Tired of the everyday routine? Ever dream of a life of romantic adventure? Want to get away from it all? Escape, designed to free you from the four walls of today for a half hour of high adventure. Tonight we escape to the waterfront of Havana and to an underworld where men live and sometimes die by a bizarre code of their own. As Burnham Carter tells it in his exciting story, Night in Havana. Something was wrong. Halfway of the street of pious works that evening I realized that something was wrong. None of the regulars were to be seen. Assuredly something was wrong. But what? And then I remembered that since I had decided to live an honest life it was really no concern of mine. As I came abreast of the Why Worry restaurant I saw the Dutchman Maru who owned a cabin cruiser and who was consequently a man of some prominence in the smuggling trade.  Ah! Tio, do you have a penny? A penny? See, see. Thanks. A hundred and ninety-three pounds. When I reach one ninety-five I'll switch from beer to coffee. When I hit one ninety I go back to beer. It's like the rice and fall of the tide in Havana Harbor. Two pounds of beer to go. If you're headed for the Cafe Mosca I'll join you. Well, bueno, bueno. Hey, come along. Eh, Tio. Got a few dollars to lend me? Aye. I have not seen any money in weeks. I was hoping to find a purchaser for my gun this evening. Oh, Tio, you're kidding. You're going to sell your gun? How will you live? Honestly. Honestly, Maru, please. Please, you are attracting attention. Quiet, you have attracting attention. Quiet, quiet, quiet. The police are please, Lieutenant Molina, across the street. Oh, here they come. Who's that with him? This newly promoted, very ambitious... Oh, no, I missed the highlight game last night, Maru. I remained at home and read in station. Stand where you are. Oh, it is, Lieutenant Molina and Sergeant... Search this one, Sergeant. Nothing, Lieutenant. This one conceals nothing. You see, I lead a dull and peaceful life. See that you do. Use your handkerchief, Tio. Your lips bleeding. She seemed upset. I bear the noble will. They have a hard life, too. It is still bleeding, huh? Yeah, yeah. You can doctor it at the cap in Moscow.   I was afraid you might be carrying your gun. I was, but when I saw them coming towards us, I put it in your pocket. May I have it now, please? Here. That is not what I would call honorable, Tio. Maru, there is solely honor in this world that one is just not able to toss it about indiscriminately. Tell me, does my look look bad? Yeah, just beginning to swell. Ah, Amalia will be very upset. Amalia? Who owns the Moscow? Say, say, we are betrothed. No, I did not know that. Congratulations. Come, come, come. I buy you a beer. Come on, step up to the bar. Later, later, Brad. Thank you, but now I must see Amalia. Tio, little one.  Oh, there's nothing, Amalia. Nothing that the police were in a tetel and more. There is nothing. Oh, the police, yes. They were here. A lieutenant. Wait, he gave me his card. Yes, Lieutenant Molina. Ah, tell me, did he mention the cause of all this ill-feeling? An American tourist, a very rich one.  She has been robbed. Ah, and so soon after the police chief let it be known that American tourists were not to be robbed. See. Hey, Amalia, what did she lose? An emerald. An emerald of great size. She had it in her purse, left the purse in the Café Diablo. Tio, you had nothing to do with this matter, did you? Oh, no, no, my friend, nothing. I wish I had. You promised, Tio, you promised. Ah, I know, I know, but... Ah, Amalia is not easy to be honest. I would have been honest years ago. Could I have earned money? Little one, listen to me. After each of your transactions they have kept you in the jail. It is like my account books. One malefaction, one payment in jail. See. Now you owe them nothing. There are no debts. But you have a debt. The mortgage, up $2,100 on this café.  But there is my pride to consider my dove. To this marriage you bring the Café Mosca. I bring nothing. Ah, Tio, you bring your love. Is that nothing? Oh, in your eyes it is much civil. In the eyes of the bank is nothing. Who is this American tourist who carries emeralds in her purse? Oh, Mrs. Turner. She is here with her husband. They have great wealth. They came from Miami in their own cruiser. It is called the seabird. And it is almost as large as Maruz. Tio, promise me you will not become involved. I promise that if I do, it will be within the law.  It will be within the law, I swear it to you. Did Lieutenant Molina mention a reward? No, he said that if I acquired information concerning the emerald I was to inform either him or Mr. and Mrs. Turner at the hotel florist. Ah, well, perhaps it would be best to learn whether there is a reward and of a size sufficient to warrant time and effort. I am going to see Senior Turner Amalie. Tio, little one, I am afraid. There is nothing to fear. An honest man need never have anything to fear. Let me have Lieutenant Molina's card, my doll. It may help me to impress the Americans. Yes? Hey, Senior Turner. Yeah. I am Lieutenant Molina of the secret police in my car. Oh, well, come in, Lieutenant Molina. Tell me, is there another detective with that name? It seems to be... Well, well, it is a common patron name, like in your country, Smith. Oh, oh, yeah. Have you found the emerald? Regrettably, sir, no. But I would like to talk about it. And I call my wife. She's the one who lost it. Knucklehead Lucille. Here I am. Don't yell so, Ducky. Ah, this is Lieutenant Molina of the secret police. Ah, it is a great honor, senora. Just see that, Ducky. It's just my hand. I saw. Now, sit down, Lieutenant. Care for a drink. No, thank you. Not while on duty. Well, what's on your mind, Lieutenant? My wife and I have a dinner date tonight. It has occurred to us, the chief and I, that although we will, of course, eventually catch the thief, it will take some time. You understand. There are a thousand rat holes and rats of many nations and much cunning. We don't have much time. Mrs. Sternner and I are leaving Havana tomorrow afternoon. And that brings me precisely to my point, the matter of a reward. Oh, no, no, no, no, no, no, no. Not another nickel. If I can't grease the wheels with 750 bucks, they just won't be greased. I'm a reasonable man, Lieutenant, but I'm no sucker. Of course you are not. Seven hundred and fifty. Ducky, I'm sure the Lieutenant knows his business. If he thinks the reward ought to be boosted, then boost it. I want that emerald back. Now look, Lucille, I... Would a thousand be better, Lieutenant? Oh, indeed it would, Senora. Then offer a thousand, Ducky. You see, Senora Sternner, whether I go between... Yeah, yeah, yeah, I know how it works. Okay, call it a thousand. A thousand dollars? It would not pay the mortgage on the Café Moscow, of course, but considering that virtue is said to be its own reward for being, it was handsome. I spend the next two hours seeking knowledge of the emerald, but the activities of the police had forced the best of the regulars into hiding, and I learned nothing. And then, as I was walking along the street of Solz in the old city, a voice spoke to me from the darkness of the gateway to the ancient convent known as El Corazón. Teo Martinez? Your servant. I got a matter we'd like to talk to you about. We? Him and me. Oh, your pardon. I could not see your friend in the shadows. We got a mutual acquaintance. Georgie Young. Georgie Young? Oh, see, see. I have not seen him since the profitable years of prohibition. Before we left Miami, he said to look you up if we ever needed someone to work an angle. He'd give us this letter of introduction. Oh, thank you. Strike him much, please. Yeah, a little closer, please. Ah, dear friend Teo, these muchachos are okay. I have a deal with them just like in the old days, Georgie. Well, you know my name? What is yours? More, Gus Moore. My name's none of your business. Stop it, Leo. His name's Leo Wach. We only been here in Cuba a little while. You said that you needed somebody to work an angle. Yeah, we got a shipment on a Florida coast just beyond Key West. We want it brought here tonight. There's 400 rifles. Ah, please, please, please. I am not interested in the nature of your cargo.  Eh, is it Georgie Young who provides you with the merchandise? Oh, yeah, yeah. Just like in the old days. Eh, it can be done, gentlemen, but it will cost money, of course. $1,500. That's highway robbery. Let me rough him up a little. Stop it, Leo, will you? Okay, Martina, it's $1,500. Cash on delivery. Uh, half of it now. Uh, $300 now. I'm sorry, no. Well, that's all the cash we have. How do you propose to get more? How much is this worth? The American tourist assembly. What's so funny? Just the ways of fortune. And sorry you would not understand. You talk too much. Yeah, see, one of my great failings. I can fix that. Kind of, Leo. Well, how about it? That rocks worth more than $1,200. Oh, much more? But the question is, what can one unload it for? Tell me, how did you acquire this? Oh, a day and I picked up at a bar while the husband was dead drunk. Well? I will make the arrangements. You give me $300 on the emerald. You get the $300 now as a binder. I'll hand you the stone when you deliver it. Very well, I agree. Now, you listen carefully to me. You will both meet me at the Café Mosque at 11 o'clock. We shall drive from there to a certain beach where we pick up the boat. Is that clear? Yeah, Leo won't meet you. And you? Oh, I'm staying at the van. I gotta make arrangements for the truck. I'll meet the boat when it comes back and hand over the emerald. Just tell me where. The beach at Marienau. There's a dirt road. You cannot miss this. Two hours to Key West. 15 minutes to load. Three hours back to Cuba. The boat will arrive at Marienau by five tomorrow morning. Is it well? Yeah. Here's your $300. Much thanks. You know what? I do the chapies who talk too much. Stop it, Leo. Will you? You'll meet at the Mosque at 11. It will be my pleasure. Well, is that coffee you drink, Maru? Oh, sit down, Theo. Sit down. Yeah, it's coffee o'clock. I hit $195 a half hour ago. What's up? If you're not too busy tonight, there's a little shipment to be moved in from Key West. Oh, how much? $1,000 for you. And for you? Ah, only $100. Go on. What do you think?   