Hi guys, it's me from the future. I just wanted to pop in before this vlog really starts and just say I love you guys um in this vlog I'm really anxious so I am like cut out a clip where I get kind of like emotional and talking about like all the Families that are affected and I can't even imagine those who are like losing their jobs who have like families to support and stuff But I don't want this to be like a super depressing negative vlogs I know a lot of you guys are watching vlogs to kind of escape so I decided to keep that clip out of it But just know like I love you guys I am so sorry just to everyone like this is just such a horrible horrible Time and it's really hard, but I think The best thing that we have is community and I think that it's been cool to kind of see like weird bonding things out of this I don't know. I'm trying to look on the bright side I think it's also a good time to like if you have a new project You want to start from home and do all these things like do it, you know I started my YouTube channel and I didn't really really have to leave my house for a long time just things like that I'm gonna be posting as much as possible To kind of just I don't know be a distraction when I even started my YouTube channel That's because I watched YouTube videos growing up to be a distraction not during a time like this, but yeah I don't know. I just love you guys and Life in the world is really overwhelming right now and everything is confusing and like obviously most of us I haven't left my house in like four or five days like And we probably will be here for a long time But I'm just really grateful for like my friends and my family and things and times like this and for our community I think now this is why I want to even come in here Like if you have the means to help someone help someone I'm trying to look into different ways that I can help people out or like trying to come with different ideas and stuff Like outside of like the close people in my like in my life But I think as a community we should band together and help each other in any sort of way That doesn't need to be financially it can be if you can't do financially like if someone's small business just needs promotion or things like that I don't know if get creative if you guys have any ways that we can help each other or help our world Obviously social distance we've all besides that I feel like most of us are doing that at this point And if you're not you need to be social distancing But yeah, that's what we that is what this community is for It's for supporting each other and helping each other So I'm gonna have a Facebook group which we guys have always been in I know a lot of you guys are gonna want to just like comment in like a YouTube comment The Facebook group is way more of a community and we're just like a lot tighter there So yeah, that's gonna be open all you to do is join. I accept very very quickly I accept pretty much everyone. This is just scary right now guys I'm really grateful and I just keep thinking about like how many people I there's this organization baby to baby in LA that I love and if you guys can help them out if you're in LA like 100% whenever this all kind of calms down definitely getting in contact with them, but They really open my eyes to how easy it is to like go into poverty and lose your house and How people like you don't really ever imagine yourself as homeless but like that is the reality for so many people is that like they're two paychecks away from being homeless and Like I said, I was gonna cut the clip out and I'm just saying it all in here But it's a little bit let more light hearted than what I was talking the other day So yeah, my heart like it's just heavy for everyone and it just sucks And I want everyone to stay home be healthy But yeah, if you guys have any if you guys just want like not even help just a community people to talk to Whatever Facebook group is where it's at podcasting your normal YouTube videos are gonna be normal all about stuff So you guys can come to our channel to our channel. It's like all of ours And yeah, but I love you guys and let's get into the vlog right. Hi guys. Um, I just landed I ended my national vlog two seconds ago We were at dfw terminal d the best terminal um, anyways, I We ended our trip early and we're back I actually just said bye to Keaton because she flew shoulda connecting flights We're on the same flight and she is off to the portland And now I am home So basically we're gonna go grocery shopping get all the things that we have to get And then not leave the house for who knows how long so we need food. I already have so much toilet paper Thank god for Costco when I moved like I got so much stuff. Yeah, so we're fine on that stuff But yeah, that is what we're doing today. We're staying safe social distancing But we do need to get groceries and stuff before we do so um, also I'm gonna make a bucket list of things I want to do Yeah, I just have some fun ideas to keep myself occupied. Keep in mind This is not much different than my normal schedule programming besides the fact that I can't go work out elsewhere Because like I don't really leave my house anyways But now that it's like not my option to not leave my house I feel like I'm gonna go stir crazy, you know, but anyways, this is what needs to be done So That is what we're doing. If you guys want to hear subscribe I'm gonna be posting a lot probably this week because I have nothing to do other than film edit and upload videos So Hope you guys uh like them anyways my outfit for today. This coat is misguided. It is so freaking cute I wore it in Nashville and for the 30 minutes that we left our airbnb. I got so many compliments on it Um, and then my gray american eagle crew duck guys this thing I think it's like pretty often on sale for like 25 or 30 dollars I did a deal with them a few months ago Maybe like last month actually get into the same deal and we picked out the same crew neck and we were talking about How soft they are in Nashville. It is the softest crew neck I've ever worn in my entire life And it's from american eagle. I will link it below. Anyways, we're going to Trader Joe's and then we might go to other grocery stores if we need to I don't know. We're stocking up All right guys, so we're at Trader Joe's He was getting the necessities which is wine. Um the rest of it We are only getting a lot of what they have a lot in stock Like I'm not going to be the person that like if you see a lot of food It's because there was a lot here. This is really stocked up. I'm gonna get some chicken and salmon This is everything we have. It's so crazy All right guys grand total for the Trader Joe's trip 205 But we got a lot of food that like isn't gonna go bad that I just normally don't keep in my house because it's not the healthiest Like I have mac and cheese five boxes of it for the first time ever because they have like 300 So I was like why not? Um, but we made sure to leave things Which I think is important if there's not that much of it Don't take that much because other people need to get it So I may couldn't just put back some things so the other people could take it No, you're being so ridiculous. Okay, so we gotta go to the car I don't know if there's actually even an elevator. We're at the Knox Henderson Trader Joe's I don't know why we keep going here. It's not even my closest Trader Joe's, but Anyways, we can go there too. Yeah, we might just Trader Joe's pop just getting we're going home and like not leaving So that's why All this is happening. You I think that you have to just spin the car around. Okay No, we're gonna.  I don't have weights And I literally almost bought the ball bangles and I really want them Now I'm kicking myself and I'm buying them But I only have resistance bands in the mat But I think I can do I mean I can do a lot of like Pilates stuff Which is not going to be as good of a workout But it's better than nothing, you know Really hard at work here engaging the core engaging the core. Yeah, so we'll be working out every day Or else I will for sure go crazy. See everyone is laughing at me because I want to make my gym Um, like my garage part gym and everyone's like you don't need that you I need a gym for times like this. Okay, so just saying Everyone is making fun of me and now Everyone will be hitting me up even though they can't come over because no one is coming or leaving that is the plan so, um I don't know what's happening Anyways, I can't wait to go home and shower Okay guys, we're back. We have a ton of groceries I don't even want to do a grocery haul because he already started unpacking He was staying here this weekend with cocoa Um, and just watching the house and cocoa. Uh, we have so much like random politics That's all it was available like this one had never had this kind But there was so much of this so I took this instead of like taking, you know I'm saying if they have none of this whatever So we got three because there's a lot of them. I'm gonna have this right now. I'm really hungry But yeah, we had just this is what I'm currently using as my pantry, which I'm gonna I don't even really care For the next however long I will organize this and keep it organized But my pantry is gonna get built up. Hopefully sooner rather than later. So That's why all this is a mess. He just came with the last grocery bag He has stuff for mac and cheese. Also, they had a lot of this pasta We're basically gonna live off fosta. That's what he thinks. It's the quarantine 15 Which is like not funny, but like, you know making the best of it So I'm gonna have one of these I found out about these from brooks vlogs and they're so freaking good all right, so Quantic is trying to keep boxes out of things on the counter Oh my gosh, I had the worst hiccups on the counter like that's just not happening. So that stuff's all up there Um, we have this stock. I cannot wait for a real pan. Oh my gosh I keep picking me and counting for a real pantry. Sorry guys frozen food this is the My hiccups. This is the first time I've even had frozen food at all and then um My pantry or my fridge is very stocked. It's not one of those like Pinterest organized fridges I would love to do that, but I need to go to the container store. So anyways This is what we have ship, which is an app that Allows you if you guys listen to me and she works with them I