Hey there. Hello. Thank you Chris Tara. All right. I'm going to turn the mic over to you and you're welcome to get started Thank you so much for joining us Thank you Good afternoon everyone It is my great honor to present the Republic of Korea's view on space threats at the fifth summit for space sustainability Let me start by briefly over viewing the Republic of Korea's space activities and plans As an emerging space-faring nation Korea began its development of satellites in the mid 1990s and is increasing its range of space development programs from satellites to space exploration With a 30-year history of space development Korea now operates various satellites ranging from low Earth orbit to geostationary orbit Last year we successfully launched our homegrown space launch vehicle and Became the 7th country to explore the moon with the launch of Korea passfinder lunar orbiter Based on recent technological achievements The Republic of Korea aims to continue space exploration and advance its space industry to drive robust economic growths our ambitious space goals as well as our focus on promoting the participation and engagement of the private sector in space activities are well illustrated in the space economy roadmap and force basic plan for space development promotion for 2023 to 2027 announced last year. I Would also like to highlight that the fourth base plan which serves as a comprehensive strategy document for Korea's space activities Highlights Korea's capabilities to protect its space system as one of the five key objectives This is the first time that this aspect has been reflected in our strategy document As the Republic of Korea's space development program progresses and its exploration activities expand It fully recognizes the immeasurable importance of maintaining a safe secure and Sustainable space development environment For us this principles of safety security and Sustainability are not merely rhetorical aspirations, but realities as they are for every space actors This explains our national position towards destructive direct ascent and ties out like missile testing We believe that such testing itself is a direct threat to space assets And the long-lived space debris created during the testing Poses a great risk and threat to space safety security and sustainability This is why we expressed our concern regarding the asset missile testing in November 2021 and welcome the US decision not to conduct destructive the asset missile testing in April 2022 Now I would like to go into further details of the Republic of Korea's decision To join the commitment following the US, Canada, New Zealand, Japan and Germany Since the announcement from the US in April 2022 The Ministry of Foreign Affairs Convinced several interagency meetings to discuss our national position on joining the commitment We examined how this will affect our and our adversaries capabilities and behaviors As you all know the Republic of Korea has never conducted such testing Yet we cannot exclude the possibility of being the victim of such testing We also considered our real need to protect our space assets and activities We also looked at the voluntary nature of this commitment and how this will affect the shaping of global norms Korea's firm position towards a safe secure and sustainable space environment and our aspiration to develop norms of Responsible behavior in outer space facilitated the process The Republic of Korea was one of the main sponsors of the United Nations General Assembly resolution On reducing space threats through norms, rules and principles of responsible behaviors in 2020 and 2021 And actively participated in the open-ended working group This experience has helped the relevant ministries form a common understanding of the dire necessity of developing space norms Ultimately we reached a whole of government common understanding The destructive DA asset missile testing is One of the most evident and urgent threats to our space assets and activities It can have an impact on any asset of any country for long periods of time as highlighted by multiple previous speakers Korea believed that it is in our national interest to develop norms on destructive DA asset missile testing To clarify that such behavior will not be deemed acceptable within the international community Following this logic the Republic of Korea joined the commitment in October 2022 and sponsored the United Nations Resolution to call on other states to join the commitment We are pleased to see that this resolution received huge support from the international community with 165 states among 177 present voting for the resolution We understand that other states which voted for the resolution but have not yet joined the commitment Need some time to thoroughly review the domestic effect of this commitment In this regard Korea is also making efforts to raise awareness of the danger and risks posed by asset testing Including sponsoring the secure world foundation to develop an infographic dedicated to asset activities and their impact on space sustainability We invite all states to join this commitment and safeguard our future together The Republic of Korea firmly believes that the growth in the number of states supporting and Subscribing to this commitment will positively impact the safety security and sustainability of the space environment and The benefits will affect every state regardless of whether it is space-faring or not Once again, we encourage other states to join us before it's too late. Thank you Thank you so much Hyren for that call to action for governments and other entities around the world the infallible logic that was used by the Republic of Korea to Create this policy and enforce it with for themselves and their perspective on how it makes everyone in the world safer is something That we had secure world could not agree more with so thank you again very much you


A new study suggests that NMN supplementation can indeed elevate NAD levels and increase walking distance in healthy participants. Not only that, but the study narrowed in on 600 mg a day as the optimal dose. While it is believed that NAD can be supplemented by precursors such as NMN, this route has a few roadblocks. For instance, studies do not always show that NAD precursors increase NAD levels. Their safety is a bit of concern as well, since NAD can provide energy to cancer cells. However, NAD precursors are generally considered safe. This new study was based on a randomized, multi-center, double-blind, placebo-controlled clinical trial that was co-sponsored by two companies that jointly produced a food-grade NMN product. The primary endpoint was NMN's ability to elevate NAD blood levels with secondary endpoints of safety and tolerability, physical performance in a walking test, blood biological age, insulin resistance, and overall health via a subjective assessment. Previous studies were often inconsistent in showing NMN's health benefits or its ability to increase NAD levels in blood. These studies were often sex-biased or conducted in people with pre-existing health conditions. This time, the researchers made it a point of recruiting 80 healthy men and women aged 40 to 65 with a wide range of BMI scores. Three different doses of NMN were investigated, 300, 600, and 900 mg a day. NAD levels were increased significantly in all groups compared to placebo and baseline. There was also a significant difference between 300 and 600 mg a day, but not between 600 and 900. Interestingly, most of the increase in NAD levels happened during the first 30 days of the study, with the second month showing only a very mild additional increase. The 6 minute walking distance at baseline was about 300 meters across the groups, which is on the slower side. It significantly improved in the three study groups compared to placebo and baseline. Here too, the difference between 300 and 600 mg a day was large, but the difference between 600 and 900 was virtually non-existent. The gains in walking distance were substantial, with both the 600 and the 900 mg a day group adding about 150 meters, a 50% increase. Participants were not required to perform any regular physical activity, and testing was done only three times, at baseline, after the first month, and after the second month. If there was any habituation to the walking test, it should have been noticeable in the placebo group. However, in this group, no increase in walking distance occurred. However, it is possible that participants on NMN began feeling more invigorated early into the study period, and increased their physical activity accordingly, which led to better results measured in the clinic. The participants were also asked to complete a 36 question health and quality of life questionnaire. The scores mostly followed the same dynamic as NAD levels and walking distance. There was some increase in the 300 mg a day group compared to placebo, and a much more pronounced increase in both the 600 and 900 mg groups. However, there was no significant difference between the two latter groups. This study appears to establish 600 mg a day of NMN as a preferred dose that significantly impacts NAD levels and physical performance. As with previous NMN studies, no safety problems were reported. However, that does not prove the safety or efficacy of NMN beyond any doubt. Therefore, there should be more studies, with larger sample sizes, different end points, and longer follow up periods. So is NMN a part of your longevity strategy? And will this study change how you view it? Let us know your thoughts in the comments below. When there's more to share, we'll have it for you here, so please subscribe so you don't miss out. I'm Ryan O'Shea and we'll see you next time on Lifespan News.