One can hide nothing from you. Very well. I receive $150. Ah, you. Easy to deal. What time do we leave? Around 11. I'll have the boat ready just off Guanabaco. You know the dock, sir? Yes. Bring the thousand with you. I will give you the thousand tomorrow, soon after we return. Bring it with you, or we do not even leave. Well? I will bring it with me. Good. Have some coffee. No, gracias. It is necessary now to see Amalia. Until 11. Until 11. I was worried, little one. You were gone so long. Oh, I had much business to transact. Amalia, I am working on a deal, a very big deal. So big that I need a thousand dollars to swing it. Deal. See, see, I have 300, and I need 700 more. Amalia, will you lend it to me? I have, but 800 in the whole world. Well, lend me seven. Oh, if this deal proceeds as planned, Amalia, you and I will be married this week, and then I shall settle down as an honest cafekeeper. Deal. For that, I would give all the money there is. Where is this boat? I don't see it. Inpatient, Senor Walsh. It's the end of the dock. A dock, he calls it. Some crummy dock. Not quite. Quiet, please. Maru. Here, Tio. Captain Maru, this is Senor Leo Walsh. Uh-huh. Yeah. You, uh, have some money, Tio? Here. $1,000 has agreed. Senor Walsh will guide you to the exact spot at Key West. And gentlemen, I wish you both a good journey. Save your wishes for somebody else. You're coming with us. No, no, I make it a point. Never to... I say your heart is this. I have a repugnance for guns between associates. And I don't trust people who talk so much. You're coming with us. Now get in. I shall do as you say. All right, now you, Skipper. Thank you. Some tub. You'll never see a faster one. How fast? 40 knots.  Now come here, it's a quiet big mouth. Oh, I was just thinking. I was thinking that the life of honesty contains almost as much interest as the kind I have formerly liked. In just a moment, we will return to Escape. But first, a favorite program with CBS listeners who want to keep abreast of the times is the news report by Alan Jackson, broadcast every Sunday morning over most of these same CBS stations. Jackson is one of CBS's ace reporters who has covered important world events in New York, London, and Berlin. Now stationed in CBS Washington News Bureau, Alan Jackson brings you the news in a clear, informed manner that has become a CBS tradition. Remember to listen this Sunday to Alan Jackson and the news, a regular CBS feature. Tune in, tune in this fall for the shows that you love best of all. Listen carefully, here's the address. It's CBS, CBS. And now back to Escape and the second act of Night in Havana. In three hours, we were passing through the Florida Keys. Maru had no difficulty in following his own urology's directions. There was a place where a finger of land got into the water, an old rum-runner's landing base which I remembered well. A truck was waiting, and three men who came forward as we tied to the dock of rotting planks. Senior Walsh spoke to them briefly, and in ten minutes the heavy crates were on board and we were off again, clearing the land quietly at a low speed and then roaring into the open.  I'm giving it full throttle now. Relax, suddenly. You talking to me? Who else? I don't like for you to call me sonny. I do it because you're so bright. How would you like a third eye? Senior Walsh, Maru, enough, enough. Carry this as far as you please once we are in trouble. Until then... I led them to the squabbley and stretched out on the bench in the cockpit, my head on my arms gazing into the sky and soothed by the powerful hum of the engines, so steady that they were in themselves a sort of silence. I thought of Amalia and how soft and comfortable she was and that we were both at the right age for marriage. Steady people who knew that what we had was worth a dozen romantic flights to the moon and I thought of how now there would be a little money for the racing of the mortgage and the Café Mosca and then I'm dosed and then... What's the matter? I'll come you cut the motor.  Leo, are you awake? Do you hear it? See. Where are you binoculars? Binoculars? What are you looking for? Coast Guard. Where are we, Maru? About two miles off the coast between Matanzas and Havana. Ah, I see them. They're to the west about the same distance out. What is the time? Four twenty. The sun will be up in less than two hours. Which way are they heading? I cannot... There's a searchlight.  They were looking for us. Somebody talked. Martinez?  It's radar. They're also equipped. I can't find them. They come towards us. Dump the cargo. When they come, we've been fishing. No, every set I got is tied up in those boxes. I said dump the gun. And I said no. We'll run for it. This boat's faster than that one. At this angle they have us cut off from the open sea. If we go west, we pass Havana. If we go east, Matanzas. As soon as we move the radio for help, we call. And I said we'll run for it.  But on, they're moving fast. I command you. No, you don't. This does. Get back to the wheel. Careful, Maru. Careful, he's frightened. Put away your gun, sonny. Stand where you are. I said... Shoot me, will you? Enough, Maru, enough. His spine has cracked like a banana stalk. Enough, I say. Teo... I hurt. I hurt bad. Ah, light on me. Let me see. Ah... Well? Ugly, Maru. Very ugly. Help me to the wheel. Yeah. Then drop the dinghy and roll to shore. I try to make it back to Key West. Ah, you need care immediately. We'll both go to shore in the dinghy. But my boat's a coast guard. I will turn it out to sea and set the throttles.  But once they overtake us, they'll find the cargo. They know the boat is mine. One step at a time, Maru. The moment we will do as I say. First we feed senior wars to the fishes. Then the dinghy goes over the side. Then you and I. I landed the dinghy only a short distance below the place where I'd left the car. Before 5.30, Maru was in good hands and I was at the Flores Hotel, knocking at the door of this suite occupied by Senor and Senor Stern. At that moment I would have surrendered all thoughts of profit for this night's labor and return for assurance that only Amalia's $700 would not be lost. Oh, Lieutenant Molina, this is a sweet time. I give you a thousand apologies, Senor. Only an emergency forces me to do this.  Yeah, sure, sure. Only we need your cruiser. You must come with me in your cruiser, and now, now. I assure you it is our only chance. There is no danger, Senor. All right, I'll get dressed. I'll be waiting for you across the street in the black touring car.  But quickly, Senor, quickly. Where are we going? We're bound for the beach called Marianao. I've never heard of it. As I wouldn't peer. It should be just about light enough to distinguish. We've arrived. Hey, you have brought the cash. Yeah, yeah. How much do I have to pay? By the maximum, Senor, $1,000. They would not do it for less, I am sorry. No, it's okay. I didn't think you could keep them down even to that. Am I on course? Yes, exactly on course. We are almost there. I think that you had better give me the money now. It's in my right hip pocket, Lieutenant, help yourself. Ah, much thanks. Senor, I am sorry that our original laxity and guarding the possessions of your wife has put you to all this trouble. No, forget it, forget it. It's one piece with the rest of the trip. I've had a lousy time. Oh, I am most sorry. We naturally prefer to have the tourists as love cookers. Oh, it's not your fault. I was mostly too drunk to see any of this. I have always tried. Senor, there is more than $1,000 in this pocket. There is $1,100. Oh, yeah, yeah. I meant to tell you, the thousands for the emerald, the other hundreds for you for all of your courtesy and attention. Oh, regulations forbid me to accept, Senor. But since it would be discourteous, much thanks. Here we are passed, come and eat us now, I think. Yes, yes, there is the dock. Hey, let me take the wheel. Hey, someone on shore is blinking a light there. Three times. I have noticed. Hey, what are you doing, heading out to sea? No, no, no. I will back in, so. And now comes the most important part, Senor. Ah, when I bring the boat to a halt at the dock, you must take the wheel and keep your hand on the throttle with the motor running. And when I say to you, now, you throw in the clutch and full speed ahead, do you understand me, Senor? Sure, but why? It's all arranged, isn't it? See, see, it is arranged, and this is the most crucial part of the arrangement. OK, whatever you say, Lieutenant. Ah, all right, you take the wheel. Now, remember, keep your hand on the throttle. Don't forget easy. Here comes my man along the dock now. Remember, when I say now.  Senor, more? See, see, everything has gone well. Where's the watch? At the wheel. Oh, good. Throw me a line to get my boys to start unloading.  You have the emerald for me? Oh, yeah. Ah, good, good. Much thanks. Here is our cable. Make fast the boat, huh? Yeah, I got it. Now, Senor, now! Lie down on the deck. What? But quickly, you will not be surprised forever. Lie down, stay there. Well, I don't understand, huh? I think we'll soon be out of range. You lead an interesting life, Molina. It has its moments. When I saw you hand him the cable, I was sure I'd rip the stern off as soon as I gave it throttle. Well, Senor, I took the precaution to untie it on our side first. Look, look, the sun is rising. Senor, do you know of any satisfaction to compare with that accompanying good job? Well done. Three days later, I was sitting at a table in the Cafe Mosco with Maru. His wound still troubled him, and he had lost much weight. Ah, a matter he was remedying as quickly as he could. Two more beers! Eh, better make it three. Visit our boat. No, no, no, don't turn your head. It's Lieutenant Molina. Oh, oh, they found my boat. Ah, you are ready with the story. I was stolen while I was ill. I know nothing of it. No, no, no. I will testify to it. No, no, no, no, Maru. I have lost my taste for reading. I now spend most evenings at the highlight games, and I find that I'm... And then, Maru? Yeah? The Coast Guard has found a boat that belongs to you. Do you have so many that one may be lost without your feeling the necessity of reporting it? I've been ill. I've not been down to the harbour for three days. I have a question to ask. Ask it, sir. They found several large crates on board. They did? Do you know what we discovered in them? Scrap metal. Scrap metal? You used it for ballast. You careful beer? Yes, yes. To