W-E-A-F, New York. Pack of Avalon cigarettes, please. Yes, sir. Oh, just a moment, sirs. Don't forget your change. You'd never guess, but Avalons cost you less. Welcome to Avalon Time with Red Foley, Jeanette, the Avalon chorus, and Bob Strong in his orchestra, and the only man in radio who believes there are two sides to every question, his side and the wrong side, Red Skelton. Ladies and gentlemen, that announcement was made by Del King. He's a peach of an announcer, too. He can tell he's a peach, but that fuzz on his upper lip. Say, uh, those are nice-looking shoes you have on Skelton. Oh, do you like them? They're genuine cowhide. Facts are so genuine, I have to milk them twice a day. But they're comfortable, though. Yeah, that suit must be genuine cowhide, too. It's kind of baggy at the knees. Yeah. How are you tonight, microphone? How's it like here? You're just... I'm thinking of all the fun I'm going to have tomorrow in my car. I'm going out and do a little Sunday driving. You know what Sunday driving is? That's carbon monoxide with detours. Well, but what a place Chicago is to drive in.  I have a nice car. It's all mine, too. All I owe the guy that sold it to me is a grudge. That car is so old. When I took it down to the license bureau, I had to get it upper and lower plate.  There's that thing I was telling you about.  In my car, though, it's a nice one, too. It has real knee action.  Sunday driving is a little different nowadays. Back in the K-90s, when something whizzed by you, it was a horse feeling his oats. Nowadays, when something whizzes by, it's some jackass feeling his ride. I met one of those fellas... Thanks a lot, both of you. I met a couple of fellas in the courtroom the other day who were reckless... Well, get ahead of me back there. I met a fella in the courtroom the other day who was up for reckless driving, and the judge says, I'll find you $100 for driving up a one-way street without any lights, going through a safety zone, hitting a pedestrian and driving while intoxicated. The guy says, wait a minute, don't that $2 safety sticker entitle me to any privileges? Well, I've been out here stripping gears long enough, so I'll park and let Red Foley dedicate a song to my car and title. We've come a long way together. Take the wheel, Foley. You've got a lot of friends out there waiting to give you the right-of-way. I know I have, and if you have two, then you know the value of a dollar. Well, just think this over for a minute. You can save many, many dollars every year by switching to Avalons, the quality cigarette that costs three to five cents less than other popular price brands. Yes, that repeated and consistent saving of three to five cents on every pack of cigarettes you smoke turns into real money in a surprisingly short time. Now, friends, you have everything to gain in Avalons. For, in addition to this big saving, you still get a superior quality cigarette. Avalons are 100% union made from an unsurpassed blend of the finest Turkish and domestic tobaccos. You'd never guess they cost you less. Supreme quality, outstanding economy, Avalons are certainly worth a trial. Why not get a pack tonight?  I'm glad you like the band, Skinny. What do you mean, Skinny? Your face is so thin it's only got one side to it. Yeah. What are you going? Oh, while I was playing, I just happened to remember something. Wasn't the tune by any chance, was it? No, I haven't had my five minutes off for a smoke. Well, I'll be seeing you later, Skinny. Yeah, Skinny. That guy may be a maestro to some people, but he's a rat-stro to me. Thank Dell. I'm trying to start a vacation service bureau, so you follow Bob Strong and tell me what he smokes, will you? Well, I don't want to be an informer rat. Well, then get off of that stool, pigeon.  Well, now listen, suppose he doesn't smoke avalanches. Yes, he wouldn't dare.  Okay, Skinny Pants. Why does everybody call me Skinny? I ain't so thin. Then how come I can see daylight through your ears? Yeah. Oh, Miss Stillwell, say, let's get down to business. I'm starting a skeleton vacation and travel bureau. And if anybody calls up, it's for me. Why don't you get an offer? My offer's in my hat.   Yes, but would you like me if I were old and ugly? Well, don't I? I mean, uh... Uh, uh, Sullivan, you better take that. Hello, this is skeleton vacation service. We cook your tours. Yes. Yes. Just a moment, please. A lady wants to know if you can arrange an overnight airplane trip to California. No, tell her no. This is no fly-by-night concern. Hang up on her there, will you? It's kind of the... How are you there, skeleton? How are you, red bully? Say, how are you feeling? All feeling full of pep and raring to go. On a vacation. I can fix that for you, boy.  How would you like to go to Rappaports by the sea? Oh, I don't think I'd like that. Oh, good. It's a nice place. Rates very reasonable. $5 a day. Well, buck and a half, you argue. What do you say? Oh, I kind of got a hacker to see the Great Divide. Okay, but still well. See what time the next boat leads for the Great Divide. You don't. The Great Divide is in the United States, between the Pacific Coast and the Great Plains. Oh, there? I thought the Great Divide was in Europe between two dictators. Who's there? Why don't I party to build a sailor? Hey, I have a lot of fun here in Chicago.  Oh, goodness, no. I haven't met anybody that even looks like anybody I know. How are you, Miss Stillwell? After those sandwiches we had last night. Yeah, what sandwiches? Oh, we drove out in the car to get some of those Sally Rand sandwiches.  Hey, Herky, how about me helping you decide where to spend your vacation? Oh, no, no, no, Mr. Skelton. I always go to Lake Quichabelli, Aiken. And if you like wildlife, you want to go there. It made a regular caveman out of me. You look kind of neeming to me. Well, you see, I didn't come out of the cave very often. But I certainly had a lot of fun up there pitching horseshoes, although it is kind of strenuous. Yeah, strenuous? Pitching those little shoes? Little shoes? Oh, you mean you're supposed to remove the horses? How to go now? I left some beans on the stove, and if I don't hurry, they might burn my beans off. Oh, Herky, the pink cyclone. Say, Skelton, you know you wanted me to find out if Bob's strong-smoked avalanche. Well, Mr. Avalon was in the hall, and you know what Bob did to impress him? No, what? He let a whole pack of avalanche and smoked them all at once. Why that show-off? Anything to hold a jump. Oh, here he comes now. Look at that guy strut. Come on over, Bob. Yeah, come on over, Don. Old Shakespeare strong. What do you mean Shakespeare strong? You're always making a play for somebody.  