I've got this TBR clue though, but I felt like, you know, let's get in the mood because today we're going to be chatting about my top 10 murder mysteries of all time, ranked! It's probably the most important day of my life. I've ranked my top murder mysteries, I've picked my favorites, so hopefully this will be a great video for you guys to come to if you're wanting to get into murder mysteries and you can know my definitive favorites. But before we get into the video, I want to say such a big thank you to my favorites, the sponsor for this video, Book of the Month! I love Book of the Month. You guys know I have loved them for so long, I think their service is absolutely incredible. I love their whole ethos, so Book of the Month is a super popular book service for readers and every month their team scours all the new releases that are coming out to find new and emerging authors, to find the books that are going to be most loved by readers and they pick a selection of books for you to pick from. Every month is your Book of the Month, essentially the clues in the name. You are served up on a blue platter, a selection of new releases, so you don't have to spend time necessarily like researching what new books are coming out. They bring you what they think are going to be some of the biggest books of the month, some of the best books of the month, and then you can pick from that what you want to read. There's no risk also from the books interest you that month because you can skip that month very easily and they absolutely have the best price for new release hard cover fiction and you can get your first book of only $9.99 using the code Meg with Books. It's only available for those of you watching who are in the US. I just want us to like take a moment to discuss how when Book of the Month sat down for March of Selection, they thought, hmm, what would Megan like? This month it feels like it was made for me, so something super exciting that Book of the Month are doing is it's not only anymore going to be only five only, only five books a month that you can choose from, they're now going to have up to seven books a month. Sometimes it'll be five, sometimes six, sometimes seven, but up to seven books a month sometimes for you to pick from. And this month in March there are seven. Let's just take a moment everyone. Look at this. Oh my God. And of the seven, these two I'm still interested in reading. Whoa. And the romance. But these five, these five, let's just take a moment everyone to discuss these five books because they sound exactly like the kind of thing I'm going to love. Also, let's just talk about how like three of these, if not a little bit more, could be classes mysteries, like for them. Maybe mysteries are the moment. Maybe I'm a trendsetter. Maybe me and Book of the Month are on the same wavelength because I mean, come on. So we have got, tell me everything. The first three I'm going to talk to you about I hadn't heard of before this, but it's so exciting. This is a memoir slash literary true crime by this private investigator who got caught up in this sexual assault or sexual violence case. Obviously a lot of the details have been changed. It's kind of her memoir, but it's kind of written as fiction. It sounds so, so interesting. I love true crime. I love this kind of thing. Then we have the Verifiers, which is like a mystery detective dating agency.   This is made for me. I mean, the description of our main character is a lifelong mystery reader who wrote her senior thesis on Jane Austen. I mean, like my world could not be more perfect. Literally me when I saw the selection for this month. Then we have The Cartographers by Peng Sheppard. This is think of the Eshwab where like fantasy slash historical magic or it's about these this girl who's a cartographer and she discovers this map and it's like magic. And like, oh, it sounds so interesting. And then they sent not one, but two of my most anticipated releases of this year. The first is a book of cold cases by Simone St. James. It's about two murders in the 1970s where the main suspect was acquitted. And in the present day, we've got a girl interviewing her and it's spooky. It's mysterious, it's crazy. And then probably my most anticipated release of this year. Barbara by Lucy Foley. You guys know this is like my most anticipated release. I absolutely love Lucy Foley. You may be seeing her in this video again. And this is set in a flat complex in Paris where you can see across the courtyard and to everyone else's. Everyone else's rooms and there's mysterious stuff going on. We've got the socialite, the nice guy, the alcoholic, the girl, the verge, the concierge, everyone's a neighbor. Everyone's a suspect and everyone knows something they're not telling. I'm so excited. I'm going to be reading this this month. And it's just like one of my most anticipated releases ever. So listen to me. You could use my code makewithbooks to get one of these books, one of these mini books for only 9.99. I would absolutely recommend you go do that because this selection this month is literally if you kind of like the same books as me, there's something for you here. I can't believe it. So make sure you check out Book of the Month down below. OK, everyone, it's time to get into my top 10 murder mysteries of all time. I just don't think this is like great for me. I don't want to do it. I want to go home. Like I can't take the pressure of it. This was very actually very, very hard and I have been very strict. There's no missing person mysteries on here. Some honorable mentions like of missing person mysteries like the project. This is probably my favourite mystery of all time, the strange case of the alchemist daughter, but it's not a murder mystery. We are talking murder mysteries, often with a detective or with a civilian who ends up acting as a detective or like a closed circle mystery, stuff like that. That is what we are talking about here. And the first two on this list are actually four stars. I don't tend to give out a lot of five stars to murder mysteries. I'm pretty picky. I love reading them, but like you have to blow my socks off to get five stars. So I haven't even got 10 murder mysteries I've given five stars. And I've again, I've been very strict. There's one book on here that's like, my autonomy feels guilty. I'm like, is that a murder mystery? But we're going to go with it anyway. So number 10 is a very controversial book. When people read it, it is The Last by Hannah Jameson.  Like it's not the best ending. Well, I don't know. It's one of the books I think about the most because this book has the most ridiculous ending. You'll be reading it and you'll be like, hang on. We've just entered a different book. Like this is not the same like tone.  But essentially the world has ended in nuclear war. You and 19 other survivors hold up in an isolated Swiss hotel.  Then you find the body.  The race is on to find the killer before the killer finds you. Oh, it's so good. No, I don't think you understand. I'm a professor. We're following the perspective of one main character.  I love an isolated closed circle murder mystery. I love it. It's my favorite murder mystery trope. I love it. I love it. And it gives me that buzz. A group of characters stuck in a place and death starts occurring. You'll see probably quite a few on this list. The vast majority of these are closed circle isolated mysteries because that is just the perfect mystery to me. It's absolutely perfect. I love the way that then clues come out and how like, you know, there's so much scope for the author to like put in a twist that doesn't make sense. With a closer close mystery, you don't have a lot of room for excuse, I feel like, from the author. And this is just one of the books I think about the most. I think about this all the time. I think it's a great murder mystery to read. There's something that's a bit different. I don't think many people will give this five stars because the ending is so out of the blue that no one will expect it. And a lot of people probably be angered by it. But I feel like the first half of this with the tension that builds and the fighting between the characters and just this like real high stake situation is what makes this book. Then we have another four star and that is An Unwanted Guest by Sherry Lapina. So I actually read Not A Happy Family by Sherry Lapina yesterday for my patron book club. And it was my least favorite Sherry Lapina that I've read so far. But I really like this one. We have a group of guests go to this hotel for this like break they're snowed in and then someone is killed. Like, can you guess, can you guess my favorite, you know, my favorite tropes here?  