celebrate my wedding, which occurs this afternoon. So I've heard you are going to lead an honest life to you? Yes, yes. I have been experimenting with it and I find that it suits me. Well, keep it confidential. Otherwise the budget for the police department may be cut to the bone. Goodbye, Lieutenant. Go with God. You heard what he said? Scrap metal. I'm not surprised. In the old, my friend Georgie Young sent to me. He wrote that he was in on a deal just like in the old days. Well, in the old days, he saw many a bootlegger barrels of whiskey which contained only water. In that way, you see, he did not break the law and may be said to have earned his money honorably. As I may be said to do too. You call owning a cafe earning money honorably? Well, it depends on the clientele. And in that respect, Maru, perhaps it would be better after the wedding, of course, if you took your custom elsewhere. Suppose I tell you that from now on I will be as honest as you yourself. Will you let me patronize this bar? Gladly, Maru. But, of course, I will be doubly watchful of the cash register. Escape is produced and directed by William N. Robeson. Tonight we have presented Night in Havana by Burnham Carter, adapted for radio by Walter Brown Newman. Featured in the cast were Tony Baratistillo, Alan Reed as Maru, and Janet Nolan as Amalia and Mrs. Sterner. Also heard were Ted Van Elks, Bill Conrad, Jeff Corey, and Jack Webb. Special music was arranged and conducted by Del Castillo. Next week. You are torturously worming your way inch by inch through a narrow pipe deep under the ground, unable to turn back, not knowing what lies ahead. Possible death? Or escape? Next week we escape with William Cochran's grim story, The Blue Wall. Good night then until the same time next week when once again we offer you escape. Young and old benefit by the community chest Red Feather Services. When you make your contribution this year, you will be helping to support the Boy and Girl Scouts, the YMCA and the YWCA, Child Guidance Clinics and Day Nurseries. So give generously to the Red Feather Services. Stay tuned now for Sing It Again, which follows immediately over most of these same CBS stations. This is CBS, the Columbia Broadcasting System.


this joint assembly is convened pursuant to provisions of JRS-9 which the clerk will now read. JRS-9 joint resolution to provide for a joint assembly to hear the budget message of the governor resolved by the senate and house of representatives that the two houses meet and join assembly on Tuesday January 26 2021 at one o'clock in the afternoon to receive the budget message of the governor and be it further resolved that the joint assembly shall be concurrently conducted electronically. The secretary will now admit the governor to the joint assembly. It is now my distinct honor to present to you the governor of the state of Vermont the Honorable Philip B. Scott. Madam President, Madam Speaker, Madam Pro Tem, Mr. Chief Justice, members of the General Assembly and fellow Vermonters. Last week Joseph R. Biden was sworn in as our 46th president. This peaceful transfer of power was in stark contrast to the violent insurrection two weeks earlier incited to undermine our constitution and overturn the results of a free and fair election. In the end the Constitution was followed and truth and democracy prevailed but we cannot pretend it didn't happen or sweep it under the rug. The violence was a sobering sign of how fragile and fractured we've become and just how easy it is for politicians and foreign adversaries to manipulate and mislead millions of Americans. We must reflect on it and find a path forward that unites us. And this isn't just the responsibility of elected officials. Every American of every race creed and color on the right, left, and in the center of the political spectrum has a personal responsibility to do their part to preserve our republic and its democratic principles. It's up to each of us to strive towards a more perfect union and constantly work for liberty and justice for all. We must face the darkness of oppression, hate, and injustice, the real enemies of America with unity, love, and truth. See differences of opinion, not as obstacles, but as an opportunity to listen and learn from each other. Seek consensus whenever possible compromise when necessary and be comfortable agreeing to disagree because that keeps us talking. So we leave a healthy democracy for the next generation. It's my hope that every American, those who voted for President Biden and those who didn't, will come to the table, put the progress of our country ahead of partisanship and not let this opportunity pass because the future of our nation truly depends upon our ability to come together. I hope every Ramonner will do their part and I can assure you this Ramonner will. Turning to the work of state government, we must focus on the challenges created by the pandemic as well as the problems we faced before, many of which have challenged us and grown for decades. We must support our communities, children, businesses, and healthcare system, care for our neighbors, and lead a recovery that reaches every county and brings equal economic opportunity to every Ramonner. Now, I remember what it's like to be a legislator, how tempting it is to seek a one-size-fits-all policy. But the regions of our state are very different, so we need a different approach. From the northeast kingdom to the Marble Valley, from Hogback Mountain to the islands, our communities are unique. And unfortunately, population growth, job creation, and economic tools also vary from region to region. Places like the Champlain Valley of Grown, while other parts of the state have watched their economic centers slowly erode. This was happening long before the pandemic hit, and it's only gotten worse over the last 10 months. State spending that grew faster than Ramonner's paychecks and pushed too many down the economic ladder is another long-standing challenge we've worked to reverse. In the two years before I came to office, general fund spending was growing at an average of about 3.75% a year. In the three years after, we held annual spending growth to about 1%, including two years without raising a single tax or fee. It's important to remember we did this while increasing investment in our downtowns and villages, housing, childcare, higher education, clean water, addressing climate change. We did it while improving the health of our UI trust fund and lowering rates for employers. And we did it while paying down debt in several areas and expanding reserves. While we slowed spending growth, helping Vermonters keep more of what they earn, the economy grew. In fact, before the pandemic hit, we are on track to see the biggest surplus in our state's history. This better positioned us to weather the economic storm that hit. Our commitment to saving lives, limiting spread of the virus, and protecting our healthcare system has also put us in better economic shape than many other states. In the federal funds we've received have been a critical lifeline, injecting billions into our economy, keeping the doors of our hospitals, businesses, nonprofits, and childcare centers open. All of this has allowed me to present a balanced budget to you today, totally $6.83 billion. It doesn't increase taxes, raise existing fees, or cut essential services. This includes $123 million over two years in state capital construction, $680 million in roads, bridges, and other transportation projects, $1.99 billion in general fund spending, and $1.89 billion for pre-K through 12 education. We've also fully funded our retirement obligations this year. You should know, to do so, we needed $103 million more than last year, for a total payment of $381 million. As we face this growing and unsustainable burden, I want to thank Treasure Pierce for her courage to take this issue on. Because we all know we must do something, and I look forward to working with her and you to save this system. This budget also makes $210 million a smart one-time investment to improve services and stimulate growth by tackling major challenges like broadband, climate change, and expanding economic opportunity. I want to thank our congressional delegation, especially Senator Leahy, for their work to help us through this difficult time. Overall, with a combined effort of state and federal officials, this budget makes enormous investments in immediate needs, and to address long-term challenges. So we can put ourselves back on the path to steadily grow our workforce, put more kids in schools, and help every region build a healthy economy, and serve their families with the best education in the country. While our fiscal picture looks better than expected, we must recognize it's mostly due to billions of dollars of one-time federal stimulus money. Remember, this isn't ongoing revenue, meaning it won't be here next year, so we need to be smart about how it's spent. That's why we must invest in areas that grow our economy or lower costs for the future. Let's not fall into the trap of using it to create new programs we can't afford after the federal money is gone. We must also remember, despite this rosy picture, COVID-19 is still impacting families and businesses. The promise of a future rebound doesn't offer much consolation to those facing devastating losses and sustained unemployment. We know too many small businesses are on the brink. Months, weeks, and even days away from closing their doors for good. So we must continue doing all we can to keep them open. The latest round of federal funding, which includes more direct grants and loans to businesses, is an important step. But there are still many small employers who don't qualify, which is why I propose an additional $10 million lifeline for them in budget adjustment. Unfortunately, this much needed lifeline for small businesses has met some resistance in the legislature. So I urge you, please reconsider. They need and expect our help right now. And we have the money to do it. The farmers, producers, markets, and co-ops that make up our food system need help too. So my budget puts in an additional $3 million in the working lands program. This will help these businesses survive and strengthen our supply chain. But the challenges in this sector are significant, especially in the meat processing. And we have to be able to feed ourselves. So