I may not get one. You get one sooner than you think. Maybe I better put it this way, then where are you going to spend your layoff? Are you mad at me, Bob? You didn't say hello. No, I'm not mad. How about a little kiss, jitterbugsy? Jitterbugsy. You're going to start that kissing stuff again this week. You're going to wear out your kisser, you don't watch out. Oh, corny Casanova.  Well, who said I was? What do you think would happen if I let every man I meet kiss me? I don't know. What usually happens?  That's all. Well, how do you know? You haven't even tried yet.  Okay, Bob, now about the vacation. Now, I can let you have an A-Tour for $10 or for $5. I can let you have a B-Tour, or you can take the C-Tour for $2.50. Well, look here. Here's $0.50. You take a D-Tour. Now, I'll fix you. Jeanette's going to go to town with, don't worry about me, from the Cotton Club Review, and you'll have to accompany her. Get start traveling, brother. Start traveling. Commercial, I have to say here that Avalon... Whee whee whee whee whee whee. Hey, Red, cut it out. Cut it out. What'll Mr. Avalon say? Listen, Skelton, this is no time for Uda to get funny. Isn't wet, I gotta do a commercial. I'm hearing that song too much on the radio lately. I don't seem to sing great either. Cut it out, Red. Now, cut it out. Now, listen. Friends, when Uda buy Avalon, who'd get the Toysers, Turkish, and domestic for backers that have been blended together with wear and kill?  Jeanette gives you an angry, mild, delightfully mild muck. Who didn't? Red, who didn't put in fully enjoyable? Red didn't tell him about it yourself, then. Down in the meadow, inna, it a bitty boo. Stop it, Red. Stop it. It's not me in an awful spot. Don't get this dirty, Zell. Remember, friends, Avalon bought you three to five cents less than other popular price friends. Friends, who'd never guess but Avalon's tossed to wet. So remember the next time who asked for cigarettes, say... No, no, no, ask for Avalon cigarettes. And don't forget your change. Uh, uh, uh, take it wet, Bully, and sing what you're done at change. Oh, sit still, and it's all that baby talk. Folks of the Avalon course, I'm gonna sing, uh, Tilva on the stage. Go ahead and play it, Bob Prong, will ya? So lie down. This is of Manhattan, at the island where New York City now stands. I'll take the part of Chief Skelton, the guy who sold it to Peter Minuit. Um, you set the scene. Will you tell while I get my feather zone? Okay, Red. Okay, Bob. Peter Minuit, the first governor of New Amsterdam, is about to perform his first official act, the purchase of the island of Manhattan. As the scene opens, we find him and his secretary approaching the Indian village. He'll be friendly. Oh, they'll be friendly. Don't look so scared. Who's scared?  Sissy. But why should I worry? The last time I was here, the big chief liked me so much, he took out an option on my scalp. And this time he might lift the option. Try it, we're coming into the village. Oh, here's the chief now. Hiya, chiefie. Hiya, chiefie. Well, why don't you answer me? That's a totem pole, ya dope. Well, uh, don't let it get around that a Dutchman was scared by a pole. Certainly not. Telling to put some clothes on. Sissy, I don't think we should go in there. It's all right, Peter. Come on in. Oh, there he is. Hiya, chiefie old boy. Old boy. Old boy. Howl. Howl. And howl. Why you bring squaw? I write short hands. Yeah? Me must chop off me squaw's hands. She write without hands too. Without that up, didn't I? Who's that?  Well, who is he? Him medicine man. Look, chief, we've come to buy Manhattan island. Yeah, medicine man make all the decisions. We ask him if we should sell island. Uh, uh, uh, uh, uh, uh, uh, uh. Uh, uh, uh, uh, uh, uh, uh, uh, uh, uh, to a whole tide, whole tide. Who do you ask for it? Who do you ask for it, Sissy? Uh, uh, uh, uh. Uh, him say no. What's next question for house? Who buy him next drink. Must war and delegate from Tammany tribe, keep quiet. We're people, these foreigners. Yeah, you keep quiet, too. We burn Peter Minuet at stake. We make Minuet steak. I think I look on Nick's face, see if more laughs. Indian plant corn, Indian eat corn. Indian talk corn. Chief, no like squalls, steel punch lines. She would. Look, Chief, I want to buy Manhattan Island. I have lots of money. What good money to Indian? To buy food. Indian shoot food. Buy clothes. Indian shoot clothes. Have fun.  Fail face crazy. Every time he sat down to eat, tried to stab face with stick. That's a fork, you dope. White squall like British Empire. Sun never set on tongue. Tawaga, woga, hugo. Tawaga. Look, the medicine man's talking with his eyes closed. Yeah, him in solo conference. Tawaga, Tawaga, Guga, wuga. You're fast. Well, you kidding? Very strange. Medicine man have dreams. Says sell Manhattan Island quick. You still want to buy it quick right away, huh? Yes.  No money. Every pocket will do. Okay. Here's some beads and trinkets and marbles and stuff. Altogether, it's probably worth some $24. Some imagination. Yeah. Me take junk, you take Island. Okay, wrap it up. And you sign right here. Me no can ride. That's all right. He can't read. Yeah. Thanks a lot, chief. Me think chief pulls short end of stick. Hey, wake up, medicine man.  Yeah, medicine man speak me translate. Uh, uh, uh, uh, uh, uh, uh, uh, uh, uh, uh, uh, uh, uh, uh, uh, uh, uh. Mm. Medicine man say have dream of Manhattan Island. Many moons from now. He see 1939. He see trouble for Indians if we stay here. Headlights and paper. He see Yankees, more Indians. He also see Dewey. Make it Hartford, Tammany tribe. A worker Mahuga, Mahuga. Oh, no. Also dies committee after reds. Okay, if we feast. Big chief happy sell Manhattan. You take Island. Me take ferry boat. Hoboken, two minutes. So long. You can get supreme quality and Avalon's for less. Why pay more? So the next time ask for Avalon's. And don't forget your change. Yes, Avalon's cigarettes, dear friends, cost several cents less than others. You two can say this difference like all of us Avalon brothers. Each pack is wrapped in cellophane. Each pack is union made. No wonder folks from coast to coast. Never guess, but Avalon's cost only 10 cents plus city or state tax. Ladies and gentlemen, Avalon's time is made possible through your loyalty to Avalon's cigarettes. And we'd like to thank you for keeping Avalon on the air. And we hope that you'll tune us in again next week. Good night and greetings to all the mothers. The Brown and Williams and Tobacco Corporation presents Avalon's time each Saturday at this same time. Del King speaking. So long everybody. Long time has come to you from our Chicago studios. This is the national broadcasting company. W-E-A-F, New York. 9 p.m. B-U-L-O-V-A. Bull of a watch time.