would love to deal with some of my podcasts, but basically it's an app where You your friends can like control your dating profile a little bit It's really actually cool, but they sent me a big box of goodies all these like liquid All these like little liquid coolers, which are cool Some candy some of these recessed drinks with cbd in them. Okay. I've wanted to try them. I think they're cbd Right. Yeah, exactly. Yes. I was right. Um, also again. Lauren loves these I've wanted to try them for forever and now is legit the perfect time to have these So there's a bunch of those. There is some wine So literally this is the first time I probably bought wine for my house and Actually ever um, and now I have three bottles of wine for a bunch of candy some bags Oh my gosh, that's so cute. Oh my gosh, and there's shirts. Yeah, it's a ship ship. Oh my gosh, so cute candy Recess I'm really sad. I'll let you guys know how I like these Guys, I'm back. It gets better. So they gave me a set of cards that are like so cute Oh my gosh, I don't even have a set of cards. So this will actually be fun. We will be playing. I know Quentin's reading it. I already read it super cute But I'm trying to make sure we go all the candy out I have all these bags, which I'm gonna use for groceries So I'm gonna use in my car when I get my car back. I know these are my favorite going out Really another box to open. Oh class pass also sent me something while I was gone. Cute little class pass Um the bag a class pass hat and then a water bottle, which is always so convenient I love I also got a package from charlie tiller. Oh, there's even more stuff than I thought So I just got added to their mailing list.  I'm really excited to try out Oh my gosh, as well as a perfume, which I love perfumes I actually don't think I've even ever been since a perfume besides through simpard. So I'm gonna try this This is what the perfume looks like okay It smells really good. It's like very fresh and kind of like I know that classic doesn't have a smell a scent, but you know what I'm saying So here are the products this moisturizer. I've heard amazing things about they also gave me the sheet mask That apparently it's an instant magic facial dry sheet mask Um you just do it for 15 minutes. So I'm gonna do this with our shower Or maybe when I have like an actual skincare spa time They gave me I'm gonna try this out. I've been you guys know I've been like really into new products lately, especially because I feel like I've been using the same stuff for forever This is a really pretty lip liner. That's like the color I go for typically With lip products. There's this little sculpt in highlight thing, which I don't even have another I have one highlighter Um, so that's really really pretty just an easy nude lipstick and then this also is a blush Really trying to open this for you. Okay. Here we go That's a really pretty color. So I will not really probably be wearing Make I don't know maybe I'll be trying new makeup looks and stuff who knows So I'm gonna bring this up and see I also want to try this when I actually go back to it So it's the wonder glow instant soft focus beauty lash Flash not lash. What am I trying to say? Um, so I really am excited to try this because it's like kind of advertised for dull skin So I need this because when I get pale, I feel like my skin is so dull So that is like basically my one skin concern that I have that like my major skin concern You know, so that is what I'm really excited for also this stuff is just so cute I just did a quick deep clean of my house. I cleaned all the floors the coffee table Quentin let roll the couch. I cleaned all over there. Are you wearing my slippers? Yeah, I put them on. Um Yeah, so I still have to put all this stuff away And then also organize or unpack my luggage and then I'm gonna shower And then another dishwasher load it and then do whatever for today This Candle is like I only have two candles left right now. So that's really tragic truly But I built my front another the house is clean. Don't you? Feels so good, right cocoa Cocoa so excited to just chill and have so much attention for the next few however long I just cleaned my bathroom pretty quickly. Now it's time to unpack once I unpack. I will shower and then Do nothing today. Actually my now my spritz. I'm gonna mess up there, but um Anyways, and also here's our Dallas people. I got the best airbrush hand in my life at the tan bar Um, I went just because I found them online and my cousin was like, they're the best you need to go She wasn't kidding. Literally the best thing ever Um, so I'm just gonna unpack really quickly because I'm one of those maniacs that unpack as soon as they get home and then um probably cute Hopefully Quentin won't make lunch, but um, we'll see if that happens and then I need to edit the natural vlog to have it up tomorrow And that's pretty much all I have for sure on my to-do list today Like it's crazy because so many things have like actually just like legitimately been canceled dropped whatever So, um, I normally have like 15 things I could be working on which I can But a lot of those things have now been canceled, which is fine Um, but it's just like a lot less workload slash I have a lot less motivation So I want to be filming and posting a lot because I know a lot of you guys like are bored at home and want to watch videos and stuff Um, but I also am lacking motivation just kind of want to lay around like everyone else But we'll be vlogging because it also kind of keeps me on a routine and stuff I'm also going to do like a morning routine when I work from home because a lot of you guys have been asking for tips since I already do it Um, so I am going to be doing that and yeah, I'm going to go ahead and Deal with this suitcase situation Okay, this light bulb is not it I had to put the cover up yet because I don't have a toolkit but I have now unpacked and then put everything in my hamper Put up my coats all of that stuff. I'm sure I'll do More organizing in my closet. Honestly, it doesn't really it can't really get more organized as of right now But I'm sure at some point like I'm sure by the end of this my house will be unbelievably organized, but Um, yeah, I got out of the shower. I put on the moisturizer some Leaving conditioner and a different lip balm on but I'm just in cozy clothes because I need to go edit and make some lunch So I'm gonna make um, I think probably salmon surprise surprise guys. Do I literally I don't need anything besides salmon Um, but I'm gonna go downstairs Oh shoot. What the heck? Okay, so making some sweet potatoes and then I'll make the broccoli and um salmon but I moved all these into here. This is like kind of great for like going to go stuff as well But I'm gonna make some coffee. I had a coffee this morning at like 5 a.m Um, when I was in national wealthy airport but now Also guys, okay. I have a new favorite cake cup from trader joe's This is my favorite. It's the ethiopian blend coffee cobs. I really like this. I like anything colombian coffee basically um, but I have some stocked up for Who freaking knows how long I also have a lot of tea. I need to take those off, but I really haven't even drank that much tea lately All right making coffee. I'm about to get to work at this table I have a nice candle burning And then maybe play some norah jones and then get to editing and hopefully finish this whole vlog pretty quickly All right guys. I finished editing my video. It's been exporting my laptop just died And we are actually gonna start watching twilight I definitely thought quinter would not watch this with me and then he said He wants to have a twilight marathon because he's not seen all of them So we're gonna watch the first one. We have pizza and stuff to make for dinner and I'm gonna make a quarantine bucket list Dallas just closed everything Dallas just closed like bars restaurants gyms everything past 8 p.m. Tonight. So I don't know My guess is that we're in 10 days 10 days 10 to 14 You just said a week. That's why I went down. I thought it was going to be too My dad said two three weeks at least and he like always knows things. So we'll see All right guys. I look crazy because I've just been laying down. So my hair is all messed up But okay, the ac is like really cold That needs to be turned off Anyways, um Everything has closed in Dallas as of 8 p.m. Tonight. Everything's shut down So we're not actually Obviously, you guys are like watching this you would know, but I'm just trying to give you a We live you watch this video like way way later or something. Um There is no official like quarantine or lockdown yet, but I can only assume that it's coming So, um anyways, also, it's just the like responsible thing to do to already have started social distancing So if you were able to stay home, it's definitely like If you were able to stay home, you should be staying home basically, but Anyways, um with that being said, we're gonna have some time at home Q and I are finally making our homemade pizza that we have wanted to make For quite some time now. I'm gonna put this on the crust Okay Pizza sauce. Yeah, we have pizza sauce. They're ready to bake though. Cheese. Oh, we don't need to make these cookies tonight I already I can't Okay, well we have flour and stuff as well to make that So that'll be fun two cheeses and Parmesan cheese All right. I'm now making it some green tea while we start Q is He's been so excited to make this pizza. You guys have no idea. Oh, wow. You're very good at this I really oh, hey Coco I saw that See that she's just trying to move her gate That's like Coco No, no It's just family pizza night. Don't move it.  