let's work together to find ways to support and rebuild this important part of the economy. I've also included a $10 million package to boost outdoor recreation in all 14 counties. Three years ago, we launched a pilot program to invest in outdoor recreation, funded initially at just $100,000. While it's helped projects like Newport's Waterfront Trail and Randall's Outdoor Hub, there are many more that could use this support. These projects improve communities, grow economic activity in the areas that need it most, and strengthen our Vermont brand. So my budget invests $5 million in this program. Through the pandemic, we've seen increased use of our trails, paths, and parks as folks look for healthy ways to get out of the house. Let's make them even more attractive and accessible while protecting our public lands with $5 million to make much-needed repairs and spruce up our recreational trails. And our capital bill will fund new cabins and improve access to state lands for hunting, fishing, hiking, and boating. As much of a draw as these resources are, we continue to fall behind in tourism. Vermont has the smallest marketing budget in the Northeast, spending about half as much as our next closest competitor. To fix this, let's kickstart our marketing fund with a $1 million one-time appropriation. In future years, any rooms and meals tax revenue over and above what's projected would go into this fund to keep us competitive. Let's not forget this economic activity benefits Vermonters. In 2019, visitor spending supported over 31,000 jobs and injected $3 billion into our economy. The $373 million in tax revenue this generates pays for about $1,400 in services for each Vermont household. An increased investment here will not only deliver a strong return, but will give a much-needed boost to our hospitality sector as they struggle to recover from the pandemic. Investing in our downtowns and village centers is essential to growing regional economies in every county. Last year, we proposed relief funds for a Better Places pilot program to help communities and businesses create outdoor spaces that met public help and local development needs. Unfortunately, it didn't make it to my desk. However, nonprofits raised $90,000 to help towns and needs. The nearly 100 requests from places like Wilmington, Swam and Rupert far exceeded funding and show there's much greater need and opportunity. So I propose an additional $5 million into this initiative, making it easier to access funding for revitalization and put this money and more of our monitors to work in our communities right away. Let's also increase funding in the successful downtown and village center tax credit program by $1.75 million, bringing the total investment to $4.75 million. There's never been a more important time to support communities across the state than today. We can do that with initiatives like community's grant program in the capital bill. Now, I admit I'm partial to this program, having helped get it started when I was sitting in your seats 15 years ago. It supports much needed projects that create new opportunities for recreation, health and safety and culture. Like ADA compliance work at Martha's kitchen in St. Albans, doubling the childcare capacity to center in Crosbury, constructing a new park in Pultney, rehabbing community and rec centers in places like Bennington, Groton and Isle and Pond, and supporting Jenna's house in Johnson to provide recovery and other services for those impacted by substance abuse in the opioid crisis. It funds small projects make a big difference and can be just the lift house need to attract new Vermonters, which we desperately need. Each year, demand far exceeds available funds. So my capital bill includes a total of $4.2 million over two years to help more communities. Another area where we can make an incredible impact is cleaning up brownfields. Places like the Jones and Lambson building in Springfield, the Bellows Falls garage, the Linwood crown farm in Derby, the Calderwood site in St. John'sbury, the upper valley landfill in Thetford and so many others around the state. All of these facilities once supported their local economies, but now sit vacant. And the environmental cleanup makes it too costly to redevelop. This is a barrier to the community's economic growth, as many are in prime locations for development. So I propose 25 million in one-time funding to turn these old industrial sites from eyesores into opportunities. This will greatly benefit our environment and our economy. And it's exactly the kind of lasting impact we should be striving for with this one-time money. Infrastructure investments also strengthen our communities and economy. We'll continue that work with our transportation bill, including an additional 3.5 million, paired with 1.5 million from the General Fund to boost our downtown transportation fund. This money goes to projects that make downtowns more accessible, bringing more customers to their businesses, reducing carbon emissions, and improving safety. And I'll propose legislation to expand eligibility, giving nearly 40 more towns and villages the opportunity to make these improvements. And our transportation bill includes record funding for our paving program, covering 360 miles, and in our bike-ped program, which will finally complete the 93-mile Lamoille Valley Rail Trail, spurring economic activity for some of the most rural parts of the state. It also funds 40 traffic safety, 71 roadway, and over 115 structural improvement projects, which includes finishing the Middlebury Tunnel. It provides $1 million to support a major infrastructure project at the high-gate airport and industrial area. And to help lower carbon emissions, it adds seven electric public transit buses, 228 parking ride spaces, and $5 million to accelerate our transition to electric and more fuel-efficient vehicles. Beyond roads and bridges, we can also address important needs like broadband, weatherization, as well as major IT and cybersecurity work that will better serve our monitors. Broadband is an area we know will grow the economy, increase equity and quality in education, and expand our workforce. We've long agreed on these goals but struggled to find a way to get there. Let's face it, if it were easy or affordable, we would have done it by now. Though the state has made steady progress, the reality is that it's hard and very expensive to get to last mile. Even with the additional funding we have, it won't be enough. So we're going to need the federal government to treat high-speed internet access the same way they did electricity in the Rural Electrification Act of 1936. The good news is they're talking about it and I believe with the help of our influential congressional delegation, they'll follow through. So let's invest the dollars we have to put ourselves in the best possible position to hit the ground running when federal money arrives. The 20 million dollar broadband package I'm presenting includes 2.5 million to extend internet to more Vermont homes, 1.5 million to help our local utility and communications districts plan for broadband build-out, and then nearly 60 million dollar fund for grants and loans to expand broadband in the areas that need it. I believe this is the best way to get ready for future federal dollars. And just like broadband, weatherization is an area where we've had a lot of agreement, but not enough funding. Our current programs help families and municipalities reduce energy costs, improve housing, and move us towards our carbon emission goals. So I propose investing an additional 20 million dollars to accelerate weatherization of homes for low and moderate income families. And the Vermont Housing Finance Agency will add private funding to expand the reach of these programs. And we'll put another 5 million for the state energy management program to help towns make efficiency upgrades that will save taxpayers money down the road. In total, we believe this 25 million dollar package will generate nearly 70 million in weatherization projects and significantly increase the number of homes we weatherize.  Let's put another 10 million towards an initiative to bring affordable clean energy to those with low and moderate incomes. These initiatives will lower energy costs for homeowners and municipalities and move us significantly towards our climate change goals. We know housing isn't just about energy costs, because from our biggest city to our smallest downtowns, a lack of affordable housing places a burden on families makes it difficult to keep people here and for employers to recruit workers. Last year we established the Vermont Housing Investment Program, or VHIP, to revitalize old rental units. This program is putting Vermonters into better housing as we speak by helping small landlords rehab older units. Like Penny, who renovated two vacant properties in Westminster in desperate need of repair beyond what she could afford. She used VHIP funds to renovate rooms and improve efficiency with window and heating upgrades. One of these units is now home for a single mom who is in need of something safe and affordable. Jim and Rutland and Tyler Londonville, who both use VHIP to rehab 12 total units and work with local community groups to help the homeless and others at risk. These are powerful results delivered quickly. So let's build on it by devoting an additional $2 million to this program. As you might remember three years ago, we worked with VHCB to borrow $37 million, which has since leveraged almost $200 million to build new housing. This year instead of borrowing, let's direct $20 million to VHCB, which will help build even more homes. As well, let's invest another $1 million to help families purchase and rehab homes and communities that need the investment most. And as we work to ensure our state becomes more welcoming and diverse, let's dedicate a portion of these funds to attract new homeowners who are Black, Indigenous, and people of color. We're starting to move the needle on the housing front, but we have so much more to do. Let's act now on a goal we all share. All of this work together with the Act 250 reform and the expansion of TIS to smaller communities means we can finally deliver on our commitment to expand growth outside Chippin County and help more economic centers, more families, and more businesses in every