Hello students, welcome to this course on Modeling Stochastic Phenomena. An important concept that emerges out of the study of Fourier transforms is the concept of a function called delta function. This function is not a regular function. In fact, it has to be understood in a slightly different way and not as a normal function. The idea of such a function was introduced by Dirac in his study of quantum systems. And this concept has found great applications in many engineering problems, subsequent to its use. It has been accepted now as a mathematical function, but in the sense of what is called as a distribution not as a function. That is a quantity which is meaning only when it is inside an integral and not as a standalone function. We will examine some of its properties, some of its uses and study its characteristics. Let us first examine in which context such a function is likely to be useful. We have earlier seen a quantity called Kronecker delta. For example, Kronecker delta is defined as delta let us say n, m Kronecker delta. We used this concept in Fourier series which basically means this quantity exists when n equal to m or else it is 0. There is no problem with this concept, it is regular and you can always use it when n and m are discrete. But however, if my variables are not discrete and I want to describe the situation when the quantities say n and m are very close to each other and their continuous variables then this concept of Kronecker delta is no longer valid and that is where delta function becomes a matter of use. There are many applications today which involve delta functions. For example, if you want to describe a point charge in space and you want to find out its distribution the effect of the point charge on the potential distribution in space then delta function is a very useful concept. Similarly, if I want to solve the problem of heat conduction due to a point source the concept of delta function can be used to describe that point source and then you can operate it with like any other extended source once you have understood the meaning of this delta function. So, we will now explore some of its properties. One way one simple way to understand delta function is as a limit of a rectangular distribution. Consider a distribution whose width is b and height is h such that breadth into height equal to 1 that is it is a shape of unit area. Now I half it in the base so it becomes b by 2, but I want to conserve the areas as 1 so I will have to double the height. So, it will be approximately it will look like this. So, it will have the height of 2 h and the base area of 2 b by 2. So, increasing I extend this process at infinitum then I will arrive at a situation when my function could look for example, very very narrow, but very very tall it is a w will be very very narrow and it is a height will be very very tall, but still it is area equal to w into h remains 1. So, in the limit when when w tends to 0 this function would look one cannot really describe it, but it would look like a very steepish rising function and in the limit it would be actually a line vertical line of height infinity and width 0, but we are supposed to assume its area is 1. So, such a function which conserves the area, but whose width is 0 and that is called as a delta function. So, delta x is a limit of the rectangle function as its base b tends to 0. This is one simple physical way to understand this quantity called the delta function. Another way to understand the meaning of delta function is the so called limit of the Gaussian distribution. For example, let us let us consider a Gaussian distribution described by a function f x equal to 1 by sigma root 2 pi e to the power minus x square by 2 sigma square, where sigma is a measure of width half width of this distribution. Now, supposing I take the limit sigma tending to 0, we would expect that the function its height at x equal to 0 will increase, because height at x equal to 0 f 0 equal to always 1 by sigma root 2 pi. So, as sigma tends to 0, f 0 will increase reciprocally. Since Gaussian always conserves its area, because integral f x dx minus infinity to infinity is 1. Hence, as we continue to decrease the sigma, the curve will go on becoming sharper and sharper more and more peaked, still conserving its area every time, thereby tending to the delta function that we saw in the case of a rectangle. Hence, another representation of a delta function is it is actually the limit sigma tends to 0 of a Gaussian distribution that is 1 by sigma root 2 pi e to the power minus x square by 2 sigma square. The third representation is from the Fourier transforms, which is of course, little difficult to visualize, but it is a very well known result that comes from the study of Fourier transforms. It cannot be actually proved very easily, it deals with convergence of highly oscillatory functions at large values of the variable and we merely assume the result, which says that the Fourier transform of 1 by 2 pi will be a delta function with respect to the parameter k. This basically is a Fourier transform of a variable whose value is 1 by 2 pi. We have used the definition Fourier transform as simply the integral of the function with respect to e to the power i k x extending from minus infinity to infinity. We can exactly not prove it, but we can see why it should diverge at k equal to 0 because that is what the property of delta function is. It is infinity at k equal to 0 and it is supposed to be 0 elsewhere. If you put k equal to 0, this becomes an integral of simply dx and integral dx at minus infinity to infinity is going to be infinity because it is the area under this rectangle, area between these 2 lines. So, we can see why it would be infinity. However, as for any other value of k, it is an integral over psi n and cosine functions, highly oscillatory functions and it is assumed that the limit value of these functions will tend to 0. In fact, it is taken as 0. So, this is a very useful representation at times to resolve the delta function in terms of its so called Fourier components. In stochastic phenomena, the importance of Fourier representation is manifold. It basically means if I want to see the various frequencies that is contained in a sharply peaked function, then it will have almost equally valid or equally strong frequencies of all values. That is why sometimes it is connected to what is called as a white noise problem, white meaning all frequencies occur with equal intensity and delta function therefore, is a conjugal representation of such a spectrum. We now discuss some of the properties of the delta function which is useful to apply it to different contexts. One important property upon which as I said the very definition is based is an integral of delta function from minus infinity to infinity dx that is equal to 1. Basically, the area under the this function is 1 and you cannot plot this function. We again like to emphasize that you cannot plot a delta function, because