Oh my gosh. This is specifically pizza sauce from trader joes. Bonjour. No What am I Italian all of a sudden? I really wish I was but 23 and me if you know, you know I went in is so precise And I've organized and oh my gosh. I want to go back to Italy and take it to him. See if I was again Actually, I'm seriously so fun mark makes a dream work Oh my gosh, it's almost fell guys all my merch is available on amazon the old stuff for the obvious so much podcast like all this stuff and then um My yeehaw crews those are all available if you guys wanted to shop, but anyways, okay, so um, I'm gonna go through like my Quarantine like rules bucket list honestly just like the way to make the best of this because I know I know a lot of you guys have anxiety and I know this is like a really hard time for a lot of people Because they're either not like working not getting paid combination of both Maybe they're like away from other like family loved ones all that stuff So I have gathered a bucket list of things to do in ways to like kind of just be the best in the situation I will still be working just not As much as I normally do like a lot of things have been canceled obviously I can always Make youtube videos and stuff. That's just like one part of like my overall job though Um podcasting is pretty much staying the same But I have like a lot of in-person meetings and recordings and things like that that obviously you're not going to be happening So I'm really like obviously grateful, but it's just going to be like a little bit slower And I don't know I think obviously this is like a really Not the best time, but I'm going to try to make the best of it and also Hopefully this channel will be somewhere where you guys like feel better throughout all this I have my bucket list of things that I want to do This is basically using this as a time to do things that you wanted to do That you feel like you just don't have enough time for or to like watch shows and series and like new music New podcasts all that stuff that you guys feel like you don't already do Okay, I really want to learn more photoshop I'm not passionate about photoshop and I need to know the bare minimum like it's ridiculous. I don't know that already um, so I want to learn that Um, also, I want to have like the whole iob switch instagram plan to a t tour like the temples around everything's done for like Quite some time I'm going to film and edit a ton and I have a lot of video ideas and like instagram posts And I'm going to like shoot for and videos and stuff That's another thing that you guys will see you'd be boring to go through that But I have a lot of ideas. So I do definitely plan on doing that Um, okay I reported with Megan and Liz when the podcast they were telling me that the Elvis Presley book was like There was this one Elvis Presley book that was like incredible So I want to read that um I'm gonna set up my office if my furniture comes. I don't know how that works like it was already ordered So we'll see Um reorganize my pantry right now. I have a closet that's going to my pantry. That's not set up Um, I want to find new things on skillshare like skillshare is really a place. This is not an ad Um to learn new things and hobbies brainstorm for this like idea thing that I'm working on Cook more. I also want to be like a some sort of organizational like not necessarily like a physical recipe book But I want to put all the things that I normally cook in one because I'm a very visual person So I would like to have some sort of almost recipe book even like a digital form to where I can just go And be like, oh, yeah, I forgot because there's meals that I used to always make that I always forget And then I get stuck on the same meal for like a year basically. Um, and then also I mean I'm leaving off like more detailed stuff I want to have an entire day where I literally just like don't do anything at all. I feel like I Don't really ever have days like that like I'll like do something and then go You know what I mean? Like I feel like it's kind of hard to have a whole day to do nothing And I think we're gonna have some time so With that being said, we're gonna make the best of it If you guys want me to vlog more, let me know Or want me to film normal videos or anything to sleeve your request down below because I will probably have time. I'm feeling very I'm motivated right now Not gonna lie, but I will definitely have the time So, oh also I'm gonna do at-home workouts And at-home workouts that I normally do are Shannon from hot pilates You can just like like search it on youtube pops up really quick Or I want to try like Melissa would help. I hear about her all the time I get like really into certain vloggers. I find people that I'm like obsessed with and Um, I don't know. I just like always get like kind of uninspired when I don't leave my house actually Coco really wants to come up here, but yeah, we This is my little chat with you guys Um Anyways, also if you guys are not in my facebook group You guys should because everyone's chatting in there and like making each other feel better and also kind of getting More info and just talking making friends or even like getting distracted and stuff So that's always linked down below. You guys are obviously the the group is for you. You're obviously more than welcome to join It's very fun. Um, I Didn't understand the facebook group hype until I made one And now it's my favorite thing ever. I'll look literally over instagram over twitter over Everything that I do my facebook group was my favorite part like form of social media So I'm gonna drink my green tea And hopefully our pizza will be done soon and then we're gonna watch twilight. I'm so Exhausted I think because I didn't go to bed till 1 30. I woke up at 5 30 something so yeah, I think I don't know. I didn't really sleep much last night Now it's catching up to me. Also. I'm making a lot of instagram tv videos lately. Oh my god. It's calling me So Maybe I'll make more of those but I'm gonna come right back Here's our pizza. It actually looks beautiful Clinton is insisting on buttering the crust with his garlic butter garlic butter So anyways that is probably gonna be it for today's vlog We're gonna eat and watch twilight and I'll probably fall asleep So hope you guys enjoyed today's vlog if you guys have not already be sure to subscribe follow me on instagram Hope you guys are feeling as good as possible. Love you guys and I'll see you soon. Bye


Autolight and its 96,000 dealers bring you Mr. James Mason in tonight's presentation of Suspense Tonight Autolight presents the story of the terrifying escape of a dying man an adaptation of odd man out Starring Mr. James Mason Half I tell you they're an unbeatable team Who Harlan why the generators starting motor coil distributor and all the other important parts that make up the complete Autolight equipped electrical system on your auto light equipped car You mean all those parts work as a team Harlan they sure do have because Autolight electrical systems are designed and built by Autolight to give you the smoothest performance money can buy That's mighty important too because your electrical system works for you Every second your car is running every time you turn the ignition switch or turn on your lights radio or heater It sounds like a family team to me Harlan right you are half so friends Be sure to specify Autolight original factory parts for the electrical system of your Autolight equipped car if Replacements are needed. You'll find it pays and remember from bumper to tail light. You're always right with Autolight And now with odd man out and the performance of mr. James Mason Autolight hopes once again to keep you in South Spence Four of us on the raid Murphy Nolan part outside waiting with the car and myself So there's no need to be frightened Johnny It's been all planned months to go slow down The safe will be open. It always is 330 The wages clock will be picking up the money. There's the office at the end of the hall to slow down Murphy looks frightened. I don't like the smell of this place Here we go boys. This is the one.   You as you're told the safe Nolan Let's open it is what did you expect? It's all in come on Don't get nervous now 20 steps maybe 25. I can see the car. Good mark, but we'll be away in a minute. There's the car Pat Murphy Nolan. It's like a dream. I can't move from here. It's like a dream I'm with you Johnny All right, get to the car The car they're waiting for me. I can't I can't and get in Hold on to me boys. I'm heart And I'm waking up for a dream. I have had No, something has happened to me. There was a car and Pat and Murphy and Nolan and I've got to get away Can't jump with them Wait for me boys Johnny Johnny, are you all right? Look? He's dead. No, he is not. No, we'll have to carry Listen to that man. There's no time. We can't even Johnny. Oh, we'll never make it. Leave me be gone. Oh, no, Johnny There's no time. They'll all be out looking for us. We can't get away.  Oh, no, it'll be the end of you. We'll get through somehow. Oh, no Why doesn't Pat back it up yours? I say I'm John for get out of it. Oh, Johnny. No, no, I'm the chief. It's an order All right Is the shelter across the road you had what I said I thought every was you I'm staying it's a dream because I'm crossing the road with Murphy and this people stand and watch on us Then I am wanted and it's not a dream because of the pain. I Killed a man. That's how it was and he wounded me So it's all up with me now. The police are coming and it's all up with me and Murphy. Here's the shelter Sorry, Johnny. No, it's dark. I'm heart. Yes, it's dark No rest for a bit. Oh, yeah, rest You're outside looking for us. Yes, I Killed him. That was in self-defense. He shot first. We got the money, didn't we? Hi, we got it No, let them battle have it safe No, let's have a look at that arm. I can't feel it But it hurts. No. Yes. I know. I know. Hey, wait a minute. I'm afraid to match This is bad Yeah, right. We'll have it all right. I'm finished.  Oh I'd like to lay me down for a long I'm tired Tired No, no, this work to be done work for an island fight Oh Johnny oh, I was so worried so I was you'll have to go without me. I know I won't the pain is killing me Murphy I couldn't make it They'll have the whole police out. It was a killer. It's me or the organization He helped me, but we'll get you out. It's a waste. We haven't enough men enough guns Can you hold on Johnny? Can you hold on if I go to Dennis and get some of the boys? Don't do it I find a doctor. I'll bring him here. No No Right over there and they wouldn't have come in here. It's the cordon will be fine to them when I start to throw us in Johnny murders the one with wounded He's the one we want You keep your eyes peeled man It'll mean a promotion this night Sorry, Johnny. I am I'm sorry Now will you listen to me are you listening Johnny? Now you keep yourself here. You'll be safe now. They searched don't move Something went wrong Murphy We had a plan. I know I know but it doesn't matter now. We can listen man The organization will have to lay low for a bit now. You know that We'll pass the word things will quiet down. All right, all right, but first we've got to think of you Now I'm going to get out go out and bring help While doing I will get you away, but you mustn't move from here. What's happened to Pat? Norton are they way? I told you Johnny. I told you What's that if someone got in the back? I dropped my gun at the mill.   