region recover and thrive. I'm confident with continued focus and effort, we can also grow our workforce. That's why my buzzard includes $500,000 to help the Department of Labor create a new tool to better connect job seekers with training programs, educational opportunities, and good jobs. And we'll continue to support VSAC's advancement grants, with an investment totaling $2.9 million, giving workers practical hands-on training for good jobs that don't require college degrees. We've also come together to recognize that a major contributor to the economic challenges we face is our aging population and shrinking workforce. The best way to increase revenue is to add taxpayers, not new or higher taxes. And due to our success managing the pandemic, our state is even more attractive to people looking to escape big cities or a safe and healthy place to live. So let's strike while the iron's hot and expand our remote and new worker programs. In just two years, they've helped recruit 550 new Vermonters, growing our communities and schools. This represents $25 million in income and approximately $3 million in tax revenue, all with an initial investment of $1 million. That's a 300% return. That's why my budget proposes another half million dollars to make this program even more valuable by expanding our reach to a wider range of jobs and to put people from diverse backgrounds into Vermont. Recruiting and retaining members of the National Guard and veterans is another way we can build our workforce. We've taken steps like college scholarships for those who enlist and simplifying licensing so those leaving active duty can easily transition to new careers. We're also working on a recruitment campaign for veterans who are not only looking for a good place to live but a good job after their active duty ends. Moving forward, we'll create a permanent recruitment program highlighting work and educational opportunities as well as the outdoor lifestyle many are looking for. And in partnership with the National Guard Legislative Caucus, I propose we give any Guard member on state active duty the same tax exempt allowances they receive when on federal orders, letting them keep more of what they earn. Unfortunately, one thing we haven't been able to agree on over the last four years is eliminating income tax on military pensions. We're currently one of only seven states that taxes this benefit. This makes it difficult to compete because service members look at this when deciding where to spend their retirement years, which often begin in their 40s. My budget includes 1.4 million to make this important change this year. We can take a similar approach to attract more nurses, which we know we desperately need. Every year approximately 300 RNs and 130 Elkians graduate or receive certification from programs across the state, but not all of them stay. So I'll once again propose 1.25 million dollars for a nurse retention incentive to keep more of them right here in Vermont. Our efforts to grow the workforce are tied directly to building the best education system in the country from cradle to career. From affordable childcare for working parents and the draw of a great public schools for young families with kids, to giving Vermont us the skills and training needed for good jobs, which helps businesses grow. As I said in my inaugural address, we work together to make childcare more affordable for lower income families and invested relief funds to keep our providers open and safe during the pandemic. And with my proposal to reorganize the Child Development Division, we can take steps to improve the system from within as well. With a change to our tax code, we can level the playing field for pre-K providers. Currently, some programs are housed in tax-exempt schools and others in taxable properties, putting them on unequal footing. So let's exempt all of them from educational property taxes. This will reduce budget pressures on these programs, which can result in lower costs and more options for parents. And I'll once again propose we expand the lottery, generating as much as three million more each year by allowing new games like keymail and bars and restaurants, which we should dedicate to childcare. As I said two weeks ago, we have a moral imperative to rethink how we're spending nearly two billion dollars that goes to pre-K through 12 and transform the system to give every student the same chance to succeed. This year, that has to include tackling the impact of remote learning on our kids. So the Agency of Education and Department of Health are working with school leaders on how to measure and then address the developmental gaps students might have. This is critical work and we can also invest one-time money for longer-term improvements. There is no doubt our schools have and will continue to experience budget pressures in response to this pandemic. However, with the federal funds we have, I'm confident we can fully fund school budgets and make progress on school building maintenance that's been deferred over the last two decades, all without raising statewide property tax rates. We have the resources to do this and we have the obligation to follow through for the people we serve. We all know our state college system has been on an unsustainable trajectory for many years. Since coming to office, I've repeatedly proposed increased funding for the Vermont State Colleges. Sometimes these requests are met and other times we settle for less. But you, the legislature, have shown a renewed interest and I'm a willing partner. So my budget proposes $20 million in addition to their $30 million base appropriation, giving Vermont State Colleges a total of $139 million in state and federal funds in two years. We do, however, have to be realistic. Like our pension system, our state colleges need restructuring as well. It's my hope the state college board in partnership with the legislature will continue working to address the underlying structural issues we face. Because this is a one-year bridge with one-time funding. And we know this level of funding is not sustainable. Healthy families and safe communities are critical to a strong economy. So seeing Vermont struggle with food insecurity during a pandemic has been troubling. While we have more work to do, we act quickly. And it fed over 64,000 households through the SEOC and delivered over 1 million meals to those experiencing homelessness with the help of many partners. One of those initiatives, the Everyone Eats program, has not only supported those in need, but has provided income for hard-hit restaurants, farmers, and other food producers. That's why I've requested an additional 3.2 million in budget adjustment to keep this important program going while we continue to recover. Stronger, safer communities is also the goal of my team's work with the legislature, judiciary, and community partners. On an initiative to help us better address and better invest in our justice and rehabilitation programs. This includes an emphasis on helping people reenter society, contribute to their communities, and rebuild their lives after incarceration. Because this makes our community safer. So I've devoted nearly $1 million to this effort, which includes more mental health and substance abuse treatment for those under state supervision, increase transitional housing options for those reentering our communities, and provides counseling and treatment in the cycle of domestic violence and reduce repeat offenses. We'll also continue our efforts to further improve policing. But this has to be a partnership built on trust and accountability between lawmakers, law enforcement, and their communities, which is why I'm funding initiatives that help us build on years of work by the Vermont State Police, local departments, and advocates. This includes body cameras for all state sworn officers and in corrections, and help us modernize and implement our use of force policy to protect our monitors and officers alike. This work can't just be about policing and criminal justice. So this budget also expands our office of racial equity. And we put forward several proposals to address inequity and systemic bias in the areas of housing, education, workforce recruitment, and more. Now, these are just some of the policies and programs that are included in my proposed budget. There will be more ideas and details in the days to come. And like every year, this is just the start of a conversation on how, together, we can support our people and our state with the resources we have. But we know that this year will be different. And this pandemic has tested all of us. It's created new, immediate challenges we never thought we'd face. It's exposed in deep and older problems we've grappled with for decades. And it's presented us with a rare opportunity, giving us more of the resources we need to make meaningful progress on both to my friends in the legislature. How we choose to proceed, the priorities we choose to set, and how we spend one-time money, will have a lasting impact on Vermont. If we're cautious, we can solve problems and fund projects that have been solved for years, improving communities, services, outcomes, and state government itself, and lay the foundation for an economic resurgent around the state without having to ask more from taxpayers to do so. However, if we don't learn from past mistakes and choose to use one-time money to create ongoing obligations that we can't afford in the future, we'll be forced to increase the burden on working Vermonters, slowing their recovery, and missing an incredible opportunity. But if we can work together, put the politics and partisan agendas aside, we can pass a budget that could truly transform the future and set us on a path that supports all Vermonters in every corner of our state, one that helps to create equal opportunities in our schools, supports our communities, upgrades to our infrastructure, builds more affordable housing, grows our workforce, and so much more, all without any new taxes, fully funded with what we have available to us today. The challenge and the opportunity before us are great, but I know as we always have, together will rise to meet both. I thank you for listening and I look forward to working with you in the future. Thank you Governor. There being no further business, I do hereby declare the joint assembly dissolved.