it is then infinitely high curve with 0 width. We can slightly generalize this property supposing I have defined a delta function at the point x equal to a, I just represent it by a line although it cannot be represented by a spike. There is a significant conceptual difference between a spike and a delta function. The area under a spike is 0 whereas, area under a delta function is 1. So, such a function which it is only represented for want of better way is denoted by delta x minus a. In fact, since the function is 0 outside the point a you can actually integrate from any value outside of the point a say some point a minus epsilon to a plus epsilon of the delta function and still you would get the same value 1. It is not necessary it is not wrong, but it is not necessary to have it as minus infinity to plus infinity. A small deviation from the point where the delta function is defined that is sufficient for us to carry out the integral. So, that is an important property. So, this we can say as property 2 if we denote the previous property as property 1. Now, the third important property of the delta function is that it is an even function delta x equal to delta of minus x. This comes from the symmetry of the limiting functions that we have chosen for representing the delta function with respect to the point where the delta function is defined. So, it is like therefore, like any other even function which is very important utility. Most important advantage of delta function is its ability to select values. So, if I have a function fx and if I multiply this function by delta x and then I integrate over the it is an entire domain then this will give me f of 0. So, what I did is like this I had a function fx. Some function fx which was going like this I had a delta function at some let us say value x equal to a then if I multiply by this delta function with this function and integrate it selects the value at a that is I generalize equation 4 that is minus infinity to infinity delta of x minus a fx dx this selects the value of the function at f at x equal to a. It is a very very important property there is selection property. Next useful property of the delta function is to solve some simple algebraic equation. We have always been told that one should not divide 0 by 0 supposing you have to solve an equation x into phi x equal to 0 then we would often say either x is 0 or the function phi should be 0. More elegant way of writing the same function is to write the solution phi x equal to some constant c into delta x this almost carries the same implication. So, this is a possible solution to the problem. So, it is one of the advantages of delta function because it helps you to generalize and write through a common framework. The next important property is delta of some constant a into x is 1 by mod a of delta x how do you prove this supposing this is we will just check that this is really true how do you do that supposing I have to integrate delta ax dx minus infinity to infinity this will be 1 by mod a of minus infinity to infinity delta x dx as per the above relation. This will be 1 by mod a because this integral is 1 whereas, if you do it by transforming ax equal to u we can note that same thing will come to minus infinity to infinity delta of u then dx will be du by a which again will be 1 by a and since delta was an even function sign of a did not matter. So, it will be replaced with 1 by mod a. So, we see this are equal and therefore, this representation is valid. Another important property of the delta function is supposing the argument of the delta function is not just x it is a function of x supposing there is a function g x which looks like this. So, it cuts the x axis let us say at 3 points and I have to define this function delta of g x. So, what it basically means is how will this function contribute to an integral because delta function is defined only within an integral. So, as a standalone representation it is written as if there are let us say n roots and none of which has its derivative 0 then it is g prime x n. So, g prime should exist otherwise that function cannot be the argument of the delta function it has no meaning physically it is very easy to understand this. So, it uses the property that delta function it is integral need not be from minus infinity to infinity it will give you a value of 1 when you are just to the left of the point to the right of the point supposing my function that I have spoken become 0 here and I have multiplied the delta function. So, when I integrate this function from supposing this is my x naught is the 0 point then my delta function integrated from x naught minus epsilon to x naught plus epsilon dx will also yield 1. Let us first consider the case of a g x function which has 1 0 at point x equal to x naught that is g of x naught equal to 0. We can then write the function g x as g x equal to g x naught we are actually Taylor expanding x minus x naught g prime x naught plus higher order terms x minus x naught whole square by 2 factorial which is 2 g double prime x naught plus higher order terms. Since g x naught is 0 this will be rewritten as x minus x naught into g prime x naught plus x minus x naught by 2 into g double prime x naught. Let us assume that g prime x naught is not 0 then we know that when x approaches x naught the second derivative term will not contribute especially within a delta function. Hence for our purpose we can stop the expansion at the first term itself and write this as g prime x naught with this we have delta of g x will become delta of x minus x naught g prime x naught. We have seen earlier that delta function of a product can be reduced to delta of x minus x naught divided by mod g prime x naught. So, this basically proves the result for a case where we have 1 0. So, if you have a series of 0's finite number of 0's then this Taylor expansion will be true for every one of the around every one of those 0's and one get the result that we wrote down in the just before this. Let us take an example of a function g x equal to x minus a into x plus b. So, it is basically a quadratic function it has 2 0's if a and b are 0 it has 1 0 at minus b and another 0 is a. So, it could be a function like this. Now we want to express delta of g x. So, from the formula that we derived it will be delta of x plus b divided by the derivative of the g function at that point b of course modulus similarly plus delta of x minus a and the derivative of the function at the point a. So, g prime x equal to you can show that it is a 2 x plus b minus a. Hence g prime b modulus is going to be a plus b modulus and g prime a modulus is also going to be a plus b modulus and the solution that we are looking at the delta representation for the function delta of x minus a into x plus b can be written as delta of x minus a plus delta of x plus b divided by modulus of a plus b. So, you have now reduced it to delta functions with respect to the normal coordinates. Thank you.