No, no, they won't Johnny now easy man here There now you can beat them up with a stick don't you hear me?  What's happening to me? I've never had such a nightmare before I've got to get out of this the thing is killing Johnny boy Johnny murder It's me. It's sure Don't answer me.  Get out of my car. Can't you see the look of a mr. Fancy the police will be here They're everywhere tonight. He mustn't have Johnny Martha found in here. It'll ruin me. So we wondered any did you can't help that? Think about mr. Fancy. He's a pride creature What thousand a couple anyway, you'd sell him He's wounded the organization would get you for a sort of work. They might and they might not who's to know Power little rat Am I not? Now which one of us will go to the police? Will it be you mr. Fancy or will it be me? Bringing you mr. James Mason and miss Pamela Kalino in Ought man out tonight's production in radio's outstanding theater of thrills Sells spends my car is equipped with auto light electrical system Ah good half auto light electrical systems are used as original equipment on many leading makes of our finest cars trucks and tractors and Every unit and component part is related by auto light engineering design and manufacturing skill to give the smoothest performance money can buy Well, I depend on my auto light electrical system every time I start my car You sure do happen every second your engine is running too as well as every time you turn on your radio Heaters switch on your lights or blow your horn. It pays to keep the electrical system running, right? Hey, Harlow, it sure does have so friends treat the important electrical system of your car to a periodic checkup See your car dealer or your authorized auto light service station to quickly locate your nearest authorized auto light service station Look in the classified section of your telephone directory under automobile electrical service or call Western Union by number and ask for operator 25 and remember from bumper to tail light. You're always right with auto light And now auto light brings back to our Hollywood soundstage. Mr. James Mason in Elliot Lewis's production of odd man out a Tail well calculated to keep you in South spend There is noise and this voices and The pain hurts me eyes. I heard some talk. It was talk of the police And I heard them that close to me that they might have been the same room But I couldn't see for the pain Don't be me. We'll see the police left him tonight. Oh then mr. Fancy nor you nor you I want more trouble, but I'm not turning in This is out with Tommy guns and all. It's a big sweep Sooner or later, they're getting but I want to ignore you Stay with him. I'll get whiskey. Oh For goodness sake What's to become of you Johnny boy? You're not wrong for the joys of life. I'm thinking Work my fortune She'll Johnny Johnny, I'm your friend. It is your friend shell. What can I do for you? I was supposed to stay where I was they coming back for me Murphy's bringing them as Dennis Pat Nolan Seamus I was to be where Murphy left me even preserves the whole organization altogether. It'd be worth Where Johnny? Where were you to be? What's the place? I've got to get back. They'll be looking They'll be trapped you help us. You're in a desperate state. You can't walk the street you can't You need someone you couldn't move in the city alone, I'll take you and I'm telling you straight. Do you hear me Johnny? Rest on me Johnny. We'll find them Mr. Fancy We'll go to the back way out. Are you up to show? I told you sure I'm helping is all he asked to meet his friends to save him Now we're going out the back put your head out mr. Fancy and see if the coast is clear and get a pint. He'll need it Why am I cold again? The wind I don't want to walk in it's shell it is and no one trusts shell What am I doing with shell? Where is he taking me?  Oh dear helpers here I'll get a hold of this take a good poor you'll give you strength That's enough come on close to the walls man Keep out of the light. Where are we going? Where? Sure, and you were going to tell me as soon as you could remember offers coming back. Oh, that's right. Dennis The rest where where will they be the shelter the air a shelter back there. Oh, we're going the wrong way come on Johnny boy We'll go back She'll take care of you If I could get away from the thing the pain I Could leave it behind But I don't It's wet it is on my face. I can feel it like snow They said it would snow tonight It is snow I Must rest Johnny Johnny come on. You can't rest here in the middle of the street. There's police all about it. They'll see you sure Johnny I had a gun. What's happened to my gun merciful. He's raven Johnny Others people coming now. We're done for you hear me Johnny. I didn't give you up. You hear that? I tried There he is. Oh Murphy it's me. It's shell. I was trying to save the poor man taking him to the shelter He'd wandered in his delirium. He had Johnny Johnny is that you'll be all right now. Dennis is bringing the car around Found them. Listen, they're all around. We haven't a chance. Don't yell. He take him along.  Oh, dear helpers the police. Oh, they're turning into the street. Look hurry Quickly Johnny quickly. Come on Johnny. Come on Hey, we'll all be killed. Take him to my place. Where is he? It's not too far No, no Agnes. I've got a gun. They won't lift us. There's killin No, I'm going back. Give me the gun Agnes. He wouldn't do any good.  We'll start again Johnny. Hurry Been killed. I killed a man.  I'm alone now It's a bitter wilderness But why am I running? It's cold so it is but why am I running? I will not run. I will not Can't you keep him quiet if he wakes him up in the tenement we're done for that's for sure We'll almost there Johnny then we get a doctor. You must be quiet when we go upstairs the shelves place. I Am quiet. I haven't said a word What happened to the boys all right Don't make any noise. Your woman downstairs will give us away in a second if she knows Where's no one Murphy Johnny darling? No, you must be quiet. I can't walk anymore. I want to Few more steps you can do it sure you can Johnny. Yes in a minute in a minute. We pulled it off. I did Got away with it. I know side with him for the dear sake quickly There's a bit Johnny now you can rest. Oh, it's Oh, it's that's the blood that's gone out of him. That's what it is. He shouldn't be alive Maybe this will stop the blood turn over help me help me show Can you better doctor show I don't know I don't know what I've done this night. I should have turned him in I'd be rich Now I'll probably lose my life See what I can do when keep him quiet quiet. Don't let him make a sound I Know that face dear face Agnes and it is her own Why she I Remember now we raided the mill Murphy Nolan parts and me I Killed a man. That was it. I must have been hurt. We were hiding Who was it Murphy? What was I doing with shell? He's an informer the organization will have to take care of him one of these days The city is too dangerous We'll go to the country Then when it's quiet, we'll start again. Yes Johnny. Yes, but you must try not to talk now You mustn't talk Why did you why did you come to look for me? You should have stayed at headquarters. I wanted to who let you do it Dennis He wouldn't I had to I feel better Good now lie quietly It's nice isn't it. Yes, we're safe here Where is it? She'll please No No, you must be mad he'll inform. No, he's gone to find a doctor to find the police No, he helped me bring you here. You can't trust him. It's all right. It is my back now I'm sorry Don't talk. I'm sorry because because I was afraid. I think I've been afraid tonight. No I'm not now. You've never been afraid of anything From the first day I saw you even before that when I heard about you. I knew you were not afraid Will you give me a kiss? I love you I've never had time for that The organization I love you The organization has had you let them leave you to me now.  You wouldn't understand It's my life They must have recognized me on the street or are someone informed I ran but they're moving down the street You'll get out the back way then. No, no, no, it's a cordon. You can't He's the one that informed. He's the one no Johnny. No, no, I swear Oh, I'm sorry. I'm sorry Johnny Sure, I did my business. Give me the gun shoot him. He's the one Informer informer. No, it's all right.  They won't take it Then this is how it might have been But the fight and what about the fight they can't beat us they won't They won't open the door Give me the gun give me the gun I'm the chief give me the gun Give me the gun give it to me Agnes I Love Spence Presented by auto light tonight stars. Mr. James Mason and miss Pamela Calino This is Harlow Wilcox speaking for auto light world's largest independent manufacturer of automotive electrical equipment Auto light is proud to serve the greatest names in the industry That's why during the early months of 52 the auto light family joins together in saluting the leading car manufacturers who install auto light products as original equipment our Auto light family is made up of the nearly 30,000 men and women in 28 great auto light plants from coast to coast and Instill other auto light plants in many foreign countries Our family also includes more than 18,000 people who have invested a portion of their savings in auto light as well as 96,000 auto light distributors and dealers in the United States and thousands more in Canada and throughout the world Our auto light family will salute the Kaiser Fraser Corporation on the next auto light suspense television program If you live in a television area check the day and time of suspense on television So that you'll be sure to see this program and remember be with us next week for another thrilling auto light suspense program on radio Next week on suspense our star will be mr. Richard Widmark in a radio adaptation of Walter van Tilburg Clark study in panic The track of the cat in weeks to come. We shall also present miss Barbara Stanwyck and mr. Herbert Marshall all on South Spence Odd man out was adapted for suspense by Anthony Ellis from the book by FL green Suspense is produced and directed by Elliot Lewis with music composed by Lucian Morawak and conducted by Lud Glaskin featured in the cast with Ben ride down a hurley he William John Stone Joseph Kearns Charles Davis and Raymond Lawrence James Mason may currently be seen in Lady Possess. This is the CBS radio network