So, what we so far discussing is just like we know, in some applications we will be interested in knowing a priori whether this state is going to be positive recurrent, null recurrent or it is going to be transient like in the insurance kind of applications I will just discuss last time we said that it is important to know whether this is a recurrent state like if I am going to go bankrupt that the state that I am going to become bankrupt is going to be recurrent I do not want to be in that business right. So, a priori if you know or if you could model all the system you want to quickly identify whether this state is going to be positive recurrent, null recurrent or whatever it is. So, we are just now stating what are these results that you can what you can say about that just by looking your Markov chain right say like all this characterization was only simply based on your Fjj values right Fjj values which you can compute from basically your transition probability matrix. So, your transition probability matrix is basically characterizing your Markov chain. So, if I tell you my Markov chain that you can quickly come up with your transition probability matrix from that you can compute this Fjjs and try to identify what are the states how they behave like. Now, based on this you can try to understand how your things are like for example, your transition probabilities you may start with on transition probability matrix and for that you observed these states are transient and these are recurrent and all. But you notice that I am going to be in recurrent state most of the time so this is a bad case I do not want to get into this. So let us change my Markov chain changing Markov chain is equivalent to saying that okay let us consider these parameters you want to look at a different set of parameters that is like you changing your design or like you changing okay let us instead of starting with this much capital maybe let us start with another set of capital or something like that and then you analyze based on that okay what are the states which are going to be positive recurrent and transient then you will see that okay in this I find that my recurrent states are less maybe this will looks like a bit safer bet and then I want to maybe go with this. So, what we are just trying to do is given this set of inputs given one description of Markov chain see Markov chain as long as somebody tells you okay these are the transitions happening you have a Markov chain you are identifying but that Markov chain it can change if you change the parameters right your transition probabilities once you have a new set you can do the same thing. So you can do this analysis for any given transition probability matrix and based on that you can say maybe like you say that okay this transition probability looks like it will lead to a situation where things are less risky and maybe as a designer or the guy who is actually going to implement them maybe he will see how he can get those parameters okay fine. But for this computation of all this set null transient and null recurrent we are really no don't need to worry about my initial trans probability I only need to worry about transition probability matrix because as you see when I was trying to do all this calculation I state given this state I kind of fix my state and from there I am trying to analyze. So the initial distribution is not affecting much right because you are already fixing a starting state and after that you are analyzing initial distribution only affects in which state you are going to start from but that you are already fixed now what is only going to decide how your future is going to have always your transition probability matrix. So that is why I am saying to analyze this all you need to know is only transition probability matrix okay. Now we know that we have this different states as of now we have classified them into two broad categories transient recurrent and further positive recurrent and null recurrent. Now is it possible that all my possible states I can group into these classes or like let us say I can group it to some classes and what is the property of each of this class. So let us try to understand that through what we call as communicating classes okay let us define something. So let us take a Markov chain where the state space is some countable set okay as we always denoted by s as I mean I am again denoting it by s which is countable. Now we are going to say that so let us take a DTMC with countable set of states now take any state j and take another state i I am going to say that j is reachable from i that means I am going to hit state j starting from state i if there exist n positive such that this n step transition is the probability of n step transition is going to be strictly positive and in that case I am going to denote that i goes to j that is from i I can reach state j. See as long as what all I need is to reach j from i is at some point at some time I should be able to reach it with some positive probability okay right now this n could be very large but that is fine as long as I can do it in some time that is fine. Now we are going to say that you take two states i and j if I can reach j from i and vice versa like also if I start from j at some point I am go back to state i if that happens then I am going to say that state i and j communicate okay. So if i is reachable from j and j is reachable from i then I am going to call i and j communicate okay and that communication symbol I am going to use it as this double arrowed line okay when I say j is reachable from i I am going to use single arrow taking i to j and when i and j are communicate I will use this one okay. Now this is a relation right what is this relation this is a communicate relation. Now we can show that this relation equivalence relation so how many of you know equivalence relation what are the properties reflexive symmetric transitive property what is reflexive property self okay. So reflexive property does this relation satisfy reflexive property so then for that what I need to check i go into itself right. So if this is the case what I need to show i is reachable from i and again i is reachable from i to show that what I need to show there exist some n positive sorry n greater than or equals to 0 so that I can go from i to i with positive probability what is that n 1 or 0 0 right because p i i equals to 1 by our definition if you are in i you go to there in 0 time with problem this that is the meaning of p i i 0 that is what we have defined like right. So that means with probability 1 if you are in the state in your state you will be there in 0 at the round so this trivially holds. Now second is symmetric what does this mean j then this implies is this true so what is sign this is so what is i let us take i communicates with j means right i is reachable from j and j is reachable from i so that means there exist some positive probability some positive finite time in which I should be able to reach one from the other right that is same as so here you start from i and go to j and then when you look at the other direction you start from j to i and just do I mean the same thing here yeah this is this is also other direction if you this is also but in this case we just need to argue that this implies this if you start with this then you can say that this implies this now what is the next so what is how to show that is this true so how you are going to show this so let us show one direction like i communicates with k. So i communicates with j means in some finite steps I go from i to j and again j communicates with k means in another finite step with positive probability I go from j and k so if you take this product I know that in this many rounds at least with positive probability I go from i to k and similarly the opposite direction so this relation is an equivalence relation so what is the property of an equivalence relation this relation partitions but the thing is it partitions your class so you have this state space S this relation if you apply it is going to partition your S into what you are going to call as equivalence classes so that means all the states within when you have partitions and if you take one particular partition in a way all the states in that partitions are going to be equivalent that is what we call as equivalent class and here all this we have so many classes so we are going to call them as equivalence classes okay fine we have some more definitions once you have a equivalence relation this is clear that it partitions now equivalence class is just like the classes we have in this partition we are going to just call them equivalence classes okay so proof for this work out yourself just ensure that you will not get any overlapping sets okay now let Xn be on S so let us say I have a DTMC with transition probability matrix P okay then so we are going to as we said S is going to be partitioned by this equivalence class and the different classes we are going to get we are going to call it as communicating classes okay because we know that each pair in that particular class or a particular partition is going to communicate with each other right so we call them as communicating class classes now you take a communicating class so if you are going to take one communicating class and you are going to take one element in that communicating class and take state which is outside this communicating class okay and if let us say I belongs to that communicating class C and J is outside this communicating class if Pij is equals to 0 then we are going to call this communicating class C as closed that means the probability of you going from any state of this to outside state of this class is going to be 0 that means you are not going basically outside of this class to any state so you are that is the kind of close class you have and if this does not hold this property then we are going to call it as open communicating class okay. Now we further say that the transition probability P is going to be irreducible if there is only one communicating class that means entire state space is just one communicating class at and in this case so and this case we say the Markov chain itself is irreducible okay again to define this communicating classes all what I need to know is this enough if I know my transition probability matrix yes right because to define this I need to know all this instep transition probability matrix but we are in the case of time homogeneous Markov chains for this this Pij's instep transition mobility can be obtained from my one step transition probability matrix so all this properties that I have defined that this relation whether it the and whether my class is going to be communicating and whether my class this communicating class is closed or it is going to be open all I need to know is just my transition probability matrix from that itself I can define all these things and now whatever the transition probability I am going to deal with I am going to call it as irreducible if that leads to me only one communicating class so that my state space is not going to be a partition it is just going to be one communicating class so once I say one communicating class I am should be reachable from one state to another state within that communicating class so that is why I am going to call it as irreducible you cannot partition into two parts that is the case so in that case we are just calling our Markov chain itself is irreducible okay a quick example on this let us say I have some states like this where can go from here go from here go from here go from here and also go from here so the points where I have put this arrows that