So it's safe to say that regardless of the pound for pound ranking this, that, this and that, you the big fish. Yeah, in America, I'm him. H-I-M. Hemathy, damn. Facts. Hemathy. Oh, that's not the, just look in the camera when we say that. It's him. Hemathy. Hemathy. That's it. He's gonna be here with me.  Not the little one. July 29th, why are you gonna win? Man, I just feel like if we talk about No, no, no, no, no, no, no, no, no, no. I'm asking you, July 29th, why are you gonna win? I ain't actually, I can feel like Cause I'm gonna break him. Cause I'm gonna break, I'm gonna break, I'm gonna break his will. I'm gonna break his mental, his physical. Shit, I probably broke his spiritual. I'm gonna break everything.  He's gonna be the first time. It's like, it's like them trying to, it's gonna be like them trying to break Couta Quintet. Damn. Couta was a rough nigga. Yeah.  You just put a cut off all of that. You know what I'm saying? It's gonna take some times. Definitely gonna be some work, but I'm definitely gonna break him.  I was definitely gonna be rough. I like, I respect Terrence. I know, I know. I respect that. Yeah, I know how he coming. And I know his mentality is to kill. And I know he a doll, but like I say, I'm gonna break him. When I see y'all at the press conference, I see a clear size advantage. I see it clear.  He look, he look.   I'm just saying. I know y'all both got a way that was 47. I'm a country boy. You trying to say you was from New York or somewhere? Yeah, I was born in New York. Oh, okay. Like I posted, I'll say, yeah, New York birthed me, but Texas raised me. You know what I'm saying? I'm a country boy. My daddy from Jamaica, he a country, he a yard man in Jamaica. We from the country. My mama family from Alabama. Like we from the country, man. Just, just sitting here listening. Your accent definitely say you're from the country. My daddy, my daddy. He is from New York.    Wait a minute, y'all, you're saying that. So your accent definitely say that you're from the South. What I'm saying is we got this fight with Bud. I'm pretty sure.  Then after that is 154. Yes, sir.


Well, we've been walking the Freedom Trail and this is one thing we wanted to sort of especially came to see and unfortunately it's closed today. That's right, the USS Constitution closed. The nation's oldest warship is shuttered because the ship of state has run aground and there's little action on Bunker Hill because of the lack of action on Capitol Hill. The National Park Service employees that run these landmarks have been sent home as part of the shutdown. I'm very disappointed because we came here in the Freedom Trail to see it. Now we cannot, so it's very disappointing. I started the Freedom Trail yesterday and I took only half of it and today I was in great expectations of seeing also the USS Constitution which is downhill here and Bunker Hill and I was a bit disappointed by the fact that it was all closed so like and it's a pity also because I'm standing here only for a few days so I would have loved to take the most of Boston today, it's a pity. While the park housing the Bunker Hill monument officially remains closed, that has not deterred many tourists from visiting this historic site. Luckily for out-of-town visitors, many of Boston's landmarks including Fanuel Hall and the Old State House are run by local nonprofits. That means their doors are still open and for those willing to pony up, private tours of the Freedom Trail are still available. Do you think you'll get more business today given the government shutdown? We hope so, that would be great. For the Boston University News Service, I'm Jamie Bologna.


Hi, I am Jan Reardon and I am here representing the Jennifer Reardon Foundation. The foundation has been designed by my brother for my sister-in-law who passed away and we are carrying on her legacy of being kind, loving, caring and sharing. And four of the pillars of the Jennifer Reardon Foundation are women's empowerment, community vitality, early childhood education, and financial literacy. So many thanks to Town Meeting TV here. I have the pleasure once a month of bringing somebody in from our community here in Vermont to talk about one of those pillars. And without further ado, I would love to be able to move on to my guest today who is Mark Redman. He is the Executive Director of Spectrum Youth and Family Services right here in Burlington. And I have known Mark for many years. He's been the Executive Director of Spectrum for 20 years actually. We were talking earlier with the staff here what a gift Mark is truly to this community and to the state of Vermont and beyond. I know children that have come up through and had their lives changed and have moved on to other states to do good things. So it doesn't just stop in our state. So I'm going to turn this over to Mark Redman and say thank you so much, Mark, for being here. Jan, thank you. So great to see you again. So good to see you. I've been on the show. Thank you. Thank you so much. I am very excited and I'm really probably not going to say very much, not that I'm saying you talk a lot at all. I know you're shy. I know you're so shy. I do. I'm an extrovert. Right. I have a few questions, but I would just love to hear the beautiful stories that you can share as no one else I know can. But basically, if you could start with the mission of Spectrum for those people that might not be in this area and don't know about Spectrum. True. Spectrum was started in 1970 by a group of ministers. It was called BEAM, Burlington Ecumenical Area Ministers. It started very humbly 53 years ago as a shelter for homeless youth. And then it has really grown tremendously since then. Has it ever? Yes. So now, like last year, we worked with 1,240 different youth in our different facilities. And you say youth and young adult as well, correct? Yes, exactly. Up to 26 or?  We're mostly known for working with homeless youth, right? Right. So here in Burlington, we have 26 beds where young people can stay and we start with like a warming shelter to get kids out of the cold and the snow. They do well there. They move to what we call the landing. And we have tutor room there. And then we start to work on getting you into school, getting you a job, teaching you how to drive a car. And if you do well, you move on the next phase to apartments we have in that building too. So now, you get the key to the front door, the key to your own room. We're going to really teach you, you know, how to open a bank account, how to shop, how to budget your money, all of those things. So thorough. And then they move on to independent living from there. So it's like different. Nice transition. It's transition. Yes. You know? And our thing is you meet children, you meet young people where they're at. Exactly. Children who's really at the ground level struggling with mental health issues, struggling with addiction issues. So we'll meet them there. Other kids have come and they're really ready, you know, but for whatever reason, they lost their place to live or their family had