Maen nhw'n dechrau i'n mynd ychwanegwro ac fel bydd ychwanegwro, felly oedd y gallwch i ychwanegwro a'r Llyfridol yn ymdillog i dri o edrytu i'r Llyfridol. 29 oed yn gennyddol, Rwy'n rhaid i'ch blwyddyn. Rwy'n rhaid i'n fath gyda'r ffocwsgrif, a gennym, fel ai ddim i gweithio cyflym yn yma. Rwy'na ddim yn eistedd o'r cyflym, ond efallai'ch cyflym yn amddangos, oed o amddangos cerddol, Ond ychydig am ychydig, wrth hyn cymryd y bydd ychydig yn gweithio ar y cyd. Ond rŵr y的是nod. Dwi'n cael eu cyfnod, dwi'n cael eu meu rhydd o'n cyd. Gweithio, ymfaith ond hyn yn rhywbeth iddynt. Mae'n wedi'i chyfnod arnynnau ar yr halenau hynny, dwi'n ceisio ar hyn i unrwydd cymryd ar y newydd gyda'n cymryd ar Newydd Cymru. Rhywpeth oedd eich cyfryd ar y ddam ar y club. Am peth yn weithio ar ddwaith. Pŵrwch i ddiweddno'n dod i gael y preffran hwn wedi rhoi tu beth y swith ar ei fod yn chryfodol. Roedd hynny'n ffordd am hyd yn y ffordd i'r sefylliant, i dda i'n credu'n ddifu seemu. Ac mae gennym ni'n gyfolio! Mae'r euffiliad a'r wnaeth, mae'r roasted ymgyrch iawn, mae'r awbiad yn allwch, fe fe'n gyda, maen nhw'n teimlo i'r pethau. Byddai ni yn mynd i gweithiorif fel. Ac mae'n rhoi iawn. mae'r pierd wedi'u clywed ymryd o'r ddwyad, efallai mi'n chebyn o'r llygwch, a'i ffrif. Yn gyfan nhw hefyd, mae'n dod peidio'r llygwch pethau ond o ddigon, phyn nhw'n dysgu'r cwpio? And the last question he asked was if you could take something from another club that not manager or anything like something that they do, what would it be and my answer to that was they treat this city like there isn't another football club any so they don't give them keys they go round they put stuff all over the place they put murals all over the stuff. we don't we go we go the matching all this stuff why don't we go round the city and treat it like its ours that we own this city Rwyf yn fan gweddill ar cyfans sydd y niferon. Wrth gwrs, mae rhai mae'r rhai gair o- –rhyw ddechrau ar gyfer y ddechrau, ac mae rhai gair o- –rhaid wnaeth bod y fan yn dechrau. Mae rhaid bod yn dechrau. Rhaid yo byddwn i'r fan, nad rhaid? Côr, mae'r pan i'r môl, mae'r panwn dydd y cyfan, mae rhai gyd angen a'r fewn i'rio. Gallen ni'n meddwl. Gallen ni'n meddwl, mae mae'r fan yn meddwl... Mae wedi'r fan yn meddwl. Rwy'n deimlo y fawr i'n gwrs honi a'r ffaith yr FFF. Mae'r ffaith ddechrau'r tyffydd dechrau. Roedd yn rofo'r Thysbydol, rwy'n gwrs ac'r thysbydol a ddechrau'r thysbydol. Rwy'n gwrs ymdweud aristinct gyda'i gyda'r Thysbydol. Mae'r ffaith dyna ffordd a'r llwy a oedd ei gwybaros. Mae'r ddechrau a'r ddweud o amser am y gwaith. Mae'r gwaith mor amser am gael yr ysbydol. Mae gwybod sicrhau ar hyn. Mae, y gallwn y gweithio chi eich perfost sydd angen i Ysgol, ynddyn nhw, yn yr allan o gwnaeth hen yn wasbeth. Kaidwn diolch gael ardwaith. Mae'n gwybod i'n gwled. Mae'n gwybod i'n gwled, wrth gwrs, mae'n cael ei mynd i'r ddweud o gweithio. Ysgrif demonstr. Mae'r gwybod i'n gweithio a chyfl pancakes. Mae'n amlwg y dylai a'r llog. Y nifer, mae eich ddweud o gefnonaeth. Mae'n gweithio. Mae'n ddweud, oherwydd mae'n gweithio. Mae ei chweithio i ymlaen o'r ffordd. Mae eich chweithio ar y ffordd. Mae gennym, mae rhaid, mae rhaid i ddweud y ddechrau. Arsul a London. Tottenham a Llyfrgell. Rwy'n gwed wedi cael eu Llyfrgell a Llyfrgell o'r ffordd. Mae'r bach yn ddweud. Mae'r llawd o'r ddweud o'r Llyfrgell. Felly, mae'r ddechrau, ac rwy'n gweithio i ddweud. Mae'r ddechrau i ddweud, mae'n gweithio ar y ffordd o'r cymddiant. Welcome to Liverpool, or welcome to the city of Liverpool, home of Everton. You can have that, that could be our thing, couldn't it, all over the city. Welcome to the city of Liverpool, home of Everton football club. You could have that everywhere, and straight away it's telling people what it is and where we are, without going, without like saying putting Liverpool in it, the city of Liverpool. And you can have that campaign everywhere over the city, cos I was saying we, why aren't we... That should be done by the ferry terminals and the railway station. Everywhere. Billboards everywhere. All the stations and everything. Just having one big hit, one big hit all of it. Like you said on the M62. On the big screen as you come out of the, out of the limestream, on those two big screens, as you're coming into the city, as you're coming over the flyover, even like painted onto the flyover as you go under, going down edge lane, going up Queen Stav, big billboards saying welcome to the city of Liverpool, home of Everton football club. You don't want family moors built, you need to start now cos you've got to have a... Yeah, you've got to start. That should be the ultimate rebrand. You should start now, of course, but realistically what the dock front there, the city is in. But we're in the city. Yeah, yeah, we're in the city. That's when, by that time, you should be at a point where you've already pushed it for long. Liverpool are in the city, that's where you've got to be. But what we can't be shy doing, and I've got this point of question, don't be worried that the city's called Liverpool and they're called Liverpool. Like, don't get me wrong, we don't want it on our backs like you've just said, the West Ham. But make people understand that we're here because I think we've been very, very shy about that, like you're saying. And this whole L4 thing, we've got to break out of that. It won't be an L4, it will be somewhere else. It will start somewhere else, yeah. We'll be right, but we'll be by the city. Weirdly the closest thing to Taurus. Exactly. So why can't... That's something that we've got. But by having a phrase like that, there's a level of pettiness that would drive them mad, I think, which would be great. But it's also not rubbing their nose in it, is it? Cos it's the city of Liverpool. If you live abroad, right, and someone says to you, where's Everton? A lot of people wouldn't have a clue, they'd just go, oh, Everton, they're playing the Premier League. Now, somewhat, some of them are... Is that by winging? By saying challenge. The neighbors of Liverpool or whatever, right? But a lot of people would go, I suspect, a lot of foreign people would be like, don't know, if they know they're a football, they'll go, oh, they're a football team. But I don't know where they are. Have you got that campaign going? City of Liverpool, Everton or Fron Liverpool? Yeah, even a poster in 1878, did it have a little stream at the bottom of your reader bar? Just something like that, you know, with originals. No, we're not confined. Why have we allowed ourselves to be made smaller, smaller, smaller, smaller? So, yeah, we're in within Liverpool, but just in this L4 postcode. Cornershop, run like a cornershop. They've got a store in Chester. What have happened if people, the club had come out before the other decidons to set cars and said, should we do this? People would have gone, no, they would have gone, no. But because it was done and people were like, oh, my God. And that's what I'm saying about the live-abade thing, it's like, the live-abade was ours and I know people have campaigned in the past.  And then you made, I don't know, I bet you'd sell it under T-shirts because people would do it just to wind them up for a start. But if you brought it back into the consciousness that this is part of our history, you know, then... Let's do a T-shirt with the live-abade on top. Do you know what I mean? And the live-abades are blue at the top or something. If you bring that back into the consciousness, then people will go, or even if it was like, I mean, it's a bit early now, but let's say you start building the stadium and you could have almost like an aerial view of, like, from the live-abade down there and straight away you're encapsulating the two things in one and you're making that part of our history and you're not taking that away from their history, but you're saying actually that was ours first, like one of the things you've seen, well, any off-limit, I don't know, I feel it was ours. There's nothing they have that wasn't ours first. We had the live-abade on our kit first. Okay, it wasn't a... It wasn't, yeah, but it would not... To stolen from us as well. But the point being is that the live-abade was on our kit first. Okay, it wasn't an emblem. It was to show that we'd won something within the city, but it was their first. There's nothing they've got. And I couldn't... And I'm glad everyone agreed to me on the thing. It was like, we've got to own this city. And it's not for them, by the way. We're not trying to beat them. This is the point I'm making. We're not trying to say we're the biggest club in the city. We're saying we're from this city. This is just as much as our city as their city.  Listen, the claimant stays in bricks on the stadium. Do you know what I mean? So it doesn't... If they claim anything, if they'll claim a course to it, but do you know what? They can't actually claim anything. They own the colour red, you know. Christopher Bayer was wrong. Dave Facescape was blue. Exactly. Cos it was our reject kit. So... I'm amazed they don't have a gold or silver. They also took the owner's office as well. So that's what the whole premise was. We've got to take back this city. Simple as that. OK. Next.


Here's the Samsung S7 Edge right here. It has some edges on the edge and on the other edge and it looks like the S6 Edge but it's different. Now it's the S7 Edge with a faster processor and the slim design, giant display, Samsung UI, better camera, a fast focus, every pixel is a focus, that's what it said. And then there's a S7 with a mirror on the back, kind of buttery, nice edge. Price, you know, is there any price? It's probably 700. The same price as a used car. There's a nice display and it's very shiny, shiny and nice and thin. It's the main point and it's still not a USB Type-C so you'll have to wait for S8 if you want USB Type-C. This display is good. But I guess it's the same display. Maybe. That's the four.