means these transitions are possible with positive probability where the link does not exist that means that transition happening with zero probability so let us call the states 1 2 3 4 so how many states are there here 4 states and how many communicating classes are there here what are those communicating classes 1 2 because I am able to reach 1 to 2 and also 2 to 1 and what is the other communicating class 3 to 4 right this also I am able to reach 1 from each other here and now is this communicating class a close communicating class so this is one class let us call the C1 and this is called C2 is this communicating class a closed one why is that yeah we can go with positive probability from a state in this class to a outside state C2 whereas this C2 is going to be a close communicating sorry it is a closed one yes right because we are not going anywhere outside so what if if I add another states here and allow this to happen so in this case if I add this will this 5 becomes goes inside C2 but can I reach from 5 to 1 so I can go here with positive probability from there I can go here right in that case I will include 5 also in that but suppose if I remove one link here it is a that can become another itself class right with just one element in that and in this case 1 and 2 is going to be closed one it is not going to be closed one right because it can escape from this 2 to a state which is outside this class now we are saying closed this one we are going to say closed if it do not have any escape route to go outside outside a state it is some positive probability so in this case when I wrote this right if do you have any escape route to be outside go outside to state other than 1 and 2 you go 1 and 2 and then 1 and 2 you may just be like that right you will not possibility and here there is a possibility for you that you escape from being in C1 you can go to here right because there is positive probability now if I just say there is I add one and then let you go there then you are able to escape from this class so that is why it is no more closed can now comes it is called class property theorem if there is only one thing what is there to define closeness it is just going to be closed right there is only one state in that and the definition becomes kind of a vacuous here it is just one state and you have to going to that state okay so when I say it comes to this state and it is not going anywhere means it remains in this state always it is just staying in that not moving anywhere okay now the states so this theorem is stating that the communicating class we have will be such that all the states in a communicating class are going to be just one type they are all other going to be all transient are going to be all null recurrent are going to be all positive recurrent it is not possible that you have a communicating class that will have some states to be recurrent and some other states to be positive recurrent or null recurrent it is going to be either one of them okay so let us quickly look into this why this is the case suppose let us take on communicating class and let us take let us C be a communicating class and take ij belongs to C so if ij belongs to C what I know what I know there exist R and S such that pjk of R is going to be positive and pkj of S is going to be positive right this is by definition if i and j belongs to the same communicating class it must be the case that I am going to go from j to k with positive probability in some finite steps and similarly I should go from k to j in some finite step with positive probability now suppose now I want to claim that if I assume that j is transient I want to prove that then k is also transient and if I assume j to be positive recurrent then I want to show that then k is also positive recurrent and if j is null recurrent then k is also null recurrent so let us see why that is the case okay let us first look into the case where I want to go from j to j in this much steps R plus n plus S this R and S I am going to be take this whatever this value n is variable for me any n now what is this this is probability that X R plus n plus j or S is equals to j given you are going to start from X naught equals to j right this is the meaning now what I will do is I will instead of going from state j from the beginning to again state j in R plus n plus n steps I will want to reach this while going to while going to some other some states in between also so let us further condition this by saying that R n plus S equals to j what is that conditioning I want to add X R equals to k then X R plus n is equals to k given X R equals to j so what I want here is yes instead of directly going here you I also want it to be first reach state k in the first R steps subsequently in the next are n n steps that is R plus nth round I want to again hit state k and from there in the next n steps that is R plus n plus S round I want it to state j so is this probability is going to be lower bound for this yes right because I added these two extra conditions on this I basically asking this Markov chain to go through this special state at these steps not just looking at the final step here so if you just now apply the chain rule here what you will get is so already applied the chain rule and the Markov property here and by definition the first term is going to be jk R rounds then pkk in n rounds and then p going to be jj now this is k here this is going to be kj in s rounds it is correct so I have just applied chain rule and apply Markov property and this is just by definition and now what we have basically done is pjj R plus n plus S is going to be less than or equals to pjk R pkk in n steps and then pkj in steps now I want to use this property first thing you can check that pjj of n you are going to be pjj s this I am simply going to take n this is not equals to 1 here and this one here so this is because I am just taking this this guy is a lower bound sorry this guy I have lower bounded and then summing it over all possible n's and now further I am going to lower bound this by this inequality which I have obtained here so here s is the variable this R and s are fixed and I know that this pjk R and pkj s both are strictly positive quantities so now let us use the property that we have stated in a theorem suppose if I assume j is going to be transitive let us say let us assume j is transitive sorry transient so suppose if I assume j is transient what I know about this quantity as n goes to infinity we just know it is to be finite right I know that if my state j is transient this is going to be finite right and I know that this quantity here is some positive quantity and this quantity here is again some positive quantity if that is the case then what I can say about this quantity this is also finite right then what would we say in that theorem if that the theorem when the summation pijn is finite if and only if j is transient right that theorem was if and only if case so now if I assume this quantity to be finite let that is following making assumption that j is transient then I already concluding that this guy is also finite what does this imply what is transient so that implies k is also transient so what if I take any two states j and k in my class what we are concluding is if j is transient so is k so it must be the case that any states in this communicating class they must all be transient right okay so now let us look at the case where I is sorry let us look at where j is now we want to look at null recurrent let me see what I want to look at okay now let us say null recurrent now we are going to say that if j is null recurrent then we were going to show that k is also going to be null recurrent okay suppose let us say j is null recurrent and if I assume that that implies k is transient but then I am contradicting my first statement right suppose k is transient then it must be the case that j must be also transient but I am assuming that j is null recurrent so by this counter positive argument this cannot be the case so j could be either null recurrent or positive recurrent so let us see what it is so we have to then use the another result we have used where instead of looking at the sum we look at the averages okay so let us try to look at the average and then this I am going to write it as 1 by n which just we have few steps will be done okay what I am doing here is basically I am taking this 1 by n and summation of n terms here that is the average of the first n sums here and then in this I am going to drop out all the terms before r plus s and looking at the sum beyond that so basically here I am assuming that n is large it is at least r plus s right that is why it is starting from this I am only having few terms here that is why it is a lower quantity and then I am just re-indexing it so instead of starting from m r plus 1 I am going to start from n l equals to 0 and then re-indexing this quantity and also okay when l is 0 this is going to be yeah I am just re-indexing this quantities equals to this quantity and then I am going to use this relation that I have already got that is l equals to 0 to n what I have got pjj of r pkk of n and pkj what did I do so this we have already shown that this can be lower bounded like this now I am going to do a little bit manipulation I will pull out this term and this term outside so this should minus r plus s right pkk of l times n by n times pkk so there are how many terms here l goes from 0 to n minus r plus n r plus s they are just we are done now what I am going to do so is this clear I have just pulled this outside and that is the reorganized I have taken 1 by n and this quantity here but now look at this this is summation of how many terms here l equals to 0 to n minus r plus s right so what I will do is simply this n I will keep in the numerator n minus r plus s and n minus r plus s I have just multiplied and divided by this term now if you look at this quantity here this is now average of n minus r plus s terms right so there are how many terms there are n minus r plus n terms and I am also dividing by the same number of amount okay now let us now let us apply our result what would we say as n goes to infinity if I let n goes to infinity and this guy j is null recurrent what did I say where does this quantity goes to limit n tends to infinity where does this go so this goes to 0 as n goes to infinity and now as n goes to infinity you look at this there are n minus r plus s divided by n what does this ratio go to that ratio goes to 1 right because r plus n this is a constant term that does not add much and what does this ratio will go to this ratio is going to be same as limit as p k k l as sorry n as n goes to infinity right because you are just skipping some finite terms here if you look into the infinite summation of this they added by that same number here this is going to be the limit of so this quantity is going to be a limit of p k k of n so this is going to be let us say goes to 1 to m or may be m goes to infinity m goes to n equals to 1 to m okay so this quantity is just going to this so now we are what we have shown is as j goes to sorry as n goes to infinity this term is going to be 0 and this term is 1 here but this is going to be positive quantity this plus term is also going to be a positive quantity in that case what we say this limit must be 0 right because this quantity is upper bounded by 0 so then what does this mean the j is also the yeah if j is null recurrent then k is also null recurrent so now last thing suppose now let us say if j is my positive recurrent it is because it cannot be transient or null recurrent so only option you are left with is positive recurrent.