problems so that we can meet them there and move them along. But I like that idea that we're kind of meeting each kid at different stages. Completely different scenario each time. Yeah. There's a young man on our board of directors now. He's my boss. Eight years ago. He was homeless, came into our drop-in center to get a sandwich. We helped him. He lived in our shelter. He got a job in our car detailing business. And then now he's living at home, going to CCV at night, full-time job. And I work for him. He's one of my bosses. That is one of the best stories I think I have ever heard, Mark. I love that. I love it. Right. And recently you have focused more, your organization under your guidance has focused more on the mental health side of things. Has that expanded? I would imagine the need is greater than ever. But you have expanded recently in the St. Albans area as well, correct? So we've done two things, right? So America is in a mental health crisis right now, especially for teenagers and young adults, especially females and LGBTQ kids, of whom we see a lot. And that's country-wide. The pandemic made it worse. And it very seriously increases in suicidality, eating disorders, depression. And so I think three years ago we had three mental health counselors. Now I have a dozen. Oh, boy, that's telling. Yeah, that's telling. We had a waiting list of 80 youth before. So if somebody called and said my son or daughter is depressed or they're anxious, I'd say, great, well, there's 80 kids ahead of you. Now it's down to 12. I want it to be zero. Right. Right. So we have more and more and more people who have master's degrees and license and counseling. Then we have the problem, where are we going to put them? Because we're running out of space in our building. Somebody, God bless them, donated a million dollars to us. Somebody else who had lost a child to suicide donated $400,000. Oh, my goodness. How beautiful is that? So we bought a building on Pine Street. So now that we have their own space, so I'm really, really happy we've been able to meet that need. And speaking of space, because you have that space, but you also have space where we were talking about the housing. So you have, you know, from beds right in the shelter and then going to apartments. Right. So what's good is everything in Burlington is within walking distance. That's so key. So Elmwood Avenue is where I work and some of our staff work in a few blocks over is 177 Pearl Street. Right. Drop-In Center, Community Health Center has a health clinic in our building, medical care in the building, doctors, nurses, physician assistants, 16 beds upstairs. The warming shelters. How long has that been in place that way? That's been there since like the mid-90s. The mid-90s. Even the beds. Yes, even the beds. Oh, I didn't realize that. Yes. But then we added about six years ago. We realized in the winter, those 16 beds are full all the time. What do we do if it's 2 a.m., it's two degrees out? Maybe we put a youth up, the stay will, in a hotel. In my opinion, hotels are not really a good place for a teenage boy or girl. So first we went to Bishop Coyne. Let me use the church hall, 10 beds there. And then we moved it over to South Minewski Avenue. But everything, and now our mental health council is Pine Street. So everything is within walking distance. I love it because I'm just thinking, when people see you coming, I'm sure they just say, okay, I know I'm doing something for this guy because you have the resources that you have. One guy told me the other day, I left my wallet in the car. I'm sure you've heard that before. Yes. And he's very generous to us. And that's, you know, a lot of it is the state has supported us, but the amount that we've raised from citizens inside and outside of Vermont is astounding. What is, I know some of the numbers, but what is your biggest fundraiser? Like, I know you do, you know, so many different things that's sleeping out during the night in the middle of March and the, you know, here in Vermont when it's 10 below. But what is the biggest, you have the empty bowl? So we have the empty bowl coming up on October 10th. And I joked. 12 years ago, my development director said, hey, we're doing an empty bowl next Thursday. And I was like, what's that? So we'll just show up at the congregational church and get ready to give a speech. So it was 40 of us in a church hall with donated soups and bowls. And I think it raised like $10,000. Now we have 400 people at the Davis Center and all the food is donated. So soup is donated, the bread is donated, the dessert, and people like that because they feel like, Mark, when I go to that, I know the money is going to go to your mission. You go to the caterer or the flower person, you know. Right. All those other expenses that many organizations, nonprofit or not, you know, they have to incur. But you have a few connections around town. That will raise easily over 100, 150,000. Oh, that's so impressive, Mark. But then the sleepout, we saw that on Facebook, I don't know, 12 years ago, thought, well, let's try it, these business executives sleeping outside in Manhattan. And maybe we'll get 10 people to sleep out in the Unitarian law. And we ended up with 40, and now it's hundreds of people. And they're all over the place, right? All over, you know? Because Essex did right at the high school. Well, this one little girl from Westford, the second year we did it, she was nine and her mother emailed me and said, my daughter wants to sleep out in the backyard the same night you were. Is that okay? And I said, sure. Oh, I love it, yes. The little girl did a GoFundMe page and sent us a check for $1,600. So I said to my son, it was 10. You can do it, you can do it. So now, because of that little girl, hundreds of kids, Boy Scout troops, swim teams, church groups all over Vermont, raising, they raise like 40 or 50,000. The total event raises 300 or 400,000. Oh, that's remarkable. But again, we nominated that little girl for a college scholarship and she went to college on that scholarship. She's a graduate now. She's a chef in Boston. She's all grown up. But I never would have been smart enough to have the kids do it. But because of her goodness, you know what I mean? That spread right from there, exactly. Boy, oh boy, it just took off from there. So it's a great example of how one person with a kind heart and cares about people who are homeless started a home movement. Right, talk about organic. I mean, you can't advertise to get something like that. That is so pure and what it's truly all about at the core. It's true. And the good that will come from that for years to come is remarkable. It's really true. How about there is something more recently here that was on the news that I really want to talk with you about because I just love the concept and like to get the details, but the direct cash transfer program. Okay. You can give us the highlights of that. Well, that's our latest thing. So, right. So, you know, one of our staff, she was an AmeriCorps worker. We get AmeriCorps workers