Okay, thank you. So today, I was going to address the issue of expanders and shrinkers in the case of SOP cross SOQ symmetry. So we'll look at hyper surfaces M of the form of this type evolving by mean curvature flow. So for simplicity today, yesterday I discussed all the minimal surfaces that exist in this situation. And today I want to show which expanders and shrinkers you construct. And how many there are is dimension dependent. And the most interesting case is where the dimension, the total dimension P plus Q is between 7 and 4. P and Q will always be greater than or equal to 2. And so the picture for all these cases is very similar. So to simplify all the calculations, the equations that we're going to get, I'm just going to assume for today, P and Q are both equal to 2. And so that leads to the following PDs. So mean curvature flow for MT is equivalent with always this first term. And then here you get P minus 1. So that's just 1 over X times UX minus Q minus 1. That's 1 over U. So this is going to be, today is about that one particular PD. And so the boundary conditions are this. And I'm going to look for self-similar solutions. So if we look for hypersurfaces that evolve according to this rule, I always think of this as separation of variables. The situation is very analogous to what you do with ordinary differential equations. You will undergrad your PDs. If you have a linear PD, the simplest solutions that you find, you get by separating variables. You substitute a function of t that multiplied with a function of X. So here we have a function of t multiplied. And then instead of a function of X, we have the hypersurface, which is the space dependent thing. So if you look for self-similar solutions of this type, so for plus, you get expanders. For t, for minus, you get shrinkers. These correspond to solutions of the form UXT is this again. And so this kind of function is characterized by the condition that UT is equal to 1 half U minus X times UX with a plus or minus. And this plus or minus 1 half I'll abbreviate to, so I'll call that lambda. And so if the time derivative has to be equal to this, then I'll just replace that time derivative here. So the ODE for expanders shrinkers is, okay, and if I sort the terms like this, then I have the terms that have, these terms all have units 1 over length. These terms all have unit length. So they scale differently. And so we're going to, how do we analyze the solutions to this equation? To begin, first, give it a particular height A here. There should be exactly one solution coming out of here. What we did last time for the minimal surface equation were lambda zero, so you could include that here. We considered scaling invariant quantities because this term was not there, so it made sense to look at scaling invariant quantities, bless you. So let's see. Z is X over U, so that's this slope. And W is UX, that's this quantity. So in the scale, those are the two scaling invariant quantities that you can associate to a curve. And if you know these, you can reconstruct. So if you know these functions along the curve, you can reconstruct the curve up to a scaling. That's not going to work for this differential equation anymore when lambda is not zero because this equation is not scaling invariant. So what we do is we look at a three system of three differential equations. I just add the third variable is X, which is itself. And so now prime is X d dx. And if you want to, you can say we do an exponential change of variables of the independent variable, sorry, yeah, the independent variable along the curve. And if you do this, then the differential equations you get are the same system that we had yesterday plus a few extra terms. So we had, so z prime is, so d z, which is d z, d theta, that's always this. W prime now is, it's basically just these terms. So we get, right, so you get these terms. This X squared, so this term was not here yesterday. And then X prime, what is X prime? X prime is just X. Okay, so we now have a three dimensional system of differential equations. If we have solutions to these differential equations, then we can reconstruct. So first of all, we know everything is a function of X because X is one of the solutions. And just from these relations, we can then reconstruct U, right? U is just X divided by Z. So if you, if we have solutions to these, we can reconstruct the surface. Okay, so the boundary conditions that we have are, boundary conditions at X is zero are the following. So first of all, X is zero. Z is X over U. So X is zero, U is equal to A. So we're looking at, so I'm trying to construct a solution that starts at height A and does something like this. Okay, so X is zero, U is A, so this will also be zero. And W is UX. We always want this angle to be 90 degrees because then when you swing it around the axis, you get a smooth surface. So this also has to be equal to zero. Okay, so the number A does not appear in this condition. So that it will show up a little later. So the solutions that we're trying to construct start at this point and for this three dimensional system, they had, these are solutions starting at the origin. So we have to see what solutions start at the origin. So we do the same thing as yesterday. We linearize. We have one extra equation, so we'll get a three by three system. And I long ago decided never to compute a determinant or find eigenvalues in public, but this one is easy. So the system is, I think it's better if I write WZ. So I'm going to throw away all higher order terms. So for W, the only linear term that we get, ah, sorry, yeah, near the origin, this term is singular. So I can't start at zero. That would be a problem. What I do is I'm going to re-parameterize the equation and I should introduce a new variable, another theta. What I'm going to do is I'm going to multiply all these equations with Z. I'm sorry, this is not what happened. Sorry, yeah. To deal with this singularity, we have to introduce a new variable. You look at one over Z as the new variable minus that. It doesn't solve this. Yes. Yeah, so the proper way to do this is not to use Z and W, but to use these angles and then all these things don't show up, but then the equations become a bit. I don't know if I want to spend time on this. I was trying to present you a simplified version of what we actually did, but this, so the simplification is too much. Sorry about this. Okay, so if you'll bear with me for a second, I'll just have to, sorry. Yes, but I, so, okay, so what I want to do is, so if this is a regular system of differential equations, if the right-hand sides are smooth and all the, if they're polynomials, then you can linearize and there's an unstable manifold. And if you use, so that thing exists if you use instead of, so if you take the angles, if you take the angles instead of the tangents of the angles, so we would have to do a certain amount of algebra to, yeah. Yeah, so if I, right, so at X is zero, this term should not be relevant, so if I, let's see what happens.  So if you write the equations like this, so, okay, so I've replaced this term by this and the cost is that we have to add an extra variable, X squared Z. Okay, I'm really sorry about this, X squared over Z, right, U is X over Z, so X squared over Z is X times U, but this should be just a U, yeah. Okay, so I'm, I'm going to cut this calculation short a bit, there is, yeah, so I'm sorry, I'm going to not do this because I'm not really sure how this is going to continue, I should have started with this being, with this angle calling that theta and this one phi and then work with those and then if you take the radius as a variable, you get a nice system of differential equations but the formulas are longer and I, while preparing for this, it looked like this was shorter but I must have overlooked that one over Z. Okay, so let me put it differently, so the result is, is this, so lemma, so for any positive a, there is a unique solution of that type, so to the equation, so let me call this the expander shrinker equation and now I'm still going to use that system that I have over there because once, okay, so and now we have to distinguish, so lambda and this actually, this is true for any value of lambda, so but in particular lambda, so lambda is allowed to be plus or minus one half, so for both expander and shrinkers we get a solution. Okay, so let me do expander first, so in this case the same kind of arguments that I showed you yesterday show you that this is defined for all positive values of a and again it's the, so it's the same kind of arguments, the second derivative at the origin has to be positive, at any point where the first derivative is zero, the second derivative has to be positive, so that means there can't, there can only be one of them, okay, so that you get that from this equation and the fact that lambda is positive, so they're defined for all positive, for all, sorry, for all x positive, right, so these solutions are globally defined and this is, and for all a, so we get a whole family of expander, and now what happens to these things as x goes to infinity, so as x goes to infinity, let's look at these equations, the one term, so there's one place, so x shows up here, as x goes to infinity, this becomes the most important part of the equation and so for very large values of x I'll calculate this, okay, so this is minus z prime over z squared minus w prime, and this is, now as x goes to infinity, this becomes minus one over z plus w plus, and then all those terms, one plus w squared, z minus w plus lambda x squared, okay, and now if you assume which you can prove, so you would have to, if you assume that z and one over z are bounded so that the slopes don't get close to zero or infinity and that the slope of the tangent also doesn't get close to zero or infinity, then for large values of x, the only important term is this one, you get this equation, the difference one over z minus w satisfies an almost linear equation, so there is, consider this the inhomogeneous term, this is the factor multiplying one over z minus w, as x goes to infinity, if lambda is careful now, here I have minus w prime, so all these things should have had minuses, not this one, this one needs a minus, okay, so what happens for large values of x, this becomes a really large number, this is of order one, this tells you that this quantity will decay exponentially at some rate that you can figure out, so and what is one over z minus w, well one over z is u over x and w was u x, so this tells us that if you're far away, so the interpretation is this, let's say you're on the solution of, one of the solutions of the differential equation, you're at this point, one over z minus w is the difference between, it's essentially these angles, except these are the tangents of angles, so it's the tangent of this angle minus the tangent of that angle, what this says is that those two are going to zero, the difference between those two are going to zero, so as you're moving along this curve, the slope of the tangent will rotate and it's going to align with the slope of this line, okay, so solutions will look like this, right, so if a solution is moving, you're on a solution and you arrived here, you're moving in this direction, the direction in which you're moving is going to align with the radius vector and so it'll just do this, right, so and this, so if you take this, it implies that the solution goes to a cone, in particular if you use the fact that this is of order one and that this is x squared, you can improve this using barriers to show that this is something of order one over x squared, so then since that is integrable, you can prove that the thing converges to a cone, right, and the order one over x squared is this order one divided by this x squared and so in this argument, so I'm going to repeat this argument for shrinkers, so instead of repeating it, let me just say what changes right now, so in a second we'll look at shrinkers, if so for shrinkers lambda would be minus one half and everything is the same except this lambda now is negative and instead of, so this differential equation for the difference between those two angles forces that difference to grow exponentially rather than shrink, so for shrinkers you would have the following picture, so if you're on a shrinker and the slope to the curve is not aligned with the radius vector, then that differential equation forces this slope to turn around, so if you're moving around it will, the solution will look like this, which if you plot solutions of this or if you try to compute solutions numerically and you have something that is sort of on a cone, this cone is unstable, if you're tangent at some point is deviated