The Blue Angels FA-18 Hornet. Achieving a top speed of up to 1,400 miles per hour. With a rate of climb of 500 feet per second. With a fuel capacity of up to 9,900 pounds. Powered by twin 404 GE400 engines, able to produce 16,000 pounds of thrust. This versatile machine is one of the Navy's greatest aerial assets and the Blue Angels demonstration aircraft of choice. And throughout the 2018 air show season, this aircraft and others like it on our team will travel to a city near you.


ARL Central is the latest extension of the U.S. Army Research Lab to the Greater Chicago Region. It really brings the expertise of the surrounding universities, of industry, of national labs, in a partnership with ARL so that we can advance research faster, we can make new innovative discoveries to a point where we can get it into the hands of the soldier. The Greater Chicago Area has a long history of working in science and technology specifically for the military. Now we're formalizing this in this kind of framework, ARL Central, to support critical research required by the Army. In national security, we obviously have to be sensitive about what we're doing, and that tends to limit the ecosystem upon which we can draw for the best talent and the best ideas. So the concept of an open campus at the Army Research Laboratory gets us the ability to start interacting with communities that have great science, great technology, so it drives this collaboration. The benefits of partnering with ARL are that we're bringing a lot of challenging problems in an Army context with looking at the future Army that we need to be. Mission focused research forces people to think beyond the way they would normally think. The University of Chicago has always been an incredibly strong place for fundamental science, and increasingly a place where applied science and new ventures are at the cutting edge of what's happening with technology and science, together with our researchers and how that could help the Army is a real advantage. The relationship between Northwestern and ARL Central is related to the strengths of our University. In material science, we have worked for many decades to improve the materials that are brought to the warfighter. There has been deep engagement on the Army side and on our side. There's a lot of mutual benefits between the Army Research Laboratory and ARGOM. They can come do the kinds of science and technology advancement using facilities at the laboratory that they may not have. In turn, it gives us an opportunity to take the great science we've been doing for fundamental discovery and make it useful for the military. For the soldiers, one of the most important is the proportion. So it is a really natural fit for ARL to form an organic partnership to advance the new technologies in the proportion area. We're really bringing a sense of purpose, a sense of mission. We have researchers that are working with soldiers, you know, on the technology of the future for them. It's a tremendous group of people, young people, full of ideas. It is not only important but very exciting for us to know that we're doing something that is really relevant and useful. The mission of the scientific push as well as the mission pull produces a research that is much better than something that is either one or the other. At the end of the day, we all go into the field of science and technology because we want to have impact. ARGOM scientists can now have impact in areas that otherwise they could not have this partnership not existing. My vision for ARL Central is we're going to form strong bonds with all of our partners here in academia and industry with other national labs so that we can provide the best technology to the soldier in the conflict of the future.


In Wales, our world-class training arrangements ensure trainee GPS like David enjoy dedicated learning time and support from an educational supervisor in his practice. With flexible training opportunities and more affordable living, David found everything he was looking for in Wales. From working in a friendly community he loves, to unwinding on some of the best beaches in the world. In Wales, we don't just offer our GPS a career, we offer them a way of life. And with all this on your doorstep, there's something for everyone. Find your work-life balance, just like David.


Plants are subject to various environmental stresses, including drought, which can have a significant effect on their growth and yield. Drought stress causes a number of morphological, physiological, and biochemical responses in order to reduce water loss through transpiration and maintain photosynthesis. These include stomatal closure, increased production of antioxidants, and the accumulation of osmolants and aquaporins. Some plants are able to tolerate drought better than others, and this has been studied extensively. Techniques such as breeding for drought resistance, selection of cultivars with high drought tolerance, and irrigation management can be used to increase the tolerance of crops to drought. Additionally, understanding the mechanisms of drought tolerance in plants can provide insight into how to improve the ability of crops to cope with drought. This article was authored by Dirty Kapoor, Savita Bardwaj, Mark Olandi, and others.


Yes we can hear you. Just a second so that you appear on the screen. Great. You're sharing your screen so everything's good. Okay so the floor is yours. Thank you. Okay thank you Pasco for the introduction. So every day when we read emails browse Facebook or have a talk with colleagues via Zoom. Our security is insured by authenticated encryption or AE, a tool that delivers both privacy and authenticity. The conception of AE emerged in the year 2000 as a way to add authenticity to classical encryption and it then quickly moved from theory to practice. Since then there have been a number of efforts in strengthening the original notion such as unique nonce AE or misuse resistant AE. These efforts have led into widely used schemes in the TOS IPsec and many other network protocols. Still despite the huge volume on AE works, essentially everything still falls under the umbrella of privacy and authenticity and have been a long helped believe that these notions are enough for practical purposes. Surprisingly in a recent attack known as the partitioning oracle attack, land, grubs and distant past shatter this common belief. They demonstrate a break of standard encryption in the context of password based encryption. So in this setting the adversary tries to exploit a server as an execution oracle to recover passwords. Strangely the attack has nothing to do with authenticity or privacy of the encryption scheme. Instead the adversary tries to find some form of multi-collisions on the server text of passwords in a dictionary. If we can find an S-way multi-collision it can speed up by a factor of S compared to a brute force attack and for standard encryption schemes S can be as big as thousands making the attack practical. So the partitioning oracle attack demonstrate that in contrary to common belief privacy and authenticity are not enough and it's not the only place where things go wrong. Consider Facebook message franking. So Alice and Bob have an encrypted conversation via Facebook messenger. Now thanks to you and inclusion Facebook doesn't know your key anymore. Still it allows you to report abuse but you have to send the decryption key for verification. Unfortunately there's a way to practice mechanism by sending a fake key and associated data. It can turn Alice's deny message into a malicious one and she would be wrongly penalized for that. Again this has nothing to do with privacy or authenticity because the adversary Bob in this case knows the key. Instead what we need here is a new notion known as committing security. In formal latest means that the server type should be a commitment of the key and for some applications such as message franking you want to commit the message as well. Although committing security is new it captures what practitioners intuitively think about encryption schemes. Unfortunately this intuition is not supported by any standard day schemes yet it is urgently demanded by many applications such as subscribe with Google or the OPEC password based exchange. There have been a couple of works on formalizing committing security but if you look closely anybody speak a slightly different dialects. Some wants to commit just the key. Some wants to commit everything. Some work deals with the old-fashioned probabilistic AI. Some deals with the standard the unique nonce AI. Although even defies a new type of AI with a different syntax there's currently a lack of a unified systematic treatment for committing security and partly because of this current solutions are somewhat frustrating. Earlier schemes tend to be pretty slow as slow as hashing the message. Recent schemes are faster but you consume more bandwidth than necessary say two additional blocks per message. In addition why unique nonce AI schemes like GCM receive a lot of attention from both industry and academia. There has been no effort in fixing misuse resistant AI schemes like AES, TCMS, IP. Since this work we give a systematic treatment for committing notions and based on that we build direct transforms UTC and RTC for unique nonce and misuse resistant AI. The UTC transform is a generalization of a prior work by Albertini at all