and she read an article out of the University of Chicago and sent it to me. And there was a report that said according to Merrill Lynch, 78% of American parents help their kids financially in some way from the age 18 to age 30. And, you know, with college or first car or first apartment. And I always say I'm exhibit A. My parents paid for my college tuition in Villanova. When I graduated, they lent me the money to buy my first used car. When I bought a house years later, they co-signed on the mortgage. Beautiful. My wife and my ex-wife have paid for one son to go to Providence College. No debt. He's a nurse today. Our other son. Oh, that's beautiful. We're paying for him to go to Notre Dame. So, but the kids who come to Spectrum who are homeless, most of them. They don't have that opportunity. They don't. They don't have that. No, it's a gift. It's a gift, as we know. Their parents might be missing. Their parents may be homeless. Their parents may be suffering from substance abuse disorder. Parents might be living in the pods or a home. They don't have that. They don't have that. They don't have that. So, we have kids who are kind of stuck. They're living with us. We've taught them how to get a drive a car, to start a bank account, how to cook. Kids who are in foster homes, they're kind of stuck. They can't make that next leap. So, we caught on to this with the first ones to do it in New England. Somebody donated half a million dollars. And then Peter Welch got us a congressionally directed spending. Beautiful. So, I'm grateful to Senator Welch's office. Absolutely. So, what we did is, we picked 10 youth. We did a whole. We explained it all to our staff. They nominated 60 different youth who they thought would be appropriate. And we picked 10. So, these are our kids who have jobs. Right. It's a readiness type of concept. Start a bank account. Right. We believe they're going to use this money well. Absolutely. They're going to get on a debit card for the next 18 months, $1,500 a month. Beautiful. A year from now, we pick another 10. A year from then, another 10. And we'll be able to compare. We have an outcomes specialist. She will compare how they did versus the kids who didn't get the money. Oh, this is exciting. Finding an apartment. Getting a car. Going back to school. Maybe going to a coding academy. That opportunity. Right. We think it's going to work. We're the first ones in New England to do it. If it works, we would love to expand it. But we are very, very. And we're not just handing the money out. Not at all. The criteria that's in place. That's right. And they need to come back once a month. Me with their case manager. And we're able to track because of the monies on a debit card, where the money's being spent. Exactly. Right. So we believe they're going to spend it well. And this is just the boost that they need. One month in, two youth already moved out of our shelter into an apartment, opening beds up for two kids who've been on the street. That too. There you go. Exactly that benefit. They use the money to clear out their credit card debt. So their score. It just gives me chills. I think it's an area. And I like that we have a culture where someone who's an America worker who works in a drop-in center can give an idea to an executive director. Well, that exactly. Just going back to your development director and then people, sure, that's why all of these events. Let's try it. I mean, you said on Facebook about the sleepout. Exactly. And you are so good in that sense. This kind of gets off topic a little bit. But I just want to share it with you. I drive Uber once in a while, which I love to do. And I tend to pick up a lot of people that work at Spectrum. I can't tell you. I'm a undercover boss. I don't say anything. I don't tell them I know you. The way they rave about you, Mark, as a leader of that organization. And it's been people at all different levels. That's very nice. And I just wanted to share that with you, because that's not always the norm. We love when that happens. And we have great leaders, certainly, in our state. We do. We're very blessed in many ways in the profit and nonprofit worlds. We have some great people that are so committed to Burlington and to Vermont and to the health and wellness of our state. So it's a great place to live. But you're right there on the top of that list when I hear these stories. And I just wanted to be able to share that with you. Well, we got picked in the best places to work last year. We have the number one nonprofit. And that is an anonymous survey among your employees. That can go in a lot of different directions. So when you do it right, you know, and we've won it twice in a row last year, we have, again, the top of the nonprofit. So that was really cool. So that's a good sign that people are happy here. That's significant. You know. So the other thing we talked about is St. Albans, you know, because you mentioned that. So we have had two workers in St. Albans for about 10 years working with Youth Enforced to Care. And you have a site? Is there a building in St. Albans? Yeah. Back then we did not. We just rented space. And those two workers, we worked with Youth Enforced to Care helping them to finish high school, helping them to get a job. And they, boy, those two women said to me, over and over, a spectrum needs to be up in St. Albans. A presence there, yes. We need a drop-in center. We need homeless beds. St. Albans has, there's a lot of good things about St. Albans. Absolutely. A lot of economic activity, but there's also a lot of poverty. There's as many children in Enforced to Care in St. Albans as there are in Burlington, even though Burlington has four times the population. The size, exactly. So, again, a very nice family donated $200,000 to us to get us started. Senator Welch, again, got us some money. So we opened up a drop-in center right on Lake Street. And we didn't know when we opened, you know, let's hand out free lunch and dinner. Right. Let's see what happens. You know, it's surprising. If you go to our drop-in center here in Burlington, you're going to see a lot of 18, 19, and 20-year-olds. You know who we've seen in St. Albans? 