slightly from the cone then it will turn around and it will do something like this, so shrinkers look like this or they could go the other way around depending on what the instability look like, okay, so this explains why there are lots of expanders but there are very few shrinkers and so there's a, so there's a, let me at this point mention a PDE version of this, so this is not an ODE statement, this is a PDE statement and so if M1 and M2 are N, if you have two smooth self-rinkers for mean curvature flow that are asymptotic to the same cone at infinity, so they're self-rinking solutions, they're asymptotic to a cone, both of them are asymptotic to the same cone then they must be the same, okay, so what we're seeing in the ODE here is a manifestation of this fact, okay, so we have, so back to expanders, for each, for each A, so back to expanders, for each A we have an expander, it's a monotonically increasing function, it is asymptotic to some cone, there will be some asymptotic expansion that I wrote down the other day, the question is, so which cones can we achieve by an expander, so how does the slope of the asymptotic cone depend on A, so, okay, so by looking at these ODE's again and doing more analysis you can prove that they depend continuously on the shooting height A, what I want to explain is, and this is, so what are the asymptotics for this as A goes to zero, so what do the shrinkers look like when A is a small number, so you get the ODE's gone, so I'll work with this ODE, so where lambda's plus one-half, but the same arguments also work for other lambdas, and so I'm going to, so Y is X is the stationary cone, that's a special solution, we're looking at solutions that start down here, which is a singular, so U is zero is a singular point, X is zero, U is zero is a, two terms become singular at that point, so to analyze what happens here we have to rescale, so I zoom in on this neighborhood and I multiply it with the factor A, a one over A, and so what do we get, if I do that with this equation, so let me say, let me give this variable, what happens to this equation, the, let's see, so UX, all the UX's are scaling invariant, UXX becomes, gets divided by A, one over U gets divided by A, these things all get multiplied by A, so if you do, you'll allow me to do it like this, okay, this is similar again to what I did yesterday, so you get this equation where A is a small number, and now the initial conditions are V of zero is one, and VX, this becomes VY of zero is zero, and so as you let A go to zero, the right hand side goes to zero, and what we get is the differential equation for minimal surfaces, so in the limit what we'll see down here, in the limit what you see is the minimal surface equation, okay, so where this is exactly at height one, we know there's exactly one such solution, we analyzed it yesterday, it is, it oscillates, it intersects the cone infinitely often and is asymptotic to the cone, and then yesterday from the asymptotic, so what is the shape of this difference, so this is one building block of the solution, so this thing is going to show up here, and so that thing describes the solution here on a length scale A, large multiple of A, to get the rest of the solution we will have to analyze the equation in this region, but what we see is that the, so what does the solution look like here near the origin, after it has done this, come off the vertical axis, it converges to the cone and it stays close to the cone and parallel to the cone, so now we have to analyze what solutions in this region look like, and the boundary conditions we have is that they are close to the cone here, parallel to the cone, okay, so let me not do this, the calculation that I'm showing you is, it's a typical applied math calculation, it's a formal match asymptotic expansion for the solution, and so the formal match asymptotic expansion by itself doesn't give proofs, but these are ODE's and the, so the calculations can be justified afterwards, okay, so I want to show you the results, okay, so we assume that the phi of x is much smaller than x, we substitute this, so now here we go back to the original equation again, and the only difference with what we had before is that we've given up the boundary conditions, so the boundary conditions are, no, we're not looking at these boundary conditions, but we're assuming that the solution is going to match with whatever comes out of this small corner here, okay, so, okay, so if you said u is equal to x plus a small function phi of x, you end up linearizing this equation, and so let me tell you the result of the linearization, actually most terms, so this term and those terms are all linear, so those will just stay like this, this, if you linearize around x becomes plus one over x squared times phi, and here u of x is roughly one, so, okay, so the linearized equation that we get is this, and then of course because you're linearizing there are terms of order phi squared and phi x squared, and so part of the formal part of the calculation, this being a formal calculation, means that we're going to ignore those quadratic terms for now, so this is the linear equation, so the solution to this linear equation, this is a linear differential equation, it's one of the standard equations, these, you will find them in all textbooks, it has two, there are two solutions, there's two dimensional space of solutions, if you look at, so it's not constant coefficient, but if you look at the coefficients near x is zero, it's a regular singular point, and so this means that you can find power series expansions, so phi one and two x, both are of the form x to the power r times a polynomial in x, the times a power series in x, and the coefficients are that you find by substituting here, so the characteristic equation is this, you can find the roots, and let me not calculate them, but the roots that you get are r one and two, they are complex, so this is the one place where in this story that is dimension dependent, so depending on p and q you will get different roots, if the dimension p plus q is less than eight, you get complex roots, so here you get minus alpha plus or minus i beta, so alpha is a positive number, so these are the complex roots, so both solutions, so near zero phi of x is x to the power of minus alpha plus or minus i beta, and so you would have to take, these are complex valued solutions, so you take the real part, so you get something like this, you get x to the minus alpha, that's from this factor, this constant, complex constant, you take the absolute value out, what remains is a phase theta, and then you get the real part of x to the i beta, which is this expression, so near zero, what do these solutions look like near zero, they oscillate, which is good because they have to match with the oscillating minimal surface, so near zero, what phi looks like is something that oscillates like this, and as x goes to zero, log x goes to minus infinity, so this thing oscillates infinitely often as you go into zero, on the other hand, this x to the minus alpha as x goes to zero becomes unbounded, so these solutions, they oscillate infinitely often, but the oscillations also grow, they become really large, which means that this phi, this approximation that we made, that u is equal to x plus phi, can't be valid for all values of x, as soon as this function phi becomes a little bit large, we have to switch to the other minimal surface expansion, so match with the inner expansion, how do you do that, so what was the inner expansion, that is u of x is a times v of x over a, and v is this, I keep pointing at this picture because it's almost the same picture, the minimal surface looked like this, we have to rescale this thing down, and that will give us a function, that's the function v that we have here, and we have to match that with this expression, so this leads to a certain amount of algebra, and let me tell you the result, so first of all, as you let a go to zero in this expression, this goes to infinity, right, so to match this for small a with these functions, we need to know the behavior of v for large values of a, and so v of y is, well for large values of y, the thing is asymptotic to the cone, it is y, plus, and now you need to know the deviation of v from the cone, and you can do a calculation similar to this one, and what you'll find is the same linearization, except here you're dealing with minimal surfaces, so that means that this term is gone, lambda will be zero, so the linearization is the same, so what we find is that v of y has an asymptotic expansion, and it is, so for y going to infinity, it is y plus, and then the next term is something like this, in particular the exponent alpha and beta is the same, because they come from the same differential equation, and then there are extra terms which are small compared to this, okay, so now here I have to replace y by x over a, so we get two, okay, so we have the inner expansion is u of x is a times v of y of x over a, you get that y becomes an x, everything gets multiplied with a, so we get c0 times a to the power, this y minus alpha becomes, here we get a multiplied with a to the alpha, one plus alpha, x to the minus alpha, and then we get this cosine, cosine beta log, and then here I have to write x over a, and then using the property of the cosine that's beta minus beta log a plus theta not, okay, so a large formula, and the outer expansion, okay, so there's one more ingredient that I have to tell you, I said, so for the outer expansion I'm going to use this expression, and we will compare with this one, but now here I wrote this is true for every fee for every solution to this linearized equation, but there are two of them, so what angles do we get here, and so how I'm going to select those two solutions, so before I do that I have to, so the way we select the solutions here is by looking at their behavior at infinity, so I'll come back to this here, I'm going to write u, and so we're going to look at the behavior as x goes to infinity of this, this equation, so it's a classical equation, so you could just, you could look it up, even Wikipedia will tell you, or the internet will tell you what this thing does, a quick way to remember what happens is that there are, as x goes to infinity, you could assume that either the first, the function phi and phi x, that they are the dominant term, so then this would have to be zero, and so if this is zero and this is negligible, then this, the solution to this is that phi is approximately a linear function. The other option is that the second derivative is actually the largest, and then the largest terms are this phi xx and x times phi x, and this tells you, so you integrate this, this tells you that phi x is roughly like e to the minus lambda x squared, and this, so the sign, so again lambda can be plus or negative, positive or minus, sorry, positive or negative here, you get two very different behaviors here if you change the sign of lambda, so for expanders, lambda is positive and this means that there is a solution that decays to zero really fast. If you're looking at shrinkers, then what you get is this minus lambda becomes positive and this means that most solutions blow up really fast and this is the same phenomenon as what I showed Drew before. If you follow solutions of the shrinker equation and you go out along a cone, the cone is unstable and the solutions will tend to turn around either going up or down, so it's the same phenomenon. Okay, so at this point we don't need this equation anymore. So let me stick to the case where lambda is positive, so for expanders, in that case, so what this shows is that there, most solutions will grow linearly at infinity and there's one special solution that decays exponentially, so I'll let, so I'll let phi one be the solution that has this behavior at infinity, for a solution that has this behavior at infinity, it is x plus little o of x and I'll let phi two be the solution that is e to the minus lambda x squared plus little o of x squared. Okay, so this choice determines, this choice uniquely determines phi two, this one determines phi one up to adding multiples of phi two because if you add a multiple of this to that, you still get something that is x plus little o of x, so you just pick one of these. Okay, so now we have, so this thing is like x at infinity and this one is actually, yeah, so phi two of x is like, and this one is like, has this expansion, this one. Okay, so this phi two is uniquely determined by this asymptotic expansion, so that means that the coefficients that I have here, this c, these are uniquely determined by that choice, these are specific numbers, so this phase angle theta two and this number c