter in the Tower of London that's already an increase of photo opportunity and then all the different sites and everything about walking back into history in the tower is just so when you guys go to London it is a must see, and you actually feel like you step back into history and you can just feel the presence of what actually happened back in all those years of history. Yeah, phenomenal. And waiter that you get to walk it and breathe it and live it. Wow. I know how fortunate am I. Yeah, you're so blessed. But now we want to bring it up to date and I just have to ask you this question. You know, how has COVID-19 affected the visitor account at the Tower of London or is it even open at all. Right we closed on the 20th of March. And we basically became a fortress again. There was, we were the only people who were in the tower. So, and we were self isolating as we as well as much as we could do, because we still had to carry out duties we still had to carry out our security duties. People were still having to go to get provisions and come back in and things like that so it was very, very strange it felt a bit like it just felt like like it was Christmas Day because it's exactly what we would do on Christmas Day. But it was the middle of summer. Well, it was very hot here felt like the middle of summer. And it was, you know, it was every day, and then it started to become very weird. We then, when things started to relaxing slightly, we started looking at when we were going to reopen. And because it's a charity that runs the tower and for the four other palaces in London, one in Northern Ireland. We have to keep running. We have to keep going because they were still paying us, you know, this was the government were paying the people who were unfurl all. But it was just it was sucking money out of the charity like nothing in there. So, we had to open up as quick as we could. But of course it had to maintain safety. So we are now open we opened again on Friday. This past Friday. Yeah, Friday we opened again. Yeah, for those months you were down. Yeah. And I think most of us worked. We worked kind of half and half, because we didn't need everybody on shift at one time. And we had, we opened on Friday. We have your book tickets online, preferably. You can turn up. We have a maximum of 1000 people a day. No more. Wow. One way difference. Yeah, big, big difference. But for people coming in, you know, because there's only 1000 people a day, it's like having a private tour. Yes. You know, you're seeing people laughing their heads off because they're going into the crown jewels twice, maybe three times. Yeah, they don't have no lines, nothing. It's like private tours. And because we've got a one way route, they get really close to the Queen's house now, which is one of the original Tudor buildings in the city of London. We get really close to it now. So that is really good news. And obviously we can't have everything open. But it's good. It's lovely and people are just, they're just, you're getting lots of people, locals who don't want to come during the summer. It's really good. They're coming now. Wow. Yeah, it's fantastic. Yes. And we're going to do one more better. We're going to say, okay, world, the Tower of London is now open. It's open. You'd be the first 1000 and you're in. They're going to let only 1000 in per day. But when you're in the area, you've got to see the Tower of London. It is. And the crown jewels and all the chivalry and armor of their past. It is phenomenal. And I just recommend everyone to have a stop by and be the first 1000 of the day and you get to see there. And then if you're, when you're looking out carefully, you're going to see Moira. And you're going to just say Aloha Moira. Because you have some green Hawaiian roots with her. So now we move further. And let me ask you, I know it's just a, it's a highlight every day for you, but what are the high points of being a beef eater Moira? It is the people. For me, it's the people. There's no two ways about it.  And then of course they'll see you and they'll go, and you speak to them and they go all shy or they start chatting and they start asking these questions. And, you know, it's just fascinating. People are just fascinating. People are fascinating. And then you have the, your duty is to talk to them and make them feel welcome and get to know them and then let them open their hearts and minds so they can enjoy the experience of the Tower of London. So you're, you're the perfect person for this opportunity for this position. I also know that there was one really, you got a letter. I think you got a letter from our governor. Tell us, I mean, when you shared that with me, I just thought, oh, well, another letter. Oh, I got up here somewhere. So what were your thoughts when you got a letter from our governor Linda Lingo? And what did it say there? When I received the letter, and of course it had her address on the back and I went, oh, that's really kind. You know, congratulations coming from a female governor in America. That is absolutely wonderful. And I opened it up and I started reading it and it was many congratulations. And then she's, and then it said that we would love to invite you to Hawaii to take part in an international conference. And I just couldn't speak. I just kind of went, ah, there we are. It's so funny. Oh girl, you with your leg and all. I know. Oh, wow. That was just so fabulous.  