13, 14, 15. Oh, boy. Why? Because we're so lucky in Burlington to have the Boys and Girls Club. The YMCA were used to work. Absolutely, right. We have King Street Center, Sarah Holbrook Center. Yes, so many wonderful resources and organizations. No, no. You don't have that. You can talk about something significant to say the difference between communities and the need being so great there. It's true. It's really true. So I think that is it. And so St. Albans is great. They're welcoming us with open arms. Oh, I'm sure. So now you do have a space. We have a beautiful space. And where is that located? It's gorgeous. So it's 223 Lake Street. So there's some really good nonprofits. Martha's Kitchen is up there. Oh, yes, yes. There's the Turning Point Center. What do you mean? We collaborate with the hospital, with Northwest Counseling and Sports Services. The Notch provides medical. So we're really class-a-key always, right? The collaboration. How you collaborate. You can't do it all alone. You can't just be in a community and not connect with those fabulous resources. Right. And then Champlain Housing Trust came to us a year ago. And so we just used federal money to buy an apartment complex. Would you like three of the single-bedroom apartments? Yes. Did you ever think that day would come and people are coming to you? I know. I know. Talk about a reputation. Who didn't have a place to live. So they're in there. Three is not enough, though. So now that's in my mind. We just found out we got money to start doing mentoring. We do mentoring in Burlington. We want to do mentoring in St. Albans. So what would be an example of a Burlington scenario of mentoring? Oh, man. We have about 60 mentors now. We look for adults who are, you're not the therapist. You're not the counselor. You're a stable, reliable presence in a young person's life, which for many young people, that's when they need more. That's the first time they've had that. Anything else. So Burlington High School, the guidance council will call us and say, you know, we have a young boy here, a single mom raising him.  He's starting to get in trouble. He needs a positive person in his life.  And these are people that you know in the community that are not necessarily affiliated with that group? That's right. We will recruit. We'll run ads. We'll do commercials. We'll put it on Facebook. We need mentors. We'll go to churches, national life. They're great. I met with all their executives and they've brought forth a lot of mentors. So we're always looking for mentors. Good to know. It's a one-year commitment for hours a month. It's somebody that's listening right now and is willing and interested in finding out more about being a mentor. How do they get in touch? SpectrumVT.org. We have a great website. How can I help? Oh, good. It's right there. I love your website, by the way. It's a good website. It's so user-friendly. I really like it. Yeah. Sarah Woodard, our development director. She put the website together and it is very user-friendly. You're right. So that part is good. But St. Albans is exciting and we want to keep growing there and I'll be honest with you. You know, like when Becca Ballant was running for in the primary, she visited Spectrum and she said, I'm from Brattleboro. We need this in Brattleboro. And I've gotten calls from St. Johnsbury. I've gotten calls from the city of Barry. I do believe every city in Vermont needs some version, you know, of a lot of things end at age 18 for young people, a lot of things. Right. And that's where we pick up. And even in a city like St. Albans, again, we don't have those younger things. So we need to help those kids. And that's, you know, sort of, you know, a blessing that you were able to discover that through your process and say, oh, but you really didn't know about those younger kids and the needs that they have. So what a great way to expand the program that already has such a solid foundation. It is. And I give credit to the two women who work there. And then I went up and started meeting with United Way from up there and different, I didn't come up like, we're coming in to save the day. I was like, do you need what we, and they were emphatically, yes. They were like, oh my gosh, it would be so great if Spectrum came up here. So. And as you say, St. Johnsbury and Brattleboro and other communities, what happens there? What are their next steps there where you have people that do go to identify these communities and start those relationships? It all comes down to money. Sure. And Tom, I mean, you have so much that's happening locally, but I'm glad because everybody thinks it's about Chittenden County. So it's so nice just to be in St. Albans to be considering Brattleboro, St. Johnsbury.  It is. But there's such a divide between Chittenden County and many other areas. It's different. Each community is different. So different. Franklin County is very different from Chittenden County. Very, and I love St. Albans. St. Albans. There's a lot of good stuff going on in St. Albans. Yes. You know? Beta, now the airplane company. They're opening up a battery plant there. And to have them in South Burlington is fabulous to them. I know. So thankful they're here. We're hooked into them. Exactly. And they helped us. They've taken our kids flying and stuff like that, which is cool. Very. But you're right. The key is collaboration. How do we collaborate with Lund and with Cots and with St. Louis. Exactly. So many aspects. We're so blessed to have so many wonderful nonprofits. At one time back when I was at the Y, we had more nonprofits per capita than any state in the U.S. I don't know if that's the case. And breweries too. I know. Well, that's the thing. It's so funny that you say that because when I am Ubering, so many people come here, like destination stuff from Bachelor of Bachelorette parties because they're from the state of Washington or California and they want to hear about and see these breweries that they hear about. And it used to be maple syrup or number one. That's our number one source of income is beer. So I guess that's all good. Whatever works. Yeah. Let's see. How about a couple of things that we haven't talked about here. What would you say?  And I can't even imagine the things that have changed over this course of time, just with the presence of mental illness and with the pandemic most recently. But what would you say is the biggest shift that may have even surprised you a bit when you first came into this position? The biggest shift, I think, is I'll tell you this. We now work with more BIPOC and new American youth than ever before. And I'll tell you, I was embarrassed because two researchers came to me, I don't know, six years ago, said, Mark, we did scores of interview with BIPOC youth and new American youth. And you know what we learned? Not one of those youth was going to Spectrum for services and not one of them even knew that Spectrum existed. And I said, we are missing an entire demographic. Absolutely. Because we know there are youth of color who could need help. Right. So many new Americans that have really come over here with such trauma. Sure. And anyway, I said, how do I fix that? And they said, you need to hire staff from that demographic. So now we have top to bottom about 20% of our board of directors is people of color. Fabulous. Now we have staff who speak about seven or eight different languages. What a gift that is. That is. And we have them in Burlington High School, Hunt Middle School, Essex is paying us. We have them in Essex High School, ADL, Essex Middle School. So we're working with hundreds of new American youth tutoring. Oh, life altering. Life altering is community. How to get a job, how to drive a car. In the summer, we took them all to the six, a whole trip to the six flags. Soccer, swimming, the sailing center. Oh. That is, for me, a big, big change. And, you know, my wife and I took in a refugee family 20 years ago, a Somali family, met them at the plane, single mother with five kids, ages one, three, five, seven, and nine. Lived with you? They lived with us. Brought them to our house. They lived with us. And I always joke, my wife said, this mother is going to need a lot of help. Doesn't know the language, doesn't know English. Five little kids, you need to find mentors for all those kids. And I said, that's a great idea, honey. Why don't we start with you? So, Marybeth mentioned Medina, who today has got a master's degree in education and is a teacher in Utica. And I helped Muna get a full scholarship at Champlain College, including room and board. And she just got a job at Vermont Legal Services, full-time job, you know? And so I heard from another daughter yesterday, another boy, runs a restaurant.  