but it allows a more efficient instantiation. Now while UTC and RTC works generically they still demand one additional block per cybertext and overhead for small messages is still large. To bridge the performance gap we view committing variants of a CCM and AES-TCM-SIB. Compared to the original schemes these variants introduce very tiny changes there's no subject expansion and the overhead is very little. Moreover let me mention that prior work on committing security only provide analysis in a single user setting. In contrast here we provide strong multi-user analysis for all schemes in our paper. Now let me give you a visual comparison on the performance of our unique nonce schemes CUC1 and UTC with a prior work of Albertini at all that is deployed on Amazon cloud. Now as illustrated by the picture our schemes are a lot more efficient than the prior work and for CUC1 the overhead is negligible even for small messages and the same trend continues for our misuse resistant schemes. Now in today's talk I'll introduce the framework of committing notions and I'll show you how to achieve that in the context of CCM. But before we get into that let me begin by briefly reviewing the syntax of AES schemes. So syntactically an AES scheme consists of an encryption algorithm and a decryption one both deterministic. Encryption text as input of key K, a message M, a nonce N to produce a cell taxi. Conversely decryption text the key a cell text the nonce to produce either a message M or a symbol integrating invalidity. They also take an additional argument known as associated data or AD. It is a string that for some reason cannot be encrypted but still should be authenticated. For example if you encrypt neural packages then you cannot encrypt the headers because routers need to see that to move to the next hop. So in that context your message would be the payload and your AD would be the headers. In practice the AD is often short. For example happy headers at most 60 bytes. Now let me give you an overview on the committing definitions. It turns out that there's no one size three all notion but a hierarchy of definitions. At the bottom we have the commit once notion that commits just the key K and that's enough for the partitioning protocol attack. At the very top we have the commit for a notion that commits every for all the four components in a couple KNAM and that is demanded by message franking. In between we define an intermediate notion commit three that commits the three components KNA.  In formally once you can commit KNA the message M is automatically committed thanks to the correctness of the description but there's a definitional separation between commit one and commit three and this matters not just for security but also for efficiency because it turns out that the top two notions require hashing the AD but the bottom one doesn't. Luckily in practice AD is short so we can realize commit for security in a cheaper way than hashing the message and the AD as many current solutions. Now let me describe the definitions in more details. So here the adversary needs to produce two tuples K1 N1 A1 M1 and K2 N2 A2 M2. Its goal is to produce a collision while we equip these tuples. Now to avoid trivial wins the commit one notion requires that the keys be distinct. Commit for is more relaxed it requires that the two tuples are different. Now here the adversary wants to produce a collision we can sterilize the definition a little bit requiring that the adversary needs to produce an S-way multi collision instead of a mere collision but now let's take a step backward and think about that notion it seems that somehow the sterilization is redundant in fact if the adversary cannot produce a collision surely it cannot produce a multi collision so the special definition already implies the general one so why bother to sterilize so it turns out that the sterilization gives us a trade-off for resisting auditioning Oracle attack. Now let's say that you if there's no collision whatsoever then our schemes can only deliver the standard birthday bow in committing security but if your applications can tolerate a little speed up in the password guessing then we can deliver much better security say 96-bit security now remember that there are two main notions commit for a commit one once you commit just the key the other ones you commit everything now it turns out that once you can commit the key it's easy to commit everything else and we achieve that via a genetic transform chemical hash then encrypts in particular here we have a base A scheme that is commit one secure and we want to leverage it somehow to come to obtain commit for security now remember that commit form is that you need to commit k and a once you can do that the message m is automatically committed so our first step is to hash k and a to derive a synthetic key we then use that synthetic key to include the message m not because the base A scheme commits the synthetic key so the cell attacks is a commitment of k and a if the hash function is collision resistant and as a result the overall construction has commit for security now let me stress that here the ad is processed only once because in the encryption we encrypt with an empty ad so the hash the encrypt transform has no cell text expansion and actually its overhead is optimal remember that commit for demands hashing the ad whereas commit one doesn't so here the hashing cost of k and a is somehow unavoidable security wise if the hash function is a good pf then the transform preserves both unique nonce and misuse resistant security for the a scheme and as i mentioned earlier if the hash function is collision resistant then the transform promotes commit to one security to commit for so now we have a nice way to obtain commit for if we know how to get commit one so now what's left is to see how to get commit one in a clean way and i will show you how to do that in the context of gcm so remember that commit one means that you need to commit the key somehow so the intuition is that you need to hash the key probably with something else and then include the image in the same text but how long should the hash output be the conventions would tell you that you need something like 256-bit output to prevent offline attacks why here we don't want an adversary to spend a lot of pre-computation to get a collision and then attack the committing security with anybody with constant cost per user but if you look at the applications of committing security more closely then typically one key is random sample and that rules out the offline attacks so actually here you only need 128-bit output moreover here you only deal with a very short input the key k so you actually don't need a fully flashed cryptographic hash function to now paper we instead use the davis mayer construction on a yes davis mayer is known to provide good collision resistance in the idl cipher motor and in our work we show that it continues to provide good multi collision resistance even better that's the best you can hope for from a one-to-date-bit output hash function so you have a good way to hash the key but how would you include the hash image without expanding the same text in order to answer that question let me now give you a bird eyes view of gcm so remember that gcm is basically encrypt and match so here the inclusion scheme is counter mode and the map follows the catwalkman paradigm in particular in the map you first use a universal hash function known as ch to process the seven x car c and the ad and you then use a one-time path to increase the ch output to derive tactic now let's zoom in the markers just a little bit further and the one-time path is produced by making a call to the block cipher so in our scheme chc1 instead of using a one-time path we directly use the block cipher to include the chs output it is also a very common variant of the catwalkman it has authenticity but itself doesn't have committing security to make it committing we add an extra xr turning the block cipher code into davis maya construction another tactic becomes a commitment of the key k as a result the commits one security of cuc1 can be reduced to the multi collision resistance of the davis maya construction for unique non-security cuc1 retains the security of gcm actually for short tax the situation is even better gcm is known to be very poor if the tax is short but cuc1 does that and it's an issue so even if the applications don't need committing security there may be a good reason to switch from gcm to cuc1 summing up given urgent demands from many applications we believe that it's time to upgrade encryption standards to committing ones the variance in our work allows us to do that with tiny changes there would be no server tech expansion and nowhere is little and the transition for astc cms ivy is particularly attractive because the scheme is new there's not much legacy issue and what you need here is just an extra xr thank you for listening to my talk and i'm happy to take questions now vietong yes no no i'm just asking in the room if there are questions for you so i'm like okay yeah good all right uh germy blocky Purdue University uh so uh you mentioned uh that you don't need 256 bit uh hash outputs for 128 bit security i was curious uh is this inherent if we require s equals one or is this just a feature that we can exploit when the number of multi collisions is larger than larger than two okay so uh for the actually for the base case where s is two namely a mere automated collision then we can only provide birthday bow security but if you look at the base schemes gcm or so they only provide birthday bow so in the basic setting where s is two then you have birthday by security but as i visualize in the slides let me see where it is here do you see the screen so if you allow a bigger x say at is far so then you have 96 bit of security and the bigger the s is uh the better security about you have okay okay so um i guess if we want no speed up though in password guessing we still have to go back to 256 bits um if if you want uh one today bit security then probably so okay but yeah okay thanks any other questions if not then uh let's thank uh viet Tong again