I think the atmosphere again that Sherry Lapina builds in this book, you know, they're very like in this secluded hotel. It's not very big. It's like an inn almost. Like there's probably like six bedrooms or something. It's not like a big hotel like in the last. So it's very claustrophobic and it's like rich people, which I like. Like rich people getting killed and like, you know, getting their comeuppance. We never really know who you can trust. There's a lot of really good twists in this. I just remember having like a great reading experience reading this.  Do you know what I mean? Whereas this one is a bit out of left field. This is simple, but I feel like it does it well. It's a quick reach. All of Sherry Lapina's books you can literally read in one sitting. She's like the queen of like fast pace keeps you turning and it's just a fun reading experience. So now we're getting into five stars. Now this book I think I actually initially gave 4.5 but now in my head, it's definitely a five star. I think about this one a lot as well. It is the murder of Roger Acroyd by Agatha Christie. So me and Agatha Christie have a strange relationship where I love some of her books and I hate some of her books. Some of her books have been like on my worst of the year lists. Sometimes Ms. Girl annoys me. But this one is so good. So this is the fourth in the proro series, which a lot of you know why I'm reading in order. And on the face of it, there's like a very simple Agatha Christie. One of Porro's friends is murdered and he is like meeting this group of rich people who are staying at this house, you know, his friends and family to try and figure out what happened. Now this one's really fun. There's like maps and stuff in this, which I really like. It's very much that Agatha Christie trope of like the quaint English town where murder happens. But the real pro of this book is the ending. It's widely regarded as one of the best endings for a murder mystery. I will say I didn't actually know like going into it, it made it even more shocking to me. But the ending for this and how she wove it throughout the story and how once you finished it, you're like, oh my God, like how did I not realize that? It's absolutely amazing. I'm gagging. It's absolutely masterfully done. And I just like, I think about it all the time.  I literally sat there and I was like, and it takes a lot for you to gag at me like that. And for me to think, oh my God, it's perfectly done. But this is like, that's what Ms. Agatha Christie does in this book and she does it so well. Coming in number seven is the book that I was like, I don't know if it's quite a murder mystery. It's like pitched as a murder mystery. It was like kind of marketed as one. But I don't know if it's quite a murder mystery because the murder doesn't really happen until the end, but it's got a detective duo. It's got like a mystery and there, you know, there is a murder that then carries on the story. So I'm gonna allow it. I'm gonna allow it. It is The Devil and the Dark Water by Stuart Turton. So I feel like this author is more popular for the seven and a half deaths of Evan Hardcastle. I can't remember. But I didn't really like that, but I loved this one. So it's set in 1634 and we are following the world famous detective, was greatest detective, but he is imprisoned on this ship. He is literally in a jail. He can't do anything. But his like duo, like his right hand man, his Watson is traveling on the ship. And so it's up to him to kind of discover this mystery of what's going on board. There's like strange symbols going up on the ship. It's, oh, do I say it's set on a ship? We're set on a ship. So I love, again, that like closed, we're locked into this location. We are stuck here with these people, squeaky chair. Animals are dying. There's strange sounds, there's strange symbols appearing. And it's up to our main character to kind of figure out what's going on. And I really love the historical setting in this. I thought it was very vivid, the historical setting. I loved the relationships in this are protagonists. So the detectives like right hand man, the friends, the wife of like the head of the ship and them trying to solve the murder together is very, very fun. I also just felt like it was really well written. I really loved the writing, the pacing, the way that the story unfurled. I thought it was really, really well done. It was just like a solid book. Again, this does have an ending that was a bit disappointing for me, but I think I still give it five stars because I enjoyed the rest of the book so much. But the ending again is left filled in this one. You're like, not sure, not sure about that. Not sure about that. Very strange. Number six on this list is one that I feel like some of you are going to be surprised to see, but it is one by one by Ruth Ware. So this didn't get the best reception when it came out because it was marketed like another one of Ruth Ware's thrillers, but it's not. It's not a high-paced thriller. This is a traditional clothes circle, isolated murder mystery. Let me hear you say it. You say it. Oh my God, it's so good. So we have this company called Snoop, which is like a combination of like a Spotify and Twitter. Like you can see what people you follow are listening to at that moment and listen along. Which by the way, was an idea I think I sold, like for it not sold, but like there was a competition like design app. I'm pretty sure that's the app I designed in like year seven. So like where's my coin? Ruth Ware? Did you, how did you find out about that? Anyway, it's very Agatha Christie-esque. They are snowed in at this ski resort and they start dying one by one. It's pitch doesn't add then there were none retelling, but I don't like, and then there were none by Agatha Christie, because I don't like that scheduled dying. I'll say it again.  Whereas in this one, there's not that many deaths. It's like, it's not really and then there were none. Like there's not, there's multiple deaths, but there's not like the whole cast. Do you know what I mean? Which I feel like, and then there were none retelling needs. It needs like four or five murders, which this doesn't really have, which I'm glad about. The setting again, the isolation, the tension was amazing in this book. And something that actually made this book for me is that you know from a certain point, you know who the baddie is. You know, you know from a certain point, and the tension that comes with that think no exit is really, really good. Like the tension that that builds up through knowing who the bad person is and trying to figure out how you're going to escape them is really great. So this is my favorite Ruth wear. Actually, I think I prefer it to turn the key. I really want to reread this soon. And I really hope that going forward Ruth wear, I think I next release is like more of a stereotypical thriller, but I hope she does some more stuff like this because I loved it so much. It was so well plotted. But I'm scared because the receptionist got that she's not going to. And I'm like, Ruth, listen to me. Number five is In the Market for Murder by T.E. Kinsey. So if you've watched my channel a lot, you'll know this is, well, I'm gonna say my favorite