It was a lot. I mean, it's so easy. Yes. It takes a village. And it's asking, can this person do this? Sure. But I mean, I thought you had one person that came to stay with you. I didn't realize you had a family. Oh, it was. And I'll always remember and then we can end. The first night, they all came in, they lived in our house and we'd set up in the basement all these beds and I went down to check on them and I saw the mother in bed.  Oh, my goodness. Because that's probably how they lived. That's the only way they've ever slept. Right. I know. So anyway, I always joke the oldest boy heard me. He came up and he sat on my lap and just met him a few hours early. He was seven years old and then he left my lap and then he went and I said, isn't that amazing that he felt that trust? Then I realized he had peed on my lap. And I still remind him of that. Oh, great. He's in his 20s now. Until he gets married and he doesn't stand up with the toast. And that's how we started, that boy. That's true love. That's a commitment right there. I know. But anyway, it's been a delight to host that family and I still encourage people to do that. So what year was that? Did you say? 2004. So welcome to your executive director position and then next year you have a family. I know as if I didn't have enough to say. I thought this was a one week thing, but it's been a privilege knowing that family can develop and to grow and to become leaders in our community. And you know, UVM gave the oldest girl a full scholarship. Champlain, I give him so much credit. Absolutely. I mean it was room and board too. That's unheard of. I'd say go to CCV first like your sister. Now I want to apply to a four year college and they gave her a full ride. She was a law student and may go to law school now. Who knows? Right. But she's got a great job now still. They were in Burlington. No, they moved from Essex to Burlington to Winooski and then back to Essex a mile from where they first started. With you. Yes. I love it. The mom works. She has a part-time job, you know. Is that a success story? Yeah. And she'd be married as two other children. We have mentors through spectrum for those two kids now too. So the mentors provide such a valuable role in so many ways. Right. And so natural and genuine and all the things. And it's good for everybody. Everybody wins. The mentor as well as the mentors seems right. The mentors love it. They love it. Oh, absolutely. And now it's Joe. We lure them. It's just a one-year commitment. We just kind of lure them in. 10 years later they're paying for their college. I mean, true story. It is true. We'll ask for a year but almost everybody does more than a year which is pretty cool. It sure is. So. Well, we're going to be wrapping up here and I just want to thank you so much. My spirits for the day and beyond here when I get to hear these stories it just warms my heart. It's beautiful what you're doing Mark and I can't thank you enough. I know everybody has such great respect for you and I'm proud to know to know who you are. Thank you so much. Thank you. Yeah, we've known each other for a few years and thank you for what you're doing for the Assistant and Lewis Foundation. Thank you so much. It's an amazing thing and I give you a lot of credit for that. Well, thank you. It's, you know, bittersweet to see of being kind, loving, caring and sharing booming and I'm sure she's looking down with a lot of pride. I'm sure she is. I'm sure she is. I have no doubt. I have no doubt. So I'm glad to under the circumstances be on this journey with him and to be a board member and get to make a difference and I love every minute of that. Well, keep doing your show and what's the bear all about? Okay, so Vermont Teddy Bear is one of those great collaborators that we have right here in Burlington and five years ago when everything happened incredible man. Great person. At Vermont Teddy Bear he certainly is. I, you know, gave the idea of perhaps having a Kindness Bear. Boom, bang. People were calling me from there doing sketches. He got on it like within that afternoon there was a bear in the works.  And so then they created this and now we have Sparky the Kindness Bear who is really the mascot for the whole foundation when you go, you know to the site you'll see like there's a Jennifer Reardon Spark Kindness Sports Complex that has been named after Jen out in Albuquerque and it's beautiful and my brother had this I don't know the dimensions of it but it's enormous Sparky Bear out there and it's all through like with Vermont Teddy Bears endorsement of the bears that we brought these on the East Coast tour my brother, you know gives these out to people that say, I mean it's just it's a great bear. Well now I can email Bill and ask for a specter bear. There you are Oh my I'm sure that will happen. I love that Bill will love that. He will. I'll give you any endorsement I can. I know you will I know you will You're so well connected that I it's you know we kid about it but the value and the you know the fact of what this bear can do for people of all ages Of course is really you know under those type of circumstances it's like do you want a lot of sales or not it's kind of nice when there aren't too many sales because it's like you know during happy times as well that's great Anyway we'll wrap up here get you going again Thanks for having me on It's great to see you I'd like to have you back someday and we can follow up maybe I'd hate to wait 18 months but to get some ideas on the outcomes of that program I would love that But then we'll have some good data Exactly Thank you so much Thank you


It's going to be the first sovereign that is legalized across the country. Right now in the financial service industry, there's billions of dollars being raised in the public markets for these companies. So Canada is going to be a great experiment for other countries. We have legalization in the U.S. in select states, but we're talking about the entire country making it legal, regulating it, distributing it, making sure that it doesn't go into the hands of, you know, we want to get it away from criminals. So we'll see in a couple of years, and believe me, it's an experiment that many countries around the world are watching. Canada's going to be first.