dystopian time. California officials tell CNN they anticipate nearly all salmon in the Sacramento River will die due to abnormally hot underwater conditions as heat waves continue. Now if that wasn't bad enough, the Amazon rainforest no longer is a carbon sink as it now emits more than it takes in. So parts of the Amazon are now emitting carbon rather than absorbing it, scientists say. Now I just want to remind everyone that the Amazon rainforest is often times referred to as the earth's lungs and to know that our lungs, they're not doing what they should be doing. I mean this is, it's catastrophic to put it frankly. So we're only going to read a little bit from this article, but I just want to let you get a sense of how pressing this matter is. So parts of the Amazon rainforests reduced capability to absorb carbon emissions could contribute to global warming in the future according to a new study. Deforestation of the Amazon and rapid local warming in some of its parts have decreased or eliminated the rainforest's role as a carbon sink, a place that absorbs more carbon than it releases. Professor Luciana Gatti and her colleagues have collected air samples from four regions of the Amazon roughly twice a month over the span of nine years to measure the tropical forests atmosphere. The scientists from Brazil's National Institute for Space Research analyzed the data to determine how carbon dioxide and carbon monoxide are concentrated in different parts of the Amazon. The air samples were compared with those from the South Atlantic ocean, which has been found to absorb as much as 75% of the excess heat and 40% of human-generated carbon dioxide CO2 emissions taken up by the oceans. The scientists also used the CO2 concentration gradients to estimate how much the carbon emissions increased or decreased in connection to vegetation growth, decay and forest fires. Millions of trees have been lost to logging and fires in recent years. An unprecedented number of fires happened in the Amazon in 2019 and 2020, which reduced swaths of the forest to scorched earth. Now the individual who accelerated this process, I just want to offer thoughts and prayers to this person who privatized the rainforest. Jair Bolsonaro, president of Brazil, who currently is in the hospital for hiccups. I'm not sure if you all have had hiccups that you've had. He had them for like 10 days, right? Or currently it's a 10-day hiccup fit. I couldn't have happened to a better guy. I'd rather it be cancer. No, no, no, not sorry. Or spontaneous combustion. His lungs just combusted. That just gets it. Actually, I was actually literally thinking, what's it going to take people for a genuine action on climate? And I realized, I really, I truly think that the only way we're going to get it is if rich white men's genitals just start spontaneously combusting. Something like that is going to be the only way. Pretty much any other thing they can stock up on, like enough reserves for five decades and hang out in their air conditioned underground bunker. But if there are dicks in danger of being on fire, then they have problems and we might just keep your fingers crossed that that starts happening. I kind of love that. I was always one of these people that's like, you can't wish bad things on other people. I don't believe in karma, but I mean, you put in the universe or you get out of the universe what you put into it. So I always try to avoid that. But honestly, with people like Jair Bolsonaro, who are literally killing the planet, it's like, good luck, man. Good luck. What kind of self-defense at some point? I feel like you can make that case, right? Like, I don't feel bad that he has a really bad case of the hiccups, which obviously this is probably something that is a bigger issue, right? He might need surgery.   Yeah, let's be honest, that might be what like on the totem pole of horrible things he's done in his life. That might be somewhere in the middle, you know? He's just, there's no shortage of horrible things that this man has said and done. And yeah, so I'm all for the dude spontaneously combusting. I think it would solve a lot of problems. Yeah, whoever thought that hiccups would be one of the great protectors of the environment because so long as he's not there to implement these terrible policies, you know, then hopefully we'll all be better off. I do have one more story with regard to climate change. So in the Pacific Northwest, we just experienced a terrible heat wave unprecedented. It was 113 degrees. I want to say maybe 115 degrees. I have never in my life experienced that. I mean, in the Pacific Northwest, we're not used to this at all. Going outside, it felt like the air was getting sucked out of your lungs. All of my mom's bushes were like burned. It looked like they were hit with a torch. But now we're kind of learning about how bad that was. And unsurprisingly, it was really bad. So climate change is fueling mass casualty heat waves. Here's why experts say we don't view them as crises. Wait, did I read that right? Say we don't view them as crises. So the Pacific Northwest heat wave in late June was a mass casualty event. And that sounds so weird saying to live through that. I mean, thankfully, I'm fortunate to not be without air conditioner and whatnot. It's comfortable. I didn't really have to leave home except for a couple of times. But to know how many people were suffering and, you know, I was just kind of like kicking back playing video games. It's a little bit haunting. But nonetheless, it was a mass casualty event. Hundreds of people likely died in the multi day record breaking heat and the death toll continues to rise. Officials are still investigating the cause of dozens of deaths that occurred during that time, but at least 83 people died from heat related illness in Oregon, 54 of which were in Multnomah County, which includes Portland. Many of those people were older living alone and without functioning air conditioning, according to a preliminary report on excessive heat deaths released by the county Tuesday. Before this report came out, I actually learned about this that people in old homes were basically literally being cooked to death and they had nobody to talk to to reach out to. And some patients weren't necessarily cognitively able to ask for help or dial 911. So, you know, when I think about this, it's this is like the beginning of what I think is going to be real climate catastrophe, not to get too demerit, but it's just like we're all really at this point to where we can no longer talk about how climate change is coming. The effects of climate change are here and it's really, really, it's pressing. And already in the Pacific Northwest, there are wildfires starting and last year in 2020, when I took a week off for my wedding anniversary, the air quality got so bad that I believe it got to like, it was past what they determined like hazardous. I'm trying to remember the numbers. I forgot about the system. I had the numbers of the air quality index like memorized and burned into my brain, but you can smell smoke like through your home. And so I just want to like, I'll pass this off to the panel because there's not necessarily much that I can take away from this other than we're doomed and hopefully like you all are a little bit more optimistic or have some words of encouragement, but talk me down from the ledge because I feel like I'm I've taken the black pill. I'm on the ledge with you. Unfortunately, I think I mean, we see this it's it's climate change and all these disasters right in front of our eyes and everyone's like, Oh, I don't know what I can do about it. And like same thing with gun violence, we're like, Oh, I don't know what we can do to stop all this. And like, people are dying because they're afraid to go to the hospital. Because they don't want to end up with a lifetime worth of medical debt. And our politicians are like, I don't know what we can do about these are unsolvable problems. I'm just sitting here like, I mean, I tried to run for Congress and change some stuff. But sometimes I feel really helpless, because you're just sitting here like watching it happen. And we, I mean, we're like millennial slash gen X. And it's like, I mean, we're living in these times where we can watch this all unfold with our own eyes. And I don't know what it's going to be like for people the age of like our hypothetical children. I don't know what's going to be left. Yeah. Yeah. You know, a few weeks ago when I started reading these stories of people being cooked alive or just dying from heat strokes and whatnot, I had a thought, which was, you know, like basically like, oh, great. Now, not only do we need like universal health care, but we need universal air conditioning, like air conditioning. We're getting to the point where that's going to need to be right. And I actually tweeted it out, like some version of that, but probably worded in a joke, but but somebody some some brainy, so it is a lot smarter than I am. Like like basically responded to my tweet, saying like, yeah, it's it's a great thought. But actually, you know, that that would probably actually accelerate global warming, especially, especially in like city environments. And he like showed me like some some papers on it. And I was like, Oh, no, there's no escape air conditioning or not. Um, so like, there, I don't know. Yeah, I guess I'd love to be able to talk you off the ledge. But yeah, I think we're all we're we're all there. And just, I don't know.  