two are completely determined by that choice and now I can, this other one will have, so this phi one, where we have a lot of freedom, we can choose the asymptotic expansion here at infinity, it'll be c two times x to the minus alpha cosine beta log, sorry it'll be cosine beta log x minus some other phase angle and what I'm going to do is I'm going to choose that phase angle so that it's, that this becomes a sign, I'll do that. So then the general solution is, this will be x plus c one times phi one x plus c two times phi two x and what I want to do is I want to find c one and c two by writing this out and comparing what I have with what I have up here, let me do that. So near x is zero, what you get is x plus and here you get c one, so both these phi one and phi two contain, contain this same constant c two and the only difference is that one of them has a cosine and the other has a sine, x to the minus alpha and one of them has cosine, okay so that's the expansion we have near zero so now we have to compare this with that and unfortunately that tells you, right this is a cosine and you have to compare it with this thing, we use the addition formula so you get, I'm going to write this as beta log x minus theta two plus beta log a plus theta not minus theta two and I'll apply the addition formula for cosine to these two things and what I get is one thing, one term with a sine and one term with a cosine and so you can imagine that that's going to be a long formula instead of writing all that out. Let's see which term we want. Once we know the expansion of u to be this, then we can look at, so what, what do we have? We have the solution, the expander that starts at height a, small value of a near the origin. We can follow it along the minimal surface that has been rescaled, then it fall, then near the cone it follows, it stays near the stationary cone for a while, behaving like this and then we want to know it's asymptotic behavior as x goes to infinity, so we have to look at what happens to this expression as x goes to infinity and that's, okay, so as x goes to infinity the term with phi two just goes to zero really fast, okay, so this term we can forget about. This one grows linearly, so that indicates that the solution is going to deviate from a cone and the coefficient multiplying phi one, this c one here tells us how much it will deviate from a cone, so what I want to get out of this calculation is c one, okay, so that means I have to get the term multiplying, I have to get the term with sine of beta log x, so of the addition formula, so cosine alpha plus beta is cosine alpha beta, it's minus sine, so we get minus sine of this, okay, so the addition formula gives you this and the thing to look at is what is c one, this c one x to the minus alpha matches this and almost that, so the conclusion is the coefficient multiplying phi one is c zero a to the one plus alpha divided by c one, absolute value, and then this sine, so we get a minus sine and again this is a formal calculation, so the actual value is a small perturbation of this and part of making this rigorous is to show that the perturbation is much smaller than this, but all that can be done, okay, so this implies, so this, let's, these are just other constants, so this is my, this is a constant times a to the power one plus alpha times sine beta log a plus some other phase angle, okay, so it has this form and the conclusion of this is, so now we can erase phi two, so the shape of the expander with u zero is equal to a is the following, on the scale a here, it's a minimal surface, so that's the thing that we started with, then, so it intersects the minimal surface many times, then for a large length it follows the cone and at some point it is, so when you're far away it is x plus c one times x and c one is given by that formula, okay, so in particular the asymptotic slope of this cone one plus c one of a, I'm a little over time, but let me just write down the, let me graph this, this is a, okay, what is the graph of this function, the graph of this plus a, well it goes to, as a goes to zero it goes to zero like a to the one plus alpha and as a goes to infinity this log goes to zero, this log a goes to minus infinity of this, so this sign will oscillate up and down very often and what you get is, get a curve, so the slope as a function of a oscillates up and down infinitely often and so one conclusion, so one conclusion is, so what was the question, which cones can we, for which cones can we find expanders and how many expanders can you find, so if you pick a certain value of a here then these are all the expanders that you find, right, so the conclusion is that the closer the cone is to the stationary cone the more expanders there are, so in particular the stationary cone itself has an infinite sequence of expanders, so smooth, so similar expanders coming out of the cone, the, so this is the story for expanders, now what about shrinkers and so it's the same story, so it could in principle take as long, but instead let me tell you what happens, so very quickly, for shrinkers you do the exact same thing, except at this point here what are c1 and c2, you determine c1 and c2 in the exact same way, it's when you look at the behavior at infinity you have to do different things with the c2, so as x goes to infinity for shrinkers remember this phi 2 of x doesn't decay like either the minus x squared, it blows up like either the plus x squared, so the only, the only solutions of this type that correspond that will actually give you shrinkers that are asymptotic to cones are the ones where c2 is 0, so to get shrinkers you have to do this calculation but you have to find c2 as well, right, which is this, which will be this other term here and that thing has to be 0 because otherwise you will get a solution that blows up differential equation that is not asymptotic to a cone, so for shrinkers what you find is that you don't get a shrinker for each value of a but there's an infinite sequence of a's because this thing has the same argument, it's beta log a plus that phase, so as a goes to 0 there will be an infinite number of times where this hits a multiple of pi plus pi over 2 and making this cosine 0, so you get an infinite sequence of shrinkers and the asymptotic slope of those shrinkers are given by this constant which is the exact same formula that we have here, okay, so for shrinkers what we find is that there are not, there's not a whole family of them but there's an infinite sequence of slopes, well there's a sequence of a's going to 0 and for each a in that sequence there is a shrinker with that asymptotic slope and the asymptotic slope converges to 1, so for large enough and you get a smooth, you get a smooth shrinker whose slope will be somewhere over here, so let's say it would be this one and then for that slope there are many many smooth expanders, so that's how you get the non-uniqueness, okay, so I'm sorry about the mess in the beginning, I would have preferred to spend more time on this but if you have questions I would be happy to explain further detail.


Here, looking his way, trailing Burke's separation. Burke's has it. Look out for trailing Burke's. Williams turns the corner. Look at this. He's gone. Everybody, Dave Archer again here, a Falcon radio network. And this is where we're going to take a look at some of the players we think could affect the Falcons in the draft. Certainly early in the draft. Today, we're going to look at the wide receivers. And why not? Atlanta doesn't have a lot of those guys under contract right now. Been able to sign a few guys recently. Terry Fonnell has done an amazing job of securing some guys to help play that position. But can you find a big timer? Let's take a look at some of the guys that are out there. Garrett Wilson is the first guy that comes to mind. He's six feet, 183 pounds out of Ohio State. Many have him ranked near the top. I would say the same. He's an outstanding route runner with top end speed. He ran 4-3-9 at the combine. He's a kid that has the ability with run after the catch, which is important nowadays. But I think that the ability to run routes and then explode out of routes the way he does. Take a look at the tape on Garrett Wilson because what he does is he catches the football and then it's a couple steps and he's at top end speed. I love that. Not a lot of wasted movement from him. Now, he is a little bit light in the rear end, if you will. This guy only weighs 180 pounds. And you would step into the National Football League a little bit more physical every game. Not just a couple of games for an Ohio State. Every game, very physical. But Garrett Wilson, right at the top of the class here at 183 pounds, a six-footer out of Ohio State. Number two for me is Drake London. Drake London is a guy that's getting a lot of notoriety. Only played seven games a year ago. He broke his ankle in a game against Arizona. He's a big kid. He's kind of the opposite of what Wilson is. This is a guy at six foot four, 215 pounds. Doesn't have a lot of top end speed. But if you're talking about 50-50 ball, there's nobody better than Drake London to go get the football, the USC Trojan. He high points the ball extremely well. A good route runner has some technician attached to him. But I think that the top end speed is where you get a little concerned with Drake London. And then the fact that he only played seven games a year ago and he's coming off a broken ankle. Jameson Williams, the outstanding player at Alabama. Now, how healthy will Jameson Williams be? All those that followed football, Georgia won the national championship. Jameson Williams hurt his knee in that game. An ACL tear in that game. So there's some rehab for Jameson Williams to get back in time to play in the regular season. But when you take a look what the dude could do, he is the best in the country yards after catch. 9.3 yards after catch. He is a home run waiting to happen. Top end speed. He's got all that stuff run after catch. And I thought it was interesting, if you watch the Cincinnati game, remember John Menchie, his running made at Wide Receiver, got hurt against Georgia. And he had to assume the role of maybe a little bit more of the possession pass catcher. He did that. I thought that was an interesting transition for a guy that had been a speed guy and a home run guy. Traylon Burks. This is the guy that might fit what you're looking for from a run after the catch guy, multi-facitude. 6'2", 215 pounds, did not run well at the combine, which had a few alarms go up. But he did an outstanding job at Arkansas. He was really their only threat on the outside, yet he still had monster numbers. The guy got 65 balls for over 1,000 yards. He had 11 touchdowns, and he also sprinkled in the ability to run it. And don't discount the Debo Samuel factor here. A guy that can do a lot of things, not just catch the football, line up in the slaughter outside, but can you move him around like the 49ers did with Debo Samuel? There's a lot of teams looking for those type of players. Traylon Burks is one of those kind of guys that can do that. Big physical dude, but where's that top end speed? You kind of wonder. The other Ohio Statewide receiver, Chris Alave. Chris Alave, a little bit more experienced than Garrett Wilson. Got a little bit more bank of work to look at, but they're very similar players. He's six feet, 187 pounds, excellent route runner. I don't know that his top end speed is good, although he did run a good time at the combine as it is Garrett Wilson, but they're very similar players. Very competitive, excellent route runners, and good technicians, but guys that are certainly gonna be available when you start talking about in the first round. Couple other guys to look for. I mentioned John Menchie, Johan Dotson out of Penn State is another guy to potentially look at. George Pickens out of George is a very interesting guy to me. Pickens doesn't have the numbers and the gaudy numbers to look at and say, oh wow, he had a big catch in the championship game against Alabama. He was hurt all year long before that game, but had a really good freshman year. George Pickens, a big kid, 6'3, 195 pounds. Watch for Pickens to be available in second round and maybe a steal for somebody there. Alex Pierce is another guy that I think is a winner. He's maybe limited in speed, but a big kid, 6'3, 211 pounds out of Cincinnati who has the ability to catch the football in traffic. Just a couple of guys to sprinkle in. There's a number of receivers in this draft. I think Atlanta can get rich on wide receivers in the second and third round. Not necessarily have to squeeze the trigger at eight, but there's some good ones sitting at eight as well.