Item Number SCP-775 Object Class Keter Special Containment Procedures Interaction with SCP-775 is to be carried out via robotic assistance whenever possible. Should direct human interaction be required, all staff must wear the Mark II Contained Atmosphere Armored Hazmat suit at all times. Should any tear be found in a suit, the subject must be immediately placed in quarantine and checked for signs of infestation. Containment area must be coated in a layer of plate steel, with all joints and seams made as tight as possible without compromising structural integrity. A two-part airlock seal is to be maintained as the single access point to the containment area. Airlocks will be flooded with bleach. Any unit of SCP-775 is detected within the airlock and remain flooded for five minutes, or until all SCP-775 have been terminated. Containment area is to be sprayed with bleach on a monthly basis to maintain population density. Feetings are not to exceed twice data-expunged, provided only at the discretion of Site Command. Description SCP-775 appears to be a form of arachnid of the order Ixodida, more commonly known as the tick. It is of a significantly larger size, with most unengorged adults reaching a size comparable with a U.S. nickel. Coloration varies between black, red, yellow, gray, and various shades of each. Adults possess eight legs, while juveniles possess only six. SCP-775 is capable of making small leaps and travels very rapidly along solid surfaces. SCP-775 shares the trait of a flexible body structure, but is much more robust than the common tick, capable of surviving crushing, cutting, or tearing with little to no damage, and capable of flattening out to slide through 0.25 centimeter gaps. SCP-775 is also capable of swelling up to four times its original size during feeding, although this does slightly hamper its ability to move. The legs of SCP-775 are also very strong, and are capable of damaging concrete over time. SCP-775 feeds in a manner similar to the common tick, but more extensively. SCP-775 injects both an enzyme to increase blood flow, and one that begins to liquefy other tissues. This enzyme will attack all tissues except those making up the layers of the skin. SCP-775 will then eat the blood and liquefied tissue, until it is totally engorged. It will then lay an egg sac containing 20 to 30 new SCP-775, on or near the host's subject, and then resume feeding. SCP-775 will feed on any vertebrate animals, and will continue to feed and reproduce on the host, until it is no longer capable of providing nutrients. Young SCP-775 will often burrow under the skin, and attempt to feed on liquefying tissues directly. Hosts will eventually be fully hollowed out, with only the outer layers of skin remaining. SCP-775 will then fill the skin with eggs, then depart to find a new host. Hosts in advanced stages of infestation are described as taking on a bloated or misshapen form, many times with multiple SCP-775 attached to many places on the body. Nest skins are often filled to the maximum capacity that the skin is capable of holding. SCP-775 is capable of reproducing offspring two days after hatching, with eggs taking 24 to 30 hours to hatch on average. This accelerated life cycle and ability to resist most forms of physical damage, cause SCP-775 to undergo an almost continuous population explosion. Bleach appears to be effective in controlling SCP-775, with most dying after several minutes of being submerged. Notes on Recovery SCP-775 was first encountered data-expunged. The Stull family appears to have been the first infected. Recovery teams found only an adult male and a juvenile female still living in the home, presumed to be aged and aged. Both were in very advanced stages of infestation, making identification difficult. Most members of SCP-775 appear to have attached to the hands, feet, face, and abdomen of the subjects. The remaining family members were found in the basement, being used as nests by SCP-775, with one of the skins already burst and discarded. Several thousand units of SCP-775 were present in the home, and it is unknown if the subjects used as nests were transported to the basement by SCP-775, the remaining family members, or expired there naturally. Evidence collected from the nests indicate that the biology and reproductive cycle of SCP-775 may share aspects with SCP-1655. If nests containing instances of either are discovered, they must be destroyed. Lesson complete. If you missed the previous orientation, go watch SCP-774, Whistle Bones, right now. Or for the complete course, watch this playlist.