Throughout this class, we're going to be using a series of computer models to Explore some things about climate and our consumption of energy and the economic consequences all these things And so I thought it would be helpful to begin with an explanation of what a Stella model is And so you can see how they're made and how they work and so forth. So Stella models begin with some kind of a Conceptual diagram of the system that you're going to try to represent so over here is a drawing a simple drawing of a water tub It's got water in it. It has a faucet pouring water in and water leaves a tub via this drain here So there are some Mathematical relationships here between you know the area of the base of the tub and the amount in there and the depth of water in the tub The depth of water in the tub then controls the velocity of water going out the drain that times the drain area tells you the amount of water leaves In a given period of time So anyway, there's the the conceptual model. Here's what it looks like in Stella Here this square thing here. We call a reservoir a stock. That's where the water in the tub is recorded And if I click on that You can see we've entered point five cubic meters of water, that's the starting amount of water in there This this entity here is called a flow It's just like a pipe that that allows water to flow into the tub and that has a specified rate It's point two five cubic meters per minute And that's a constant value And the drain on the other hand is not a constant value. It's defined as the velocity at the drain times the area of the drain The area of the drain is defined over here by another equation that involves the drain radius the velocity of the drain here is This thing it's something called Torre celli's law Gravity here is a constant The area the base of the tub is something that is a constant that we'll start off with as point to five and The depth of water here is just the water in the tub divided by the area the base of the tub Okay, so that's the way the model is set up So you build the thing here and you enter in the equations and values for all these things and then then you're set to go And so we can run this model now and what that means is we we kind of start the clock ticking We are going to pour in water at a given rate a specified rate through the faucet We'll let it drain out according to that equation and we'll see what happens So here we're going to run this for 30 minutes And we just click the run button here and you can see very quickly it calculates this The blue curve here is the water in the tub remember we started off with point five cubic meters And it rises up to something close to about point nine cubic meters and see how it rises up quickly at first But then it levels off and approaches a kind of a steady state And that's because the the water leaving the drain is Rising and rising and rising Until it becomes essentially equal with the amount of water flowing in the faucet So at that at that point in time the water coming in is equal to the water going out So there's really no change in the amount of water the volume of water in the tub here itself this interface to the model here includes some little knobs and Things that you can adjust so I can change the drain radius. We can see what happens if we Change the radius of the drain a little bit increase it there There it levels off at a much lower value point five cubic meters in the tub if I Decrease that somewhat Say to there You can see what happens there. It doesn't even in this time period Level off entirely. It's already up to more than three Cubic meters, so we're gonna have a lot of water in the tub at that point And you can reset those things by hitting the you buttons there So this is you know, this is how you operate these models This is how you interact with them Just remember that if you're looking at an interface like this just know that behind the scenes kind of underneath the hood There is something like this this diagram that lays out the logic and the structure of this System that the computer uses to kind of mathematically solve them the evolution of this system over time


Where's everybody at the printer? This is not funny I still cannot type anything here usually it's like a part in class and part take home The in-class part would be Setting up a problem You know the first first two or three steps Yeah, and or and or solve something. That's simple, you know do it by hand for instance A simple optimization, you know just I need to restart this computer. I know for Let me Start mentioning about this I've gotten a few questions of from from the one of the problems Problem number 15 That asked for was that textile problem That once you set it up for linear programming You run the code and you read off the you know the outputs and The lambda that came out as the shadow prices Wasn't it a little bit weird So does anybody try the simplex method instead and same weird numbers Okay, well, I'll look them Because when I ran what would have been sort of the Obvious choice for the metrics a and the constraint, you know the constraints and B The simple the large-scale method gave Some shadow prices which are Not I mean not not realistic, okay And how can you tell you the shadow prices are not? realistic if you read them from the lambda You know from a lot lambda output of that program I mean, how do you how do you test? Say you didn't have this fancy way of collecting shadow prices from that command Shadow prices would have to be Change the number in the constraint. So it's the sensitivity to the constraint, right change the number in the constraint by one and look at the optimal Value, right and And again, if you do this with a large scale, then you would see that It's not it does it's not consistent, okay With the number that's been collecting through from the Lagrange from the Lagrange multiplier So sometimes again sometimes trying, you know, I guess if something doesn't look reasonable if it if it looks reasonable, that's you know That's that right I mean you make the interpretation and you move forward, right And again at this stage it may not be very clear, you know, what is reasonable and what is not, right? But I would say that trying Maybe two different methods Would be, you know two different methods for this limb probe would be a way or As I said brute force just just make the sensitivity analysis your on your own I should also mention that The implementation of linear programming or any of this methods since simplex or large scale The in MATLAB is you know, it's it's not foolproof so In fact No implementation is like universal at this point if you have a large scale problem like a large number of variables large number of constraints You know the in theory everything should work fine, right? But when you when you ask a computer to do it you run out into these cases where things don't You know certain errors Creep up, you know certain things certain limiting behaviors appear that that you didn't expect right and So it's it's really sort of an active field in fact Just I think a couple of years ago. There's been a Kind of a new algorithm for solving linear and non-linear programming problems, so Developed I think at Stanford that is now being commercialized so various commercial packages implement that that algorithm and Comsole is one of them so so I Don't know what's going on with this but just just to show you that that there are there are I mean It's it's an ongoing process. There's nothing like This is it and now we can use it close our eyes and and not you know just trust trust the computer 100% Please get this started. I might have to live with this Resolution if that's okay. It's really small on my screen. Okay. Well, let's see if it Hopefully I can run some some codes. I posted the code for Well, you've probably seen this dirt problem that we did yesterday last time in class And I posted two more codes here In chapter four, so we're gonna start talking about chapter four dynamic models Okay, so let me Do you like pink or white? We could sell or switch between the two Okay, so. Oh, this is interesting Now I cannot write on the screen Okay, now I can write on the screen, but you cannot see what I'm writing Let's see if there's a steady state here Now you okay, it's either you or me who? Any idea what to do Yeah, so if it's on the screen, then it's a If it's on the screen, then it's it's also on my machine, but I when I write here it appears here So it's not I could use that Yes, please See now you can see I cannot see anything So see how many people had differential equations here? Okay. I wasn't a trick question How many of you had have seen systems of differential equations in your course? Oh, okay So means we can skip chapter four I guess this is six months old. It's pretty old Okay, we can get a new one. Oh iPad. Yeah, that's right Now wait hold don't Okay All right, so it looks like I can at least write on this. All right, so let's talk about dynamic models Or dynamical systems first, okay, so So you've seen differential equations that are of this form where x is Kind of the state variable or it's some something that If you know it at each time Then you know the system they did that this this equation describes Right, so that's why we call a state variable And it's typically a function of time so we can It's like that we track a certain very you know a certain process in time And all this is saying is that this is a rate of change right It's it's a lot that says how the rate of change on this of this variable depends on the current state Okay, and this system is or this equation is understood sort of Well, if you can solve it explicitly, that's great Right, but not always you can solve Even first-order equation explicitly So alternative is to have what's called a direction field, right? excuse me. Yeah, so direction field and and That amounts to doing what so in the xt plane you do For instance, well, let's say it at time zero you can do that time zero for each x For each value of x You draw a direction that has slope equal to what? the right-hand side right So you have the right-hand side So the slope Equals the f at the value x where it's where you plot it where you compute it And again, this slope can change You know, I mean that that means the function is depends on x So this can change with respect to x Could also have x negative here something like that so Okay Now this is what you do when at t equals zero and then you have to do it at all Kind of you have to fill this plane With this directions, which of course you cannot do it by hand. So you just I guess you just illustrate this on On a few and of course with a computer would be much easier to do and What you notice here when you when you do this is that If there is no t in the right-hand side if there's no t dependence in the right-hand side So so the right-hand side only depends on x then at different times T the the slopes are the same so the directions are the same So this is a situation of what's called an autonomous system or an autonomous equation right, so this is right-hand side is Time independent means is an autonomous Equation or system and that is to distinguish between the case when the right-hand side would not be Depending on would be depending on the time. So what's an example of that? So more general You could have dx dt as a function of T and x right in which case what is the It's you have a possibility of the direction to be let's say it's a simple Function like this so it has the slopes are of this but At a different time it will it would look totally different right for the same x a Different time it would look different. Okay, and then another another time it would be again, maybe different Okay, so this would be a case of Autonomous system a non-autonomous system equation in this case right okay, so What do we do this direction field? How does this relate to the solutions of our of our differential equation coming back to first example here if I have a Initial condition so at time zero I know that what x is at time zero Right to solve this system this equation would amount to what amount to do what today? I just cannot write on these computers Okay, but I want to find a Curve that fits this direction field right? so for instance That might be a curve that that Fits direction field at all points right if it starts there, but if it starts here it might actually look Like this right and I look like this and so forth Also Right, so these are different solutions for this differential equation and they correspond typically to different initial conditions right if it is Non-autonomous, you know is the same the same issue except now it's it's going to have to adjust right to Kind