I get invited to London. I get invited to Hawaii. I get invited all over the world. I just thought that was, but you actually got excited about coming to Hawaii. I mean, I have the book right here. So this is the book. Oh, it is. So the governor Linda Lingo had international women's leadership conference and this was the fifth annual that Moira was to appear at she was being summons by our governor in to speak in 2008. And that's when I met. Right. So I would, so you would say still to today that would be one of your highlights. Oh, most definitely, most definitely. And I love it when Hawaiians come to the tower because you can hear the accent a little bit and I can, I can know it now. And I'm sitting. Oh, where are you from? Oh, the US. Whereabouts? Oh, we come from Hawaii. And I went, oh, no. I've been in the big island. I've stayed in Oahu. And then that's, you know, it's just lovely for them as well. You're Ohana right there. As soon as you say that you're Ohana. And so that's what they need to hear from you. So now when you, when you, when you're at work, I know you have one of the most amazing uniforms ever. Can you just share with us a few details of the most iconic uniform that you do. Well, the blue and red uniform is when people come into the tower, they get a little bit. Oh, where's your red and gold uniform? Because they always associate the state uniform, which is on the, the beef eater gin bottle, that one there. They, do you think we wear that all the time? And it's just so uncomfortable and very, very warm. So yes, so we wear blue and. The working uniform. If you could put that photo back up of the beef eaters and Moina, just give us a few details about the poundage or what you are wearing there. Well, what we've got on our, on our heads is the, the Tudor bonnet. So that's a black velvet bonnet and it's got red, white and blue ribbons around it to symbolize a red, white and blue or flag. And the white rough is very Tudor. So that's a Tudor uniform. And it's a, you see the red, where the, on our, on our chests, we have E R and there's a Roman numerals two and that is Elizabeth Regina Secundi, which means Queen Elizabeth II. Wow. And then we've got the floral symbols of the, of the countries as well, just below it is very difficult to see. But the, the skirt bit of it, I can say it's a skirt, men wear skirts, being Scottish as well. And what it is for is because it's split into four. And that was basically so that you could get onto a horse. That's basically what it was. And then you've got the breeches and our red tights. And how much is it weigh? Does it weigh a lot? Incredibly heavy, really, really heavy. It's very thick and all the gold braid is very heavy as well. So in the middle of summer, nobody wants to be there. We're on special occasions, right? Yeah, very special occasions. It's so iconic, it's amazingly beautiful. Okay, so now we got to move on because we're going to run out of time. I know. I know there's a very important event that takes place at the Tower of London, where the Tower of London is adorned with thousands and thousands of handmade poppies. And I was there one year and I saw it. Please share with us a few highlights of this monumental occasion of the poppies. The poppies was a commemoration of the beginning of the First World War.  And we had, there was a poppy, a handmade ceramic poppy. Each for each person, each person who had been killed during the First World War, and that was 888,246 handmade poppies. Wow. So it was, it was beautiful, but it was, wow. You can understand why the country was in mourning for three years. Right. Absolutely beautiful. And even now people still come now and ask where the poppies are. I know. It's such a beautiful time to be here. Stunning. Yeah, amazing. So I wanted to ask, so may I ask you, who is your boss? My boss is Her Majesty the Queen. Not many people can say that. They are with the poppies in the moat. Yes. Yeah, what a beautiful shot, Moida. How beautiful is that? Well, Long Live the Queen. Yes, most definitely. Yes. So as a human warrior, you have many opportunities to meet many dignitaries from around the world, Moida, and very many, many treasured visitors to the Tower of London. Please share with us a few of the VIPs that you have greeted in your past 13 years. I was very honoured to welcome Governor Linda Lingle and her family. They came over. They had been doing some business in Germany, I think it was, and then they came to London. So she came in and it was just lovely to be able to, you know, say welcome to my home where that was lovely. And of course we met, and we met First Lady Michelle Obama as well. Well, that's right. She's so tall. Yes, very tall. Beautiful lady. And I know there's one