cozy mystery series, but it's the only one that I really read. I don't, I'm trying to make my way through this series prior to start loads of others. This is the second in the Lady Hardcastle Mysteries. We're following like, it's like 1908, 1909, 1910 in England. And we're following a lady and her maid, but they're busy mates. Let's not talk about the power imbalance. We don't, we ignore that, ignore that, everyone. We are going to pretend we didn't hear that. They're best friends. They've been best friends for a very long time. They've accompanied each other for a very long time. And they've moved to this town, which, you know, quaint English town, supposed to be them retiring. Turns out it's the murder hotspot of the wild. And they are basically using their former skills. They used to work kind of as spies together. They're using their skills to solve the murders in this town. And I really loved this one. This book ends up having like three separate mysteries. I don't think all of them murders. There's one where like something's been stolen and stuff, but there is at least one, if not two murders in the three mysteries. And the way that they all got solved throughout the book, and kind of like coming together, and not necessarily linked, but the way that the stories overlapped each other, I thought was really, really well done. Listen, you need to listen to the audiobooks, these books if you're going to read them. I love the relationship between Lady Hardcastle and Flow. They're funny. They're lighthearted. They're in this quaint English town. They're perfect palette cleansers. They're so easy to read. I will like speak about these books to the day I die. I think they're wonderful. I think they're perfect cozy mysteries. Again, I love the historical setting. I often like murder mysteries with like an historical setting. I think it adds a little bit of something extra. And yeah, I just love these women. I'm very excited. This is the second. I've read the first three. I'm really excited to read number four soon. But I think this is a severely underrated. And if you wanted to give cozy mysteries a try, this would be my number one, obviously recommendation. But you got to read the first one first and then get to this one. Number four is A Good Girl's Guide to Murder by Holly Jackson. So this is actually my second favorite in the series. My favorite in the series is As Good as Dead, the last one. But those, the other two in the series aren't murder mysteries. I wouldn't class them as murder mysteries. Whereas this one is. Five years ago, schoolgirl Andy Bell was murdered by Sal Singh. The police know he did it. Everyone in town knows he did it. But our main character Pip doesn't believe that he did. So she is kind of reopening this case that happened years ago and is trying to figure out what happens. She's doing it as like a school project. But this is so much fun because it's mixed media. I've got interviews. We've got police reports. We've got all that kind of stuff.  Holly Jackson can really write. I mean, this series as a whole is one of my favorite series. The journey that Pip goes on throughout this series is amazing. This for me is five stars, obviously. And then the second one is four stars. And then the final one is five stars again. And the final one was like, oh, if you watch my vlog for that, you'll know that was a lot to put me through Miss Holly Jackson. That was a lot. Holly Jackson, ignoring my pain. I just really love the kind of fun, like YA. I think this is the only YA on this list.  Well written, great characters. Again, one of the only ones in this list were like you really go on a character journey with our main character as well. I just think it's perfect. Like literally perfect. We're getting very close to the top now and the hat does not want to stay on. So number three is murder on the Orient Express by Agatha Christie. Another Agatha Christie on this list. This was I think my first proper foray into murder mysteries, like real murder mysteries. Obviously a lot of you will know on this list another isolated one. The train stops. The train is snowed in. Urquiporo is on the Orient Express. And we find out one of the passengers has been murdered. Now, I often say you are going to love either murder on the Orient Express or and then there were none and you're going to dislike the other. I genuinely believe they are four different audiences. I mean, this is like, you know, I am simplifying a bit. People do like both. But I feel like one, you really, you really think is better than the other. They're had two most popular books. And then the one is, I feel like it's more thriller-y, whereas this is more mystery. A large proportion of this book, literally like a big part of this book is Urquiporo interviewing every person on the train. Like just to sink them down and interviewing them. And you get clues from each of those interviews. And I just think it's so clever that it keeps your attention, even though literally most of this book is Urquiporo sitting in a room interviewing people. Like again, the ending of this, the reveal is just like incredible. Act of the Christie's mind. Her mind. Honestly, her mind. I think the clues are all laid out for you. It's a very fair play murder mystery.  Like nothing is spared. Nothing is really like, should be a surprise, although it is. You're literally given every piece of information you need to figure this out. And yet no one ever does. It's so good. I absolutely love it. And I hope, I feel like I've read the two Urquiporo Christie's I'm going to love most already, but I'm hoping they'll be more along the way because I'm scared I peaked. But I'm going to be rereading this soon when I get up to this point in the series again. I'll be rereading this, which I'm very, very excited to do. Now we have got my top two, which I want to tell you guys are interchangeable. Like on any given day, there's no real difference. These are both kind of number one. I have separated them. But on any given day, this could switch. Like it literally ignore the ranking here. They are both my number one. Number two is The Thursday Murder Club by Richard Osmond. My favorite book of last year. We're following these four elderly friends in this like retirement village and a murder happens on their doorstep. It's called The Thursday Murder Club because they have a murder club every Thursday where they go and try and solve cold cases together. It's kind of just their fun thing that they do together.  Now this has been very hit or miss for some people.  But it's very, it's almost a cozy mystery almost. Like it's got that fun comedic element to it. I think the way it's written is really, really good. It's kind of like more matter of fact, but it's like, oh, it's so well written. It's really, really well written, especially Richard Osmond, if you don't know, is like a TV personality here in the UK. And the fact that this is like what he's writing is absolutely amazing to me. Like it's so popular and like usually when celebrities write books, I'm like, not interested. But with this, it's so good. The characters in this are amazing.  