Yeah. Yeah. And you know, we've been talking about climate change on dystopian times, because I mean, it's one of the most dystopian things, you know, we're living through it right now. And it's only going to get worse.  You know, we don't necessarily know what the outcome is going to be. And I'm paraphrasing. But as long as we go down fighting, if we go down, then that's the best that we can do and hope for. And I mean, we can't like I try not to be too down on myself. Like I try to eat less meat. I try to not drive as much. But at the individual level, we're not really going to make a difference. I mean, what this comes down to is just a few corporations who are responsible for the overwhelming majority of greenhouse gas emissions. And what's really frustrating to me is we see the negotiations with regard to infrastructure and climate change is consistently on the chopping block. And you know, we're hearing from Joe Manchin, who's talking about how, you know, he's really nervous about the excess in this bill and how there's there's no way to pay for it. And I mean, when it comes down to it, like if people are getting cooked alive, why are we having this? Like there's no debate. It's not just that. I mean, like the building that fell in Miami, and then there was a neighboring. I can't imagine what it would be like to live in the building that was like directly next to it.  But I mean, I just think we're inevitably going to I think that like we're going to look back on that and and be like, Oh, okay, yeah, yeah, like, I mean, there's going to be like 50 more occurrences like that or more. And and that's that might be sort of like, I don't know, just just a, I don't know what you'd call it, just a day that that just everyone refers to as like, Oh, yeah, that that was the first time that happened. You know, and it just I guess the last thing I'll say is just these sorts of stories. I mean, some of these things are things that I guess we could foresee occurring. But some of them don't occur lately, you don't imagine that salmon are going to get cooked alive in a river. You know, you don't think of that. So it's just like, there are all these dominoes that topple that just we never see coming like, Oh, okay, climate change is going to turn squirrels into kaiju size carnivores that eat humans. Okay. All right.  Yeah, yeah, it's like, okay, we I didn't see that coming. Yeah, that and that's just it. That's what really makes it so daunting. It's because this is a wicked issue where it has a ripple effect, right? I mean, we're not just talking about the global temperature increasing, you know, the impact that this has on the ocean ocean acidification desertification, the impact that that'll have on a food on people's lives where we'll see mass migration wars over water. It's just like for me as a political commentator who tries to talk about climate change a lot. I don't know what to say because I feel like responsibly, I would present you with the facts and the details and then I'd say, okay, this is the call to action. This is what you can do. But at the same time, I think that I'm in this phase where I'm just accepting the inevitable and trying to work with it. Like I've moved on from just talking about climate change mitigation and now we're in the stage of now it's time to also talk about adaptation because it's here, it's no longer like a theory or in the distant future. I mean, when people are getting cooked to cooked alive, like where I live, it's it's so it really puts it in a different light in your head and it kind of messes with you. Like it feels like we're in those those apocalyptic movies. And now like in the last couple of years is the first time I feel like I've actually grappled with it.    You know, kind of kind of moment because it just I mean, that's hopefully I'm wrong. Hopefully it's like, but that's that's the way it seems. And I don't know, I think an interesting thing about it. And it's also depressing morbid, like all of this is.  Oh, it's China's fault. You know, just the whole bunch of what about isms and just pointing the fingers at other things and and not accepting that maybe we could do something about it. You know, what else just really gets to me is I feel like, you know, people like the government corporations, they keep pretty much saying like it's on us, like, oh, recycle, like, yeah, definitely recycle, I recycle everything. And this is like a five year old Gatorade water bottle that I still use. But it's, you know, it's just like, I was visiting my family in rural Pennsylvania this past week. And you have to drive your recyclables 30 minutes to the nearest like recyclable pickup center.  And that's because she cares. And I'm really proud of her for doing that. But this pollution and global warming and climate change are predominantly caused by like five corporations plus the US military. And we can't change that. So yeah, that's where I am. That's where I think the hopelessness comes in, right? Because even if, like, and this is where I need to not even like think about it in these terms. But if we did everything we need to do right now, still last week on the program, actually, I talked about how there was some scientists were saying, you know what, we already passed the tipping point. There's there's multiple tipping points, of course. But I mean, it's like the shitstorms coming. And it's just a matter of like how big do we want the shitstorm to be and accepting that there will be a shitstorm. And it's here is part of almost like the grieving process for the planet.  So bear with me. It's that I do think that humanity ultimately will survive, but it's going to be a really, really dark period. And the worst of which probably none of us are alive to see, but we're going to start to see should hit the fan.  You know, we may see our grandchildren drafted to fight in wars over water. So, you know, okay, I tried to be optimistic. And then I go right back to tumors. I'm so sorry.  Yeah.


We're down here in Naples today for an unattended d***. So about six weeks ago, I got a call from the next of Ken. This was a welfare check. The neighbors had stopped by, kind of gave us a little bit of information. A couple of weeks went by, they realized, you know, he wasn't sitting out like he normally did, I guess, and sat out front and had cigarettes and maybe a couple drinks and things like that. So it didn't seem for a while that the yards getting out of control. They called for a welfare check so they busted open the front door only to find kind of a cluttered house, but not out of control. And then to turn the corner in the hallway and see one of the most spread out, I would say, you know, vio sites that we've run it for us here. The crews in there now start on the project, just like any of these situations, a sad situation. And we made the observation to you that we seem to have found that when the unattended d***, a lot of times the person is somewhere in between their bed in the bathroom. And that's the same thing here. It looked like he was on his way to the bathroom, went down in the hallway, and obviously that's where everything ended.


Hi and welcome to our session. Let us discuss the following question. The question says if a is equal to 2 by 2 matrix whose elements are 2, 3, 1, 2, then prove that a cube minus 4a square plus a is equal to 0. We begin with the solution. We are given that a is equal to 2 by 2 matrix whose elements are 2, 3, 1, 2. Now let's first find out a square. Now a square is equal to a into a and this is equal to this matrix multiplied by itself. Multiply elements of first row with these two columns of this matrix. So we have 2 into 2 plus 3 into 1 and then 2 into 3 plus 3 into 2. And now we will multiply the second row of this matrix with these two columns of this matrix. So now we have 1 into 2 plus 2 into 1, 1 into 3 plus 2 into 2. Now this is equal to 2 by 2 matrix whose elements are 4 plus 3, 6 plus 6 is equal to 2 by 2 matrix whose elements are that a square is equal to 2 by 2 matrix whose elements are 7, 12, 14, 7 into a square matrix in which elements are, in which elements are equal to 2 by 2 matrix 21, 7 is equal to 2 by 2 matrix in which elements are 2, 0. So let's now consider matrices as we get the new matrix in which elements are 2, 45. These two matrices we get new matrix in which elements are 28, 48, minus 48, 16, minus 16, 28, minus 28 equal to a 2 by 2 matrix in which elements are 0, 0, 0, 0 and this is equal to 0 matrix. Hence we have proved that left inside is equal to right hand. So this completes the session. Bye and take care.