more slide of just a dear friend of mine and there you are in Hawaii. Yes, and that is a lovely experience. I was a commissioner for the status of women for under Governor Linda Lingle, and my opportunity was that I get to meet and greet all our guests. And we became friends, lifelong friends, and we went to the North Shore and we went surfing together and just had a good time in our beautiful state and I was so blessed to be able to share it with you. Right, so hello for that. I'm running short, but I know, I know about one final phrase our audience can use as a reminder to themselves. And what would that be from you, Moina? Well, there's lots of things that went through my mind. But it's, you should never accept disrespect from anybody, especially yourself. Exactly. You see so many people doing it. And when people, when I hear people putting themselves down, even in a joke, I say to them, there's plenty people in this world that will do that for you. Don't you do it yourself. That is absolutely right. Moina. Good to love yourself. Right now, and I just can go on and on and on with you and I just so appreciate you and all that you've done for your country and how you represent the Tower of London superbly. And when you were chosen, they chose the right one. And so thank you. Thank you to all of us, the Queen of England, and just continue doing the best job that you are chosen to do. And we want to just say mahalo to you and aloha from Hawaii, and we'll see you soon. You take care and thank you so much, everybody. Just wanted to say, I've got my Mililani Hydrogen. Her hoody, her Mililani hoody. Mililani hoody. Julie looked after her and she went to visit with them and they so loved you. And they're all nice young men and women now. We're going to send this to them and they're going to appreciate that. So mahalo and aloha to you Moina, we love you and we cherish all that you are. Aloha.


We got the number one spot right here. Solasta Tips gave us the number one blog spot, and so the embedded video on our MGMTV article about Solasta's top five tips to get you started is on the front page of Google. You can get on the front page of Google for your YouTube videos, and I will explain with Daenerys how it works. Ready Daenerys? Okay. If you search in Google for the shut in game review, what do you see on the front page? An MGMTV video of the shut in game review. Let me show it to you. What's up everyone? I'm Andrew with IndieDirect, doing this video in collaboration with MGMTV. Andrew is our newest MGM director, and he got on the front page of Google for that search because of a number of reasons. Let me explain. We use hashtags shut in review MGMTV in the video by putting them in the description here, along with a few other hashtags. Also, we practice OVov, one video, one blog. Right Daenerys? I don't think she's following, but hopefully you're following. Here is the link to our blog. Notice the title, shut in game review with the embedded video. These are all indicators to Google so that when you search for shut in game review, we got the number one spot right here. What about other searches? Here is Solasta Tips. Solasta Tips gave us the number one blog spot, and so the embedded video on our MGMTV article about Solasta's top five tips to get you started is on the front page of Google. This is OVov, the power of combining YouTube with blogging with WordPress that's very optimized for SEO using plugins like the Yaust SEO Premium plugin to give you that edge to get the top results in Google search for your videos. Isn't that exciting, Daenerys? Yeah. Are you excited, Freedom Family? I will show you in future videos more details about how this works and data like how many views we're getting from Google search, but I think Daenerys is bored and I need a quick shave. So I'll be right back. Is this better, Freedom Family? Let me show you more examples. Baldur's Gate 3 Camera. Google search gives us the number one video spot right here by MGMTV. Sign of Silence review gives us the number three spot for our MGMTV blog. Scarlett Hollow review gives us the number four spot for MGMTV above PC Gamer and in Silence Gamer review gives us the number four spot above Metacritic, above Steam Community, and above Wikipedia for that search term. Honestly, I did not expect these results so soon, Freedom Family. By practicing OVov, one video, one blog, we got many top front page of Google search results for our articles and videos. And we can do the same for you. Just submit your videos and your blogs to us and we will feature them and hopefully get you on the front page of Google as well. 