The mysteries in this is so good. Again, a very fair play murder mystery where like you can figure everything out. You're given the clues. I want to be given the clues. I feel like something that happens often in thrillers where there's a murder is it's a complete curveball. Uh-uh, I want to be given the information. That's what I want. And that's what I feel like a given in this. It's funny. It's clever. It's entertaining. Oh, I just absolutely love this book. And then the number one on this list was another book that was really influential in getting me into reading murder mysteries. And that is The Guest List by Lucy Foley. I told you we'd be seeing Book of the Month and Lucy Foley again. I told you. I informed you. I told you. This was such an influential book for me. I picked it up. I remember I went to a signing with Lucy Foley and I went to check that this is signed. But obviously this isn't the one I got signed. The one I got signed is up here. And let me tell you, I got told the plot for the Paris apartment then when it came out. And I've been waiting ever since. I've been waiting ever since. And now we've got it. I honestly can't wait. But anyway, let's talk about The Guest List. So again, it's close like a isolated mystery. We are stuck on this remote island for this wedding. You find out at the beginning of the book, someone's dead. But for the vast majority of this book, you don't know who it is. And we're jumping back and forward in time to the day before the wedding and then to the wedding night, trying to figure out what's happening. Lucy Foley does something very well. It's like archetypal characters. We've got the bride, the plus one, the best man, the wedding planner, the bride's maid. And we flip between all their perspectives, which I'm really hoping that the Paris apartment will do as well. I'm pretty sure it will. And you just learn so much from all the perspectives. But like, you've got to read between the lines and what everyone is saying because people will be like, they'll be telling you stuff that you need to know straight to your face. But you won't necessarily realize how it all links up. Oh, it's so good. Wait a minute. I'm figuring this out. I'm like Scooby and Shaggy, I'm solving a mystery. The tension in this, the, oh, I need to reread this too. I'm hoping I may reread it soon. The tension in this and the reveals. The reveals are top tier. Some of the reveals in this are like gag worthy, gag nation. We are gag nation on this. So we all gag nation on this. Honestly, shocking. And again, this was like a big thing. I read it not too long after the murder on the run express. A big thing in getting me into murder mysteries and making them my favorite genre. So I feel like I owe a lot to this. But obviously this was my favorite book of last year. These are pretty neck and neck. They're pretty equal in the standing. So there we have it everyone. That is my top 10 murder mysteries. Let me know which of these you have read. What your favorites of these are. Let me know if you didn't like any of them. Let me know which of these you want to read the most. I would honestly recommend all of these so, so, so much. I think they are all brilliant. And I'm wondering if any of the books, I don't know if any of them are murder mysteries. I think a lot of them are mysteries. But I'm wondering if maybe in the future, because I'll probably redo this in like two years or something maybe. I'm wondering if in the future, any of those Book of the Month books are going to be on this list. Because listen, I feel it in my bones. This month's section was so great. So again, make sure you check out Book of the Month down below. And if you've gotten to the end of this video, what should we comment? Comment a knife emoji down below. Thank you so much for watching, guys. I hope you enjoyed this video. And I'll see you very soon in another one. Bye!


Okay so let us continue with our discussion of studying the behaviour of a function in a neighbourhood of the critical point. So we have discussed in the previous lectures you know the how to look at the logarithm the various branches of the logarithm and various branches of the power function okay and for a fractional power alright. So what we are going to do now you start with start with an analytic function f of z having a critical point a critical point at E z equal to z0 so that is f dash of z0 is 0 the critical points are the points where the derivative vanish assume of course we assume that f is non-constant on the domain on which it is defined which includes the points z0 and this will tell you that f dash is also non-constant and well on the other hand f dash is an analytic function also and z0 is a 0 of that and you know the zeros of a non-constant analytic function are isolated. So I can write that so there exists an epsilon greater than 0 or a rho greater than 0 such that well f dash of z is not 0 if mod z minus z0 is epsilon equal to rho okay. So z0 is a 0 it is isolated it is an isolated 0 of the derivative so there is a deleted neighbourhood and of course I should not include the value z0 so I should put this greater than 0. So for every point in this deleted disk close disk centre at z0 including the boundary except for the centre at every other point the derivative does not vanish okay. So any other 0 f dash will lie outside this disk apart from z0 okay z0 is only 0 there in this closed disk and that is just due to the fact that the zeros of a non-constant analytic function are isolated right. Now suppose the order of the 0 of z0 of f dash is m minus 1 and of course I am assuming m is greater than 1 right suppose this the order of 0 is m minus 1 the reason for m minus 1 I could have taken it as m but the reason I am taking it as m minus 1 is because you will see that it helps in our argument otherwise throughout my argument I will have to keep saying m plus 1 which is not very interesting. So well so suppose the order of the 0 of f dash is m minus 1 at z0 okay we say that z0 is a critical point of f of order m minus 1 okay. So the order of the critical point is the order of the 0 of the derivative okay and of course the critical value is the value of the function at that point the critical value is f of z0 this is the critical value corresponding to z0 alright and now the reason why I took m minus 1 is because we can consider the function f of z minus f of z0 okay consider the function f of z minus f of z0 of course this is also an analytic function because it is the analytic function f minus a constant alright. So this is also an analytic function but the point about this function is that it has a 0 at z0 okay and all its derivatives the first m minus 1 derivatives at z0 will also vanish because the derivatives of these are the same as the derivatives of f okay because this differs from f only by a constant.  Now this is an argument that we have seen many times so what we are saying is that f takes the value see f takes the value z0 f of z0 at z0 with multiplicity m okay that is the same as saying that fz minus fz0 takes the value 0 at z0 with multiplicity m okay and you know we have already seen this argument that in a sufficiently small neighbourhoods surrounding z0 the multiplicity is constant okay consider delta to be the minimum over mod zeta minus z0 equal to rho of mod of f of z minus f of z0 look at the minimum value of this mod okay and so you are on the boundary of that circle centered at z0 radius rho alright I want to say that f the modulus of fz minus fz0 is always positive on the boundary circle okay and if not I want to modify it so that becomes true perhaps it is already true. Let me do the following thing let me assume let me also assume that I not only assume that f dash is not 0 on this but I also assume that I will also assume that mod of f of z minus f of z0 that is also non-zero okay because that is also non-zero on this deleted neighbourhood okay the reason I can do that is because z0 is 0 of the analytic function f of z minus f of z0 and that 0 is also isolated because f of z minus f of z0 is a non-constant analytic function because f of z is also a non-constant analytic function you take a non-constant analytic function and add a constant to it the resulting analytic function is also not constant so let me write that down assume probably it is not necessary probably it will follow but anyway let me assume it for safety assume that f of z is not equal to f of z0 in 0 less than mod z minus z0 less than rho okay if which means that you know I have already chosen rows is that the derivative does not vanish but maybe if that row does not work I can make the rows smaller I can choose a smaller rows so that this is true and that will happen because z0 is an isolated 0 of f of z minus f of z0 because f of z minus f of z0 is a