of at each time there's going to be some sort of different You know think about it like wind when this changing conditions are changing as time evolves So you could be starting I didn't say you could be starting at one point here and follow a certain Trajectory a certain solution Right if you started this moment of time, but for the same x if you start at A different moment of time, which is let's say here at time zero then you could be doing a totally different Trajectory right so there is actually a fundamental difference between autonomous and non-autonomous systems or Equations and that is when you look at the solutions At the picture of the solution what what do you see? What do you notice right there consistent in the sense that they're translating their translation invariant you take the same picture and You just kind of shift it by a little bit right the direction feel is going to be the same So the solutions will be the same right Whereas here it will look I mean my picture is not great, but If you shift if you kind of translate this in time Then you're going to see a different picture right yeah Well, it's dynamic system because because your variable is time dependent So it's dynamic in the sense that given a national condition It's going to be you know, it's going to be time dependent right individual solutions But it's a time in the autonomous case it basically says that you can start at time zero and follow your dynamics Or you can start at time Ten the same location and you're going to follow the exactly the same dynamics I mean the best way to think is in the airplane you are taking off right you're waiting to be cleared Right, and there's no wind Okay, and you have your everything's pre-programmed Well, it doesn't matter if you start now or you start in two minutes Your trajectory is going to be the same right it's dynamic, but it's going to be the same it's going to be not depending on the Time when you start motion Whereas if there's a wind if there's like a thunderstorm, right if you start now or in two minutes Even though you're you know you're following exactly the same You know Physics physical laws your trajectory will be different. Okay So You can actually Experiment this on your own I I linked here something that's quite powerful called it's a mad love code for interactively kind of plotting this kind of Direction fields it's called the field eight As usually just copy you know Copy and paste it in your in your mad love On your own your local machine And it has four thousand lines of code so you don't want to even look at this But again, it's it's sort of Code developed by a professor in Rice University, and you know he graciously kind of Gave it gave it away, and it's it's it's kind of used throughout You know the best sort of tool for Solving differential equations with mad lab. So here's an example of that X square minus T that's not Great example, but I mean it's not a function. It's not a differential equation that you might Want to solve anyway? On paper, but here's here's here's a here's a time equal zero and X is I don't know between negative two and two and this is how the direction feel looks like Okay, if you can see it, but So I'm not I'm not talking about how mad love actually feeds this curve But that's basically how it solves that differential equation With this initial condition right X equals zero at time equals zero, right? And it you saw that it solves also backward in time, which Okay Which you can do So that's that's one trajectory right, but now imagine now look at if I start at the same X equals zero about a different time Right it looks different right I Mean it has sort of and you can do this I mean With a bunch of points in fact it's not that I mean I'm just I'm just hitting X equals zero a different time so you can see that it's it's it's not autonomous, right? But the best way is to kind of to see this Fix you know fix a time zero and take different initial conditions It's actually even more dramatic to see what that There's kind of a value here for X Below and above which the the dynamics is dramatically different, right? Okay, so you cannot really do this by hand I believe You cannot solve it explicitly by hand But be my guest to try it out anyway X X five equals X square minus T Doesn't look that Bad, but it is not you know you cannot solve it by hand let me Let me switch to something that we're going to be using quite a bit called a logistic equation and It's using P rather than X, but that doesn't matter And then he has a coefficient times the right-hand side is our X times 1 minus X over K or P And if you proceed with this This is the direction field that you're gonna get oh and by the way, this is not That great because it doesn't show the whole picture, so let me let me do What do I want to do I want to do some negative negative 5 to 15, okay? All right, so now Let's identify what T equals 0 is so this is what T equals 0. All right, so now you can see An autonomous system this is autonomous because there is no T dependence in the right-hand side Okay, and again, what's that feature and the in the You know in this direction field phase portrait it's translation invariant right so it's Now individual to individual trajectories are not constant right you start you start somewhere here and With time as time evolves you kind of reach a certain You know you do something right so as dynamic but again If you start at a later time at the same point it's going to be doing the same thing with that delay time time delay, okay? All right, so it's a kind of a great little tool It won't solve the differential equations Exactly So If you need more than just the picture or the behavior then you might need to solve those equations exactly so Let's start with a logistic equation. So this is dx dt equals well, let's just Do it really simple x times one minus x? Okay, so if nobody told you when you started this first time, you know, this is a kind of a simplest type of first-order od that you you were asked to solve in in in your introductory course, right? But if nobody told you that you know, there is a direction field. There's a picture that you can relate this to Then there was probably a mistake because the picture is very kind of suggestive so think about the right-hand side so the right-hand side is a quadratic Equation in x right so I want to I want to plot these directions according to The values of this quadratic function, right now the this quadratic functions function if I were to plot the right-hand side Versus x would be a problem of pointing downwards, right? and With x intercept at zero and one Okay, so this is and what do I take out of this? You know this information Can I help? Can it can that help? Figure out the direction field. I guess by hand Well when x is zero regardless of what t is What is the right-hand side when x is zero zero, right? So so this means the direction at x equals zero is zero and I can do it You know at several points Again, it's independent of t it's the same thing at oh at x equals one right The direction field is Gonna be zero. I mean the directions are gonna be horizontal, right? What happens in between like if x is in between zero and one the right-hand side is positive, right? And you can see it's it's the maximum pause. It's It's positive. What's the maximum here is I think it's a quarter So right when x is a half a half times a half is a quarter And that's the maximum it can actually be so the slope the steepest slope is a quarter in Slope, I don't know what that angle means but let's say it's like this, right? Okay, and finally Well an intermediate are kind of smaller right and smaller. I mean less deep less steep and again less deep above You can see how inefficient it is to do By hand obviously Also, when it's when you're above one what happens with the slope it's actually gonna be Negative because that's where the right-hand side is when x is greater than one is negative, right? so Sometimes we put this pluses and minuses to indicate the sign of the right-hand side Right whether you have increasing or decreasing and then decreasing below zero again And by the way, this is kind of steeper and steeper as you go below, right? so the question is Can we actually well we can fit this with a computer we can fit the solution Curve and you saw it, right? You saw it on a computer, but the question is can you? You know is there something? Explicit you can say so for instance effects starts here. I don't know at one point It's clear that it's gonna go up, right? But how do you know it's going up all the way? I mean where is it stopping? Well, obviously you won't be able to go above one because above one things are The trend is to go down, right? but How do you know that you know so how do you know maybe I should start with this if you start with one well Because the slope is zero right Horizontal line fits that direction field Yeah, so in other words the function x equals constant one solves this differential equation, right? x equals one makes this zero and Being constant makes the derivative zero Right the same with x equals zero So these are called the steady states, right? So the steady states or equilibria solutions x star, right? Such that or our points. Let me say our points such that x is Constant x star is a solution. Okay, and that means That the right hand side at x star must vanish right since the derivative of a constant in Time is zero right, so in our case The right hand side is x times one minus x. It's nice because it's it's Factored so you could get x equals zero or x equals one, okay? So these are the steady states meaning that a solution starting at that value stays stays for all times at that value, right? So this is going to be kind of one thing that we're going to be looking at is is When you have a system and not just one equation, but a system of equations It's to identify where are the part where are the possible steady states? and then What happens with the solutions that don't start at a steady state? So as I was saying before if you start let's say at a point this this point x Obviously the trend is that is going to be Increasing, but how is it going to be increasing? Where is it going to be going to eventually and so forth? Okay? Can you trust the computer always and the answer is no if you've seen it repeatedly today? But where is it going right? How high does it does this value go? Okay, now of course from the computer from that Did I show that? Yeah, so from here, it's clear that This is this is that equation, but with the different constants so take R to be By the way in this code you can change This are the parameters right? So if I change R and K to 1 then and then I have to change this The window so I can see it right so how how do we know that it actually goes towards one? Well, there's no Well, there are ways to see this but at this point for the logistic equation we know how to solve it exactly Okay, so analytic solution or Exact solution Can be found Because this this equation is what's called kind of equation is this if you have to solve it by hand separable right so So you separate the variables This is separable and Then you integrate right so into you have to integrate This rational function 1 over x times 1 minus x and It's not Too hard To do it by partial fraction right but in case you forgot that partial fraction decomposition Then this is how it would look like right Then how do you find the constants a and b right so? you could Do the standard way and just you know find the common denominator and then identify the coefficients get a system of equations, right? But how I like to say it is is you just solve visually You kind of cook up those those those Cough those constants now That's not a way to you know teach how to do partial fractions, but If if somebody faces you and says isn't that clear that a and well a is one and B is one Satisfies this well again it takes some Common denominator, but you can see it That numerator it becomes exactly what you need which is one right so Now how you do this as I said you have to do it through you know if it's not obvious you have to do it through the methods that That you know But I'm just going to use this so right now. It's going to be 1 over x plus 1 over 1 minus x dx Equals integral of dt is just t plus a constant. So this is natural log of x Minus natural log of 1 minus x in absolute values t plus c so so you've actually solved Right you've integrated this equation because now there's no more integrals And now if you if you can do if you can find x that's great and we can but You probably remember how few well There aren't many examples where you can actually Proceed with this computation until you find x explicitly so so