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For freedom and the freedom family where we all help each other grow together as a family. And I've been getting my exercise guys. I'm on day 40, 43, where I've gone to the gym. I've started weight training now. I'm not only doing cardio and do you see the muscles at all? I guess you can't underneath the shirt. But they're not there yet. I do have a six pack. It's hidden by a lot of fat. I'm working hard to expose the six pack. So by next summer, hopefully I'll, you know, get to show it to you at the beach. Come on, Freedom Family. Get out there, spend at least 30 minutes a day getting a sweat, getting your heart up and be healthy, sweat out all your toxins. Get motivated just because I can do it. I'm a 45 year old guy who's going to the gym every day for the last 43 days. Come on, if I can do it, you can do it too. Let's get fit together and send me videos approving your exercises down there in the comments below. 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Hello, so we continue with our discussion on the classical aspects of Fourier series. So these are the odds and ends that we talked about some of these we disposed of in the last capsule now we will take up the remaining parts and we come to monotone functions. Monotone functions can be pretty bad in fact, but they also have some pleasant features some redeeming features a monotone function has only countably many discontinuities. The discontinuities are simple jump discontinuities a monotone function is Riemann integrable. Given any countable density there is a monotone function which is discontinuous precisely at the points of D. So for example, you can have the set of rationals and I can construct of monotone function which is discontinuous at every rational and continuous at every irrational. A monotone function is differentiable almost everywhere the last property is highly non-trivial you can look at the third edition of Royden's real analysis. I only studied the third edition there is a fourth edition I understand, but Royden's real analysis is one of the finest books written for the sheer joy of reading mathematics in his style and here in chapter 5 he proves this last property using Vitale covering theorem and I urge you to look at the proof there we will discuss the other properties briefly. Item number 4 says that a monotone functions can behave pretty badly because it can be discontinuous at lots of places distributed densely. Nevertheless, we shall see that the Fourier series of a piecewise continuous monotone function is quite well behaved as regard this convergence this is the theorem of deletion that we shall establish presently. So a few properties of monotone functions would take a function f from AB to R which is monotone increasing and let us take a point C in the interval AB and let us look at the interval which is half open open from the C end and closed at the B end you can actually ignore what happens near B you are going to concentrate on what happens near C only and it is a monotone increasing function. So I am taking the infimum of f of t as t varies from c to b c excluded and this infimum let us call it l and let us show that the function f of t tends to l as t tends to c plus and that will prove that a monotone function has a right hand limit at each point. So let epsilon greater than 0 be arbitrary and invoke the definition of infimum there exists a t naught between c and b c excluded such that f of t naught is less than l plus epsilon.  So if t is less than t naught f of t will be less than or equal to f of t naught.  So the requisite delta is t naught minus c. So using the epsilon delta definition we already established that the monotone function has a right hand limit the case of the left hand limit is very similar and the proof can also reveals that the left hand limit is less than or equal to a right hand limit. What will you do for the left hand limit you will need a supremum of f of t for t less than c then you will establish that f of c minus is less than or equal to f of c plus that I am leaving it as an exercise. Now suppose I take 2 points c1 and c2 c1 strictly less than c2. Now let us look at the open interval c1 c2 and let us take the infimum of f of t on the open interval c1 c2 and the supremum of f of t on the open interval c1 c2. When I take the infimum of f of t on the open interval c1 c2 what do I get? I get the right hand limit of f at c1. The supremum of f of t on the open interval c1 c2 is the left hand limit of f at c2. So we get f of c1 plus less than or equal to f of c2 minus.  So I consolidated all these inequalities into one inequality. So now if c1 and c2 are two distinct points of discontinuity that means that this inequality is strict and this inequality is strict then the open intervals i c1 and i c2 that is open interval f of c1 minus comma f of c1 plus the open interval i of c2 f of c2 minus comma f of c2 plus these are disjoint non-empty open interval. They are non-empty because the function is discontinuous and they are disjoint because the previous inequality. So for each discontinuity c I can pick a rational number qc in this interval f of c minus comma f of c plus. In the function c going to qc is a injective function because of the disjoint nature of these intervals this assignment c going to qc will become injective and it perhaps the set of discontinuities into the rationals because I have chosen this qc to be a rational number. So I have an injective map from e into rationals so the set of discontinuities