non-constant analytic function okay. So that means that on the boundary also this is not 0 I mean this is not equal to this which means that the modulus of the difference is non-zero okay so it means that this delta is positive so this will tell you that delta is positive delta is a positive number and then you know once I have this I can define for w with mod w minus w0 less than delta that is the reason I need the delta we have been through this argument many times but anyway define it to be the number of times f takes the value w okay. If you put w equal to w0 which is f of z0 I have not mentioned what w0 is probably I should do that now the critical value is f of z0 equal to w0 I should tell you what w0 is alright. So when I put w equal to w0 I will get n of w0 okay and but then I can change w inside this alright because in this disk this is not going to vanish okay and except yes f of z minus w is certainly going to be greater than or equal to delta alright on the boundary on this boundary so this integral is well defined alright. So the moral story is that this number which we have seen many times before is going to give you the number of times f of z takes the value w okay and we have seen before that n of w is analytic in w and this integer value hence constant okay this is an argument we have seen many times before so n of w is equal to n of w0 and n of w0 is number of times f of z takes the value w0 which is f of z0 and that is m so this is equal to m. So what this tells you is it tells you the following thing for every w with mod w minus w0 less than delta there are n points in mod z minus z0 strictly less than rho at which f takes the value w okay. So let me write that down that is for every w with mod w minus w0 strictly less than delta there exists n, n there exists m points counted with multiplicity counted with multiplicity means some points may be repeated okay, z1 of w etc, zm of w in mod z minus z0 strictly less than rho such that f takes at each of these points f takes the value w okay this is what n of w equal to m means right. So the diagram is something like this okay so the diagram is something like this so here is my complex plane this is the w plane I mean this is the z plane the source plane and then there is the target plane which is in the complex plane and this is the w plane and I have this well I have this disc here centered at z0 radius rho and I have this function f of z w equal to f z which is a mapping and it maps into and I am looking at the at a disc centered at w0 with radius delta okay this is my mapping and what I am saying is that if you give me a point w here if you give me a point w here then you know you get all you get m of these points here okay and they may be there may be repetitions it may it means that all the m may not be distinct point it some point may be repeated with multiplicities okay they could all be one point with m times multiplicity okay which is what happens if w equal to w0 if w equal to w0 then the only z for which f of z is w0 is z0 okay but it is not to be thought of as one point it has to be thought of as m point m points because it is multiplicity m the 0 of f of z minus f of z0 at z equal to z0 has multiplicity m so you should think of z0 as being repeated m times even though it is the same point so when I draw these points that many of them could have be one and the same but these are the zi of w okay. So you know if you look at it it looks a little like it looks a little like the implicit function theorem see I what I am saying is if you take the equation w equal to f of z okay and you take this critical point z0 then I am getting a disc such that if I try to solve for z from w equal to f of z okay then I am getting z equal to m values z1 of w etc zm of w and these are solutions of w equal to f of z namely if I plug them in f of z I get w okay so I am solving for z from w equal to f of z okay so what it tells you is that you are getting m solutions you are getting m solutions for the equation w equal to f of z in the neighbourhood of a critical point z0 that is what it says okay and you know the way you should think of these so you know there are these so we have these functions z1 etc up to zm there are these functions and the way you should think of them as well you should think of them as you know inverses of f okay. So I am putting a I am putting a I am not writing f inverse the convention is to write f inverse only when it is said theoretically an inverse at least in which case f has to be injective alright but here f is certainly not injective okay because derivative has vanished at a point you cannot expect it to be injective at all and therefore I should not write f inverse so I am putting f and putting the inverse in a bracket to tell you that you know this inverse function okay the inverse function has m solutions z1 through zm and these zis are functions of w okay okay and they all solve the equation w equal to f of right now we need we just want to understand what these functions zis what these functions are are they continuous or the analytic etc etc okay and we also want to know how this mapping looks like how can you draw this mapping or how you can visualise this mapping okay. So what we are going to do is we are going to we are going to break this down by using what we have something that we have already seen namely we are going to use the fractional power function, branches of fractional power function which have which come from branches of the logarithm there are m branches for the 1 by mth power of a variable okay and they come from the logarithm by choosing the various branches of the logarithm we will use that to break this map down okay and see how it looks how this map looks and the point is that up to conform conformal equivalence that means up to a holomorphic isomorphism this map really looks like z going to z power m this map from here to here looks like z going to z power m that is the whole point alright that is the reason why we studied the mapping z going to z power m and its branches in the previous lecture right. So let me explain that so what we are going to do is we are going to do the following thing so you know so let us study in a neighbourhood of in this neighbourhood of z0 what is happening see we have we have z0 as 0 of order m of f of z-f of z0 I mean which is f of z-w0 so what we are going to get is f of z-f of z0 is z-z0 to the power of m times some g of z okay where well g of z0 is not 0 and g is analytic in mod z-z0 less than rho okay this is very simple this is just by the Taylor expansion of f okay you have this follows from the Taylor expansion of f about what is the Taylor expansion the Taylor expansion is you know it is f of z is f of z0 plus z-z0 into f dash of z0 plus z-z0 the whole square by factorial 2 f double dash of z0 and so on you go on up to z-z0 to the power of m-1 m-1 by factorial m-1. f to the m-1 derivative z0 and then I will get z-z0 to the power of m times something I will call that something as g of z this is what I will get alright and well the you know from the power m onwards whatever is there I am taking z-z0 power m outside the bracket outside the brackets as common and whatever is inside is anyway a power series it is an analytic function so I call that as gz okay this is the Taylor expansion and mind you all these terms will vanish because z0 is a critical point of order m-1 okay so that means f dash has 0 of order m-1 at z0 so you know f dash f double dash they all will vanish alright so the so f of z-f of z0 will become z-z0 power m into g of z that is exactly what I have written here so this is this is simply from Taylor expansion alright now and mind you g of z0 is not going to be 0 alright that is because f mth derivative of f with respect to z0 is norm 0 right because it is a the order of 0 of f dash at z0 is only m-1 it is not m alright so well now what we are going to do is we are going to do the following thing choose probably it already