let's see what what is a You exponentiate both sides right so you get x one minus x is a constant e to the t Where C is could be positive or negative right? Or even zero or even infinite so the first thing here to notice is That this computation doesn't quite make sense if x is one right, but if x is not one then you can do this and Now you can solve For x makes it a linear equation. So it's x 1 plus c e to the t as c e to the t so finally x is C e to the t over 1 plus c e to the t Okay, and what is this? C is any real number and of course or There's also the solution with x is as I said equals one Right that was kind of lost in this computation But this is a general solution of that of that logistic equation. Is that right? so now Well x cannot equal one in this form But x equals one is a solution as we saw before so because we lost it We assume the x is not one so we can divide by one minus x Then it doesn't mean that x wasn't could not be equal to one right? So that that's why you have x equals one as a edit, but this is You know like if you start with a half Right one can find the constant so that this thing is at time zero so Given initial condition x at zero is x naught You can find the constant right so you get x Not is C over one plus C Yep, so you can find the constant C to be What do you get C is x naught plus x naught C? So C is x naught over One minus x naught right okay, so for instance, let's think about We're talking about what happens with solutions That start in between zero and one like started one half right if x naught is a half what is C There's a half over a half. That's one Right, so what is x as a function of t is e to a t over one plus e to a t Yep, so that the graph that you saw there for x equals a half starting with x equals a half I think you can clear these There's a way to clear all of this clear figure I clear the whole figure There's a way to just clear the the solutions only okay, so So again if you are to Between I said one half so this is the point where I want to start and this is the solution right Well, this solution is exactly that function Or there's the graph of that function right how can you see this well as Well one thing is to plot this function, but The the key thing was as t goes to infinity where is this thing going? Okay, so you can see it's going to in this case to one right I mean one way to see this is is to Divide by e to a negative t so e to a negative t plus one right and You know that e to negative t goes to zero as t goes to infinity So that's why the limit is one right and again you can do this for also for in general so in general So for general x not we said x of t as c e to a t over One plus c e to a t so that's the same as one over one over c plus Excuse me one over c e to the minus t plus one And what was one over c? Well, c was x not over one minus x. This is one over x one minus x not over x not Right so you can see the solution You can rewrite the solution as one over see one plus One minus x not over x not e to the minus t so now what do you? What do you conclude if x not is Well if x not is zero this this is not not a good way of representing right but if x not is zero the solution We know it's zero if x not is between zero and one this quantity is positive So this thing goes to One over one which is one right if x not is above one This would be negative, but this is still going to zero right so all the solutions Actually eventually go to That value one right The only thing that's concerning Some concern is what if x is negative x not is negative. Yeah, yeah, true You see the graph shows you know see everything goes to one Unless you're starting below zero in which case you go to negative infinity right so Can we see it that this goes to negative infinity this thing should go to zero Which means this thing should go to Negative one And I'm not seeing that but okay. Well, anyway, I mean the point is this is an explicit solution That tells you you know given the initial condition. That's what x is gonna be At x is given x zero. This is why the solution will look as a function of t right It should be It should be obvious, but I don't see it right now that it goes to negative infinity. Okay, when x is not this is negative all right, so Again, what I'm saying here is this is kind of a rare case That he can solve explicitly The equation right and then he can match with it with a picture okay, so This is what happens when you have one one variable one one state variable You can actually relate a dynamical system or the dynamics the equation describing that with With a with a with a direction field and you can analyze, you know what happens as time evolves You can you can identify what are the steady states and so forth But certainly one day is not enough so You could have two-dimensional or several dimensional ones and the Easiest example is Is what's called a pendulum non-linear pendulum in which in which? So the system that you're describing is actually you know plain old Pendulum that's fixed at one end and it's it's kind of oscillating in the other end right and the Newton's law, you know if you have some friction Newton's law of motion the acceleration is related to the forces So assuming this has some mass Well ends up being like a second order equation like that right and it's a non-linear Because of this term okay, so how do you? Solve how do you try to solve this system? I mean it's a it's a second order equation right, but actually it doesn't have an exact solution You cannot write theta is this function of t Okay, if you didn't have these terms if you didn't have the sign if you had something Like a theta just instead of sin theta then you could solve for for theta But because it's non-linear you cannot solve it so The the typical way to study this system is or this this physical system is to convert a second-order equation to a first-order system and There there's not a unique way in general to do this but One standard one is a standard one is to call the derivative of the variable Theta to be you know the new variable a second variable, so that would be the momentum and then Write the The rate of change or the derivative of the moment of the second variable in terms of both variables, right? so this is just of course, this is just the Second rate of a theta so it's minus sign That up minus K Omega, right? Right, so now you see a basically a system of two equations Which brings the question of how do you study systems? I mean is there something related to? similar to the case of one dimensions and The answer is yes, but it is going to be a little bit more complicated So I'm gonna I mean I give you that example. I can show you the picture that comes out of this system, but This is what it boils down to I have now a differential a system of differential equations where X is You know, let's say it has two variables doesn't have to be even more Start with the two two variables X X one X two and F could be F1 F2 Right, so literally I'm looking I'm looking at D X1 Equals F1 and DX2 equals X2 F F2 yeah, and the question is If if there is such a system, how do we study it? Well, there is Think about it as a similar concept as it was for the one dimensional a direction field But this time I have to take into account that The X1 X2 are Interconnected so this the solution so the solution Will be X of t which is X1 of t and X2 of t okay think about Plotting this Well to plot it you need basically the X1 X2 plane versus T right so You imagine that I have a solution that starts at time zero at this point Okay, and again, we're gonna consider it to be Autonomous, so it doesn't really matter So we just always gonna start our things at time equals zero. That's the advantage of autonomous systems Okay, and then this is gonna have to fit some sort of a direction curve right a Direction field, excuse me So at each point At each at each moment of time. This is gonna have some Well, I don't want to call them slopes, but there's gonna have a direction right There's gonna be a slope in the x1 direction a slope in the x2 direction, but Imagine what happens in the in the? In the XD in the X1 X2 plane Well, this is over this is in time right by the projection sort of with this trajectory is gonna describe some curve in the X1 X2 plane all right, so that's That's how I would like to understand now how I would like to think about this because Soon we're gonna be run out of a dimension that we can visualize Right if it's if I have three variables, I cannot even do this plot, right? all I'll be able to do is kind of see the trace of the Trajectory in that state state space So that's that's why even into when I have two variables. It's best to think about it as a So focus on instead of this versus time Focus on what's called a phase plane in which case? I'm just looking at the X1 X2 and Now I can talk about directions in the X1 X2 plane as being So this is gonna be DX1 Dt equals f1 and DX2 Dt is f2 so So at each point I'm gonna have to plot this Direction field again, but it keep in mind that it's different than than the one where I have one variable and time Okay, so for that Let me show you this There is another code called P plane Which again has thousands of lines of code. So you just Want to copy and paste Save it here. I think we can just quit this Okay, so here's an example. I'm gonna take the pendulum here and you probably don't like this, but it says theta omega omega theta and so forth and The first thing that's gonna do is gonna plot this Direction field in the phase plane Now take a look at this. Okay, so what are these errors? These errors are no longer slopes of I Mean, they're not slopes of the of the solution, right? so we stop talking about slopes of of the actual of the actual Directions But instead at each point So it's saying that if the solution were to hit this point, you'd have to follow this direction, right? And so forth, right? So again, you ask the computer to find sort of Given an initial condition find that The trace of the solution, right? So this is not I mean you don't really see the solution here as As a function of time Right, if you if you were to plot this as a function of time you'd have to start there and then do this If time is going this way, right? Instead you just see the state at as time evolves like the states that have been that been visited right and Of course this this arrow is also indicate like in which direction it actually is is described that that trajectory, right or orbit now This is the pendulum. So this is all this is saying is that it's Different initial conditions correspond to different trajectories, right and Because it's non-linear You get all kinds of interesting behaviors You can also get this What appears to be a repetition in theta, but The fact that it kind of repeats itself This has nothing to do with anything except that theta is an angle. So once it passes to pi So this is zero and then this is To pi right so in a way in a way the face Porter just should stop Like between negative pi and pi right depends on how you measure the the Okay, and then you just draw conclusions now. What's one thing that needs to be? Where do you start this analysis? Well, of course if you have such a tool and you draw a nice picture It's not it's not enough, right? so One thing to do is to say find a steady state or an equilibrium point and Guess what the computer will find Lots of things lots of information about this equilibrium point and we're gonna talk about a lot of these things But this is just that I zero and omega is zero meaning the pendulum is sitting Straight down right and with no initial momentum But this is not the only one. There is another critical part. There's another equilibrium which is when That I spy so this is it sits Like on top again with no moment. No initial momentum Right, and I think this is the only two and then then they keep repeating by a period of two pi And so so that's kind of the one of the tools, but certainly One has to talk about and we're going to talk about this Start talking about steady states steady states Or a equilibrium and Go from there. I think I've posted the next homework assignment, so I'll ask you to go through the first example in the In the chapter four four point one I don't have a code for that because it's not anything that you should code. It's basically just Illustrating the steady states for a System of two equations with two unknowns Okay, a concrete it well a more concrete example, and we're going to go through this on Monday, but you can have a head start and Then we're going to go and talk about the actual codes You know for not just steady state, but time dependent Dynamics Okay, thank you


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