is at most countable because the set of discontinuities is at most countable the Riemann integrability immediately follows. Recall when is a function Riemann integrable a function is Riemann integrable on a close bounded interval if and only if the set of discontinuities has Lebesgue measure 0 a countable set has Lebesgue measure 0 and the job is done. So now suppose that e is any countable dense set of ab take any countable set in fact whether it is dense or not is irrelevant let us call this countable set e as r1 r2 r3 some enumeration of this set I have taken. Now you choose a convergent series of positive terms summation 1 upon 2 to the power n a geometric series what we do is that we prescribe f of x to be summation 1 upon 2 to the power j where you pick those j's for which rj is less than x that is a well-defined function. Now if x is less than y remember that there are more j's for which rj is less than y and there are fewer j's for which rj is less than x because x is less than y. So that immediately gives you that if x is less than y then f of x will be less than or equal to f of y and if this is dense then f of x will be strictly less than f of y we get if it is dense we get something better we get a strictly monotone function it is not difficult to verify that the function f of x that we are constructed is discontinuous at every point e and it is continuous on the complement of e. So this establishes some of the properties of monotone functions that we talked about early in the first couple of slides. Now we come to the Dirichlet's theorem which goes back to 1829 is the earliest rigorous result this is the general convergence theorem for piecewise continuous monotone functions. Suppose f from minus pi pi to r is a piecewise continuous monotone function the Fourier series of f converges to f of x at points of continuity and at a point x0 of discontinuity it converges to the arithmetic mean of the right hand limit and the left hand limit at x0 at plus minus pi the Fourier series converges to one half of f of pi minus the left hand limited pi and the right hand limited minus pi. To simplify Dirichlet's argument Ossian Bonnet established the mean value theorem for integrals that bear his name. Bonnet's mean value theorem is stated below and this theorem 102 let g from a b to r b continuous and let f from a b to r b monotone. So we have got a continuous function and we have got a monotone function and you are integrating the product f t g t dt from a to b the product is evidently Riemann integrable because the discontinuities are countable. Then this is nothing but f of b minus integral from c to b g t dt plus f of a plus integral from a to c g t dt for some c in the closed interval a b. So Bonnet's theorem asserts that there exists a c with this property. Further we can see a little more if you know that the function is non-negative and monotone decreasing then we can reduce this to one piece integral a to b f t g t dt equal to f of a plus integral from a to c g t dt. If f is monotone increasing and non positive then there is a c such that integral a to b f t g t dt equal to f of b minus integral c to b g t dt. So you got these three pieces and it is not difficult to prove the Bonnet's mean value theorem but let us go further with a Dirichlet's theorem on Fourier series. We need a preliminary lemma. What do the lemmas say? Theorem 103 we have the following limit integral 0 to infinity sin omega t upon t dt is signum omega into pi by 2 that is signum omega is 1 if omega is positive it is minus 1 if omega is negative it is 0 if omega is 0. The second result is that you have taken a monotone function and you have fixed a s positive and you are integrating from 0 to s f of u sin omega u by u du and I am taking the limit as omega tends to infinity I am going to get pi by 2 times f of 0 plus. Here I have not stated whether it is monotone increasing or monotone decreasing it should not matter because if it is true for monotone increasing it will be true for monotone decreasing because I can multiply by minus 1. So I have just written it as monotone. Secondly I require in Bonnet's mean value theorem some positivity or negativity or whatever it is again that will not matter. Suppose for example I want the function to be non-positive so what I do is that once I establish this I can subtract from f a large constant and I can make it strictly negative or I can add to f a large positive constant and I can make it strictly positive. So then you have to understand what happens to the limit as omega tends to infinity integral 0 to s constant times sin omega u upon u du remember I had tampered the function by adding a constant. So what is the result of this tampering the result of this tampering is going to be simply integral 0 to s sin omega u by u du put omega u equal to v then I will get sin v by v dv 0 to infinity because omega is going to infinity and that limit is pi by 2 remember. So tampering the function by adding a constant or subtracting a constant is not going to disturb this equation. So I may as well assume that the