is true but maybe what I will do is I will choose row smaller if you want to make sure that g does not vanish in mod z-z0 less than row okay choose row smaller if needed to make to ensure g of z is not equal to 0 in mod z-z0 strictly less than row okay I want g not equal to 0 reason why I want g not equal to 0 is because you know I want to write g I want to write branches of g of z to the power of 1 by m I want to do that alright and you will see why I want to do that alright that is why I want g not equal to 0 alright and I am saying g is not equal to 0 it is already true in fact I do not have to choose row smaller alright that is because I have already assumed that f of z is not equal to f of z0 in this okay so this this sentence is unnecessary anyway let it be there so well now so we have m branches of g of z to the 1 by m in mod z-z0 strictly less than row okay. So you see this is again I proved a lemma the previous lecture saying that you know if you have a function which is non vanishing on a simply connected domain then you can find branch of the analytic branch of the logarithm of that function okay and what is what is g of what is g of z to the 1 by m see g of z to the 1 by m is e to the 1 by m log g z this is what is by definition so you know and you know if I have an analytic branch of log g z then e to the 1 by m log g z will give me an analytic branch of g z to the 1 by m alright. So well of course there are there are going to be m branches but nevertheless let me choose one branch so let me write that down we may take so if you we may take integral from z0 to z of d log g of z well this is supposed to give me log g z-log g z0 and if I want log g z I have to add log g z0 log g z0 to this as an analytic branch of log g z in mod z-z0 strictly less than row. So this is the this is an analytic branch of the log okay and of course where when I do this integral from z0 to z I can choose any path okay the integral is independent of the path this is what we saw last time. So this is a this is a branch of logs analytic branch of log g log g z and once you have this branch I can write a branch of g z to the 1 by m as e to the 1 by m into this branch okay. So that is how I get an analytic branch of g z to the 1 by m alright. Now put h of z to be well you know look at look at this expression z-z0 power m alright now this g z can be written as g z to the 1 by m whole power m and I can take m and m common okay and so I can write I can take the mth root and write that as h of z. So I am writing h of z is equal to z-z0 into g z to the 1 by m where g z to 1 by m is analytic branch of the logarithm as I it is an analytic branch as I have defined it here okay. You put h of z is this this is already this g z to the 1 by m is an analytic function and z-z0 is analytic function those is so that this is therefore an analytic function in the disk and the point about this function is that if you calculate it is derivative at z0 it will not vanish okay. So then h is analytic in mod z-z0 is less than rho and h dash of h dash of z0 is not equal to 0 because you know if I differentiate this I will differentiate it using the product rule alright and when I differentiate that is I keep z-z0 constant and differentiate that g z to the 1 by m and then if I substitute z0 it is going to vanish because there is z-z0 outside and for the other I have to add it to the this term kept constant and the derivative of this which is 1 okay and if I calculate it I will get g of z0 to the 1 by m. h dash of z0 is actually it is actually equal to g of z0 to the power of 1 by m this is what it is and mind you g of z0 is not 0 g doesn't vanish where have I written that g of g doesn't vanish at z0 g doesn't vanish at z0 so g g of z0 is a non-zero complex number so it has a logarithm and this is one of and one of the logarithms is one of the logarithmic values will be picked by this branch that I have chosen okay and that is the value here okay and therefore h dash is a non-zero number alright. So what this will now tell you it will tell you the following it will tell you that you know if you it will tell you that h dash is going to be 1 to 1 I mean the mapping h is going to be 1 to 1 in a smaller neighbourhood of z0 probably in this neighbourhood itself by the statement h dash of z is at let us differentiate that it is d by dz of z-z0 g of z to the 1 by m and this is going to be I keep z-z0 constant and then differentiate this I am going to get 1 by mg of z to the power of 1 by m-1 into g dash of z plus I keep gz to the 1 by m constant and I differentiate this and I just have to say that this does not vanish probably does not but maybe let me do the following thing if necessary let me shrink let me make rho smaller alright so that I ensure that h dash does not vanish because after all h is an analytic function therefore its derivative h dash is also an analytic function and so it is a continuous function if a continuous function does not vanish at a point then there is a disc surrounding the point where it does not vanish just by continuity okay. So you know well if needed make rho smaller to ensure h dash of z is not equal to 0 mod z-z0 strictly less than rho okay and well and you know and in fact you can even ensure that but probably I do not even need this what I really want is I want a neighbourhood where h is 1 to 1 okay so that I can invert h alright or to even have h1 to 1 okay this I can do right and I can do this because of the that is because of the inverse function theorem I am using the inverse function theorem here see I h dash is non-zero at a point so it is non-zero in a neighbourhood and then inverse function theorem says that wherever the derivative is non-zero you can invert in a smaller neighbourhood okay. So you make rho smaller if you want and h will become a 1 to 1 analytic function which you know is an isomorphism on to its image because that is what the inverse function theorem says a 1 to 1 analytic function is a isomorphism on to its image the inverse function is also h inverse will also be analytic alright. So now what is the advantage of this the advantage of this is that this diagram can now be split up okay how can it be split up can be split up like this can be split up in the following way then we can understand the nature of these functions z1 of w through zm of w alright. So what we do is we do the following thing we have this so I have this disk here in the z plane so this is disk centred at z0 radius rho and well what do I do I apply h so I put I put a new variable I call zeta is equal to h of z alright I apply h and when I apply h what I am going to get is well mind you h is 1 to 1 okay h is a 1 to 1 function h is analytic alright and I have chosen rho small enough so that h is 1 to 1 okay and you analytic function is conformal okay it is a it maps an analytic function with nonzero derivative okay is a conformal map alright in fact I have assumed that of course h dash is also non vanishing there alright and in fact that will follow if h is 1 to 1 alright and therefore the mapping h is conformal which means that it will preserve angles between curves alright. So what I will get is I will get a if I take the image of this disk I will get something like a disk alright a slight distortion if you want but then essentially it is going to look probably like this I am just drawing it like this but then the point is that z0 is going to go to 0 okay z0 is going to go to 0 because h of z0 is 0 because h of z is z-z0 gz to the 1 by m so if I put z equal to z0 h will go to 0 so it will map the point z0 to the origin alright and well I can choose you know a small enough neighbourhood here of the origin alright so this is how h is going to map and then you know now what I am going to do is that I am going to put another mapping here this is eta equal to zeta to the m okay when I do this the combined map will be z going to h of z to the power of m and h of z to the power of m is z-z0 power m gz which is f of z-f of z0 see I am trying to come to f okay so if I you know how this mapping behaves okay so you know this is what we saw last time well you have this disk and you know if I draw it for m equal to 3 you know how it is going to look like if I start with ray like this alright then the inverse image is going to be 3 of these things I am going to get 3 separated by angles of 2 pi by 3 okay and for any general m you are going to get the inverse image of our ray like this is going to be m rays alright separated by angles of 2 pi by m alright and you know that each sector here of sectorial angle 2 pi by m is mapped by zeta going to zeta power m onto the whole disk alright so this is what we have seen and you know well and we know that there are branches of this here alright we know that there are branches what are the branches the branches are zeta to the 1 by m these are the branches okay and you know what those branches are we have written down those branches zeta to the 1 by m branches are there are n branches and the branches are given like this they are given as e to the 1 by m log zeta this is the first branch zeta to the 1 by m second branch is e to the 1 by m log zeta plus 2 pi i by m this is the second branch and it goes on like this until the mth branch which is e to the 1 by m log zeta plus 2 into m minus 1 pi i by m okay these are the branches of there are m branches okay in this case m equal to 3 you will have 3 branches alright for example if m equal to 3 and this is the point 1 under the 3 branches you will get the cube roots of unity alright in general you will get the mth roots of unity if this point was the point on the real axis with coordinate 1 okay so these are the branches we have seen this okay and all these branches live they are all you know analytic on the slit plane when namely on the slit disk you have to cut out this portion of the negative real axis along with the origin then you know that these all become analytic functions and where they will all live together as a single analytic function it will be above in the in the in the Raymond surface for zeta to the power 1 by m which is the m sheet at covering over this over the punctured disk alright so we have seen this last time and then now what you do is you now you know already when you go from here to here to here you already got f of z minus f of z0 so to get f of z f of z you have to add f of z0 so what you do is you take this mapping this which sends eta to eta plus f of z0 this is just translation if you get this then finally end up with this original diagram you will get this disk well you will get an image the image will be something that contains this disk alright will be will be bigger but this point will now be w0 okay. So this is the whole map w equal to f of z pictured in a neighbourhood of small disk surrounding z0 and in its image containing a small disk surrounding w0 this is how the picture looks like alright and what one needs to do is to look at the solutions alright the solutions in the in this direction okay there are m solutions for every w here there are m points which I call z1w z2w etc zmw okay these are m solutions and I want to tell you that these solutions are also it is obvious that these solutions will be you know analytic on a slit disk okay and you can write down now what these z1w through zmwr alright in fact they are going to be so let me write them down maybe I have space here to do that so you know so this is what I should get if I take the first one I will get this then if I take the second one I will get h inverse e to the 1 by m log w-w0 plus well 2 pi i by m which you can take out and write as e to the 2 pi i by m times this and so on I will get and I will get zm of w to be h inverse e to the 1 by m log w-w0 plus 2 into m-1 pi i by m so these are the functions okay in this case for example you can take this to be principal branch if you want you can take it to be the principal branch so this log I you can take it to be the principal branch then you will get the other branches but the truth is that you need not take it to be the principal branch you can take it to be any one branch then you will get the remaining branches you can of course if you want take the principal branch there is no problem okay and so you see you get these functions the point with these functions is that these zi are analytic on you see they will be analytic on they will not be analytic on the whole slit on the whole disk but I will have to slit out the if you take the principal branch of the logarithm you will have to slit out the portion of the negative real axis from I mean the portion I mean you have to slit out this piece okay you have to cut this out alright and so I will write it on I will write this as mod w-w0 equal less than delta minus this is the line segment from w0-delta to w0 I throw out w0 and this line segment okay on this is a slit disk the slit disk okay these functions are all analytic on the slit disk and what happens is that they are all functional inverses for this so you know see you have this u f is of course not 1 to 1 it is it is m to 1 it is m to 1 alright so there are these functional inverses so that is the reason I am putting f to the minus 1 inside a bracket because it is not 1 there are so many of them and these are the zj's j equal to 1 to m they are the functional inverses okay because this followed by this will give you identity if you plug in zj equal to if you plug in for zj and here I should have written w equal to fz not fz0 should have been w equal to fz see if you plug in for zj in f okay f of zj will give you back w I mean it will give you back correct it will give you back w okay all these zj's are functions of w mind you because this is the w plane this is the complex plane this is the w plane these functions are all functions of w so you take this zj of w you plug it in f of z you will get back w so this followed by this is an identity map that means you have you have m inverses you have m solutions for this equation w equal to f of z in a neighbourhood of the critical point that is what is happening okay and the mapping looks the mapping can be described like this okay so in fact you know if I if so you know if I take the inverse image if I take the inverse image of the disc here this is after all translation again I will get this disc centred at the origin okay and then if I you know when I go like this I am taking the fractional mth power okay so you know it is going to look like this and then from here to here when I go by if I go like this it is h inverse and h inverse is conformal therefore you know this will this will result in something like this okay you will get something like this this is for m equal to 3 so more generally if you write m if you take any m you will get m you know you will get m curves centred at z0 and going out radially okay and that is how the mapping looks like okay so you know if you forget this essentially this is just a distortion of this map this map is just a distortion of this map and what is this map this is z going to z power m it is just a distortion of the power map so what you are saying is if a function f you look at a function the behaviour of the mapping at neighbourhood of a critical point of order m-1 you know up to a conformal twist okay it will look like z going to z power m that is what we have proved okay and therefore it will have m branches the inverse function will have m branches and these are the branches alright and where can you make sense of them all as a single analytic function what you have to do is that you will have to put a Riemann's surface over this which is an m sheet at cover on that all these will become analytic okay and they will become a single valued function so the moral of the story is that you get a single valued inverse for a function even at a critical point but the inverse leaves on a m sheet at covering that is the point okay that is how it looks right so I will stop here.