function is monotone decreasing and non-positive or non-negative whatever I like so that we can apply the Bonnet's mean value theorem. The first part is standard it is clearly enough to prove it for omega positive because if omega is negative sin is an odd function the both sides will change sin that is all make the change of variables t omega equal to u and the integral simply becomes integral 0 to infinity sin u by u du and we just have to prove that sin u by u du integral from minus infinity to infinity is pi but this is elementary complex analysis apply Cauchy's theorem to this holomorphic function e to the power i z minus 1 upon z this is holomorphic on the punctured plane but applied to a semicircular contour indented at the origin. You have probably seen this kind of stuff in your baby complex analysis course so I will leave it to you to verify this and we straight away go to the proof of the second part which is more interesting. So as I explained before there is no harm in assuming that f is monotone decreasing and non-positive so that I can apply the Bonnet's mean value theorem and add and subtract f of 0 plus now once I add and subtract and f of 0 plus this integral that you see that is very clear what is happening to it as omega goes to infinity I just explained this will go to integral 0 to infinity sin v by v dv that is pi by 2. So we need to simply tackle the first term so let us look into the first term in detail.   So all we need to do is to deal with this integral from 0 to r f of u minus f of 0 plus upon u sin omega u by u d u but now we apply the Bonnet's mean value theorem here is where the monotonosity of the function is going to play a critical role and by Bonnet's theorem this is going to be equal to f of r minus f of 0 plus integral from c to r sin omega u by u d u. Now we are ready the integral term of course is bounded bounded in absolute value by say a and so we see that the integral from 0 to r f of u minus f of 0 plus upon u sin omega u d u is less than epsilon in r in absolute value the proof of the theorem is completed we shall now prove the theorem of Bonnet since the theorem is usually not given in modern texts as explained to you and a proof is available in George Gibson's advanced calculus page 277 for convenience we state the theorem here in blue and we will prove this theorem in the next capsule and we will need for this purpose Abel's summation by parts formula this proof of Bonnet's mean value theorem is a beautiful application of the formula for summation by parts by Abel and this formula is used frequently in analysis particularly when you want to discuss conditionally convergent series etc you can see for example Rudin's principles of mathematical analysis already in chapter 3 you will see this result stated there stated and proved and we will discuss these things in the next capsule and I think it is a good time to stop here thank you very much.


It is a Black Monday for travellers, both drivers and owners of the Bonney vial, named Jette in Port Harcourt, city local government area of River State, as an early morning fire, several persons including pregnant women, women and a toddler were reportedly burned to death. Some of the victims were also eyewitnesses of the Jette, he said, that many fibre and wooden boats were destroyed in the inferno, which started at about 2am, sparked by a local boat that arrived in the city from Brass, late Sunday night. Counting their losses, they said the effect of the disaster was massive due to lack of emergency services such as fire service at the Jette. They appealed to the River State government to build a fire service station close to the Jette as promised to avert future currents. I've lost everything. Good day, I've seen what is happening on the Jette. I've seen people property, I've seen matured land, being naked. That is just for property alone, we did not talk about souls, souls we've lost. We can't go here this morning now, even the ones who don't pick up, they try to bring this engine to the waypoint, who don't pick so many, not be the ones that will die for night, for night, where we'll keep here. This thing started around 12 o'clock in the night, with the no fire service, nothing. It is the boys area here that struggle to quench this fire.  That is when you see most people died inside the fire. The acting manager, National Inward Waterways Authority in Port Arcut, Abram Ogwu said, there is the need to sensitize the boat drivers and travelers on safety measures. Even though the cause of the massive blaze is not known, previous fire outbreaks at the jetty had been attributed to the storage of inflammable substances, like petroleum products used in powering the speedboat engines in makeshift buildings within the slum. The jetty is usually a busy transit point for thousands of residents, traveling from Port Arcut by sea to Boney, Belay, Nimbay and other riverine communities. Hello, hope you enjoyed the news. Please do subscribe to our YouTube channel and don't forget to hit the notification button so you get notified about fresh news updates.


Stay tuned to the end of this video because it is time, once again, for the Nerdfighteria census. Good morning, John! It's the last reunion video! Ba-ba-ba-ba-ba-ba-ba-ba! It's Question Tuesday. We answer real questions from real Nerdfighters. We're gonna get right to it. It's the YouTube edition. John! Yeah? Who is your favorite OG YouTuber back in the real early days? Mmm, Community Channel Natalie Tran. Natalie Tran. My 2006 favorite was probably Lonely Girl 15. Yeah, my 2006 favorite probably, like, The Winecone. Oh, I loved The Winecone. And Ze Frank, of course. When did you realize this was gonna be your job? It's still not my job. I don't get paid to do this. So, uh, Neff, careers on YouTube tend to be very short. Like, Hank and I've been on YouTube for 15 years now, and so we've seen how really, really good careers cannot last very long. And the people who stick around tend to be people who reinvent themselves a lot. And- Which we don't do! And us! Sam asks, what do you think are the best and worst site functionality changes YouTube has made over its existence? Well, in 2007, the site didn't work. And people forget about this, but like, the site works now. Yeah, it's very impressive. The player is very good. Players excellent. People forget how good the player is. So good. There are a lot of secret hotkeys, and that's my favorite UX thing.  You can go back a frame, or you can go back 10 seconds. You can go back frames. In terms of my least favorite feature, I think it was better. When it was a five-star system. Wow! Well, because then they went to likes and dislikes, and now they got rid of dislikes. I think five stars was the best. Yeah. But I'm old, so of course I think that, and also I wrote a book about the five-star scheme. Was there any time, Jason asks, during Brotherhood 2.0 that you wanted to stop? Yes, like in March, where it still felt like a really long project. But the hard part for me is always imagining the future. That gets really overwhelming. Right. In the present, it was always fun. Jeremiah asks, when is the next X-Jokes in four minutes coming? That's the first minute of Deering of John is for. Yes. That's where I do that now. Jessica asks, who got the play button for Vlogbrothers, or do you take turns? And I do not know the answer to these questions. I don't know where it is. I just looked in the background to see if I have it. I don't think I do. I think it's on the wall at Complexly, Missoula. I remember, I was on a plane when we hit 1 million subscribers. And I remember the feeling, it was incredible. And now I never think about it. Who is your oldest YouTuber friend? Hank. Yes. I feel like Hank and I are really lucky that we have each other, because if we didn't, I wouldn't have a peer. And now I have a colleague. It's very hard to do this on your own. Who understands what it's like from the inside. Robin says, I miss video responses. Do you miss video responses? I miss them now way more than I miss them when they actually got rid of them. Because I feel like they would do better now. Some people forget this, but like video responses were widely misused. Yeah. It's really easy to forget how much we know about those days of YouTube. Thanks for chatting about it with me a little bit. It's been fun. It's been fun mostly. I had a good time. I thought you meant the entire 15 years. Oh no, I meant the last 12 minutes. Thank you to everybody who's been through it, even a little bit of it with us. And John, I'll see you on Tuesday. Yeah. On my computer screen. Yeah, I know. It's sad. I have to poo, so. The Nerdfighteria Census is a yearly, or in this case, because some things came up, a bi-yearly survey that helps us know what we should be thinking about and focusing on and how we should be understanding this community. And it's super valuable for everything that we do together. So please follow the link in the description to fill out the census. It's one of my favorite projects of the year. And I'm so, so, so looking forward to hearing everyone's thoughts. Thank you again. Link in the doobly-doo.


Hi, babe. It's me, Sarah Silverman. I know what you're thinking.  Is she gonna pretend to show us her boobs? Is she gonna make jokes about assholes? First of all, no.  Though sidebar, it's extraordinary. There's a bill in the Senate right now. It's called the For the People Act, and it's gonna do a lot of amazing things, like end gerrymandering and secure our elections. But even more importantly, it's gonna stop billionaires from straight up buying our elections.  The billionaires and the politicians in their pockets. So we gotta do this. We gotta get the For the People Act passed. Visit my friends at represent.us-asshole and make sure your representatives know you want this thing passed now. Get loud. Makes a difference. Squeaky wheel gets the grease.


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