And the cheating effect, if you pull a loop it doesn't exist, you don't lose a turn. It just doesn't come out. In case we're going to see it, I'm going to get it in the right order to see it. Yeah. Yeah. Then you get a 10-year-old suit. That's all he's got. That's all he's got. That's all he's got. That's all he's got. That's all he's got. That's all he's got. That's all he's got. Obviously, 10 and likes. Our time frame. One more. Yeah. Ah, this is going to be a big throw-in. Turn it into a ball. Three, six, seven, eight. And then the baby's going to come out. Yes. I'm serious. I'll tell you. You feel like you're going to pay? And then I'm going to throw this. She is going to throw it up in the right order. You're going to get it in the right order. You're going to get it in the right order. Let's join together in a moment of centering silence. Together for our in-gathering hymn, number 1010. Welcome to the first Unitarian Society of Madison. This is a community where curious seekers gather to explore spiritual, epical, and social issues in an accepting and nurturing environment. Unitarian universalism supports the freedom of conscience of each individual as together we seek to be a force for good in the world. My name is Dorit Bergen, and on behalf of the congregation, I would like to extend a special welcome to visitors.  Newcomers are encouraged to stay for our fellowship hour after the service. You will see persons holding teal, stoneware, coffee mugs. They are FUS members knowledgeable about our faith community who would love to visit with you. An experienced guide will be available to give a building tour after this service. So if you would like to learn more about this sustainably designed addition or our national landmark meeting house, please meet near the large glass window on the left side of the auditorium immediately after the service. We welcome children to stay for the duration of the service, but if you or your child needs to talk or move around, the child haven or commons are good places to retire. The service can still be seen and heard from those areas. This would be a good time to turn off all electronic devices that might cause a disturbance during the hour. I would now like to acknowledge those individuals who help our services run smoothly. Our sound operator is Mark Schultz. Our lay minister is Anne Smiley.  Our ushers were Anne Smiley, Doug Hill, and Gail Bliss. Coffee is being made for you this morning by Chip Quaddy and Lois Evanson. Please note the announcements on the red floor's insert in your order of service, which describe upcoming events at the Society and provide more information about today's activities. And there are some special announcements. Eric Severson ends his internship with us at the end of this month, and we'd like to send him off with our thanks and good wishes. The intern ministry team is hosting a table in the commons where you'll find note paper, pens, and markers to write a message for inclusion in a book that we will present to him at his final time in the pulpit, May 28th. For anyone who missed our fabulous cabaret, there are second chance auction items available in the commons. Please bid early and often. Volunteer signups for next fall's Children's RE classes are looking pretty thin. Please help us change that by volunteering to join a team of three others. To find out where our needs are, or for other information, please talk with Leslie Ross, our Director of Children's Religious Education. She will be in the commons following today's service. As our congregation moves closer to the June 4th vote on Sanctuary, we and congregation, Shireisha Mayim, will be hosting the question and answer sessions that are listed in your program. Please come with your questions and your thoughts about this important decision. Again, welcome. We hope that today's service will stimulate your mind, touch your heart, and stir your spirit. Opening words today come from Gretchen Hayley, that we might hear one another into being. Create here a circle of promise and trust. A safe place where we can be uncomfortable together. A place of grace and forgiveness, bold bravery, and new beginnings. Where we might still name the needful things of our hearts. The missing places of our lives. The dreams yet unfulfilled. Some barely yet clear enough for words. Come, let us be vulnerable here. Human and true and messy. Or try to be just a little. Let the real peek out from its usual hiding places where we might meet each other, see each other, see life for what it is. Still possible, still becoming, still trying to be whole again with our help. Come, let us worship together. I invite you to, let's see here, rise as in body or in spirit for the chalice lighting, which is printed in your order of service. Our own light goes out and is rekindled by the spark of another person. Each of us has caused to think with deep gratitude of those who have lighted the flame within us. And I invite you to greet your neighbors. Many people young or young at heart who would like to come forward for a story, let's do that. Come on down. So how many of you have heard the phrase, lending a helping hand? A few of you? Okay, what does it mean? It means lending a helping hand. Help people, right? That's good. Well, that's what the adult service is going to be about today, is about helping each other. Now, how many of you have done finger painting? Painted with your fingers? How many of you? What kinds of things have you painted? Do you remember? A rainbow? Squashes? Oh, splotches. Oh, yes. So today's story is illustrated. All the pictures are done by finger paint. So I wanted to be able to show you those pictures. They're just beautiful. It's not a very long story, but it's about the things we can do with our hands. And some of the questions may seem a little weird to you, a little complicated. It's about things that we can touch but we can't really see. Things like love. We know it's real, but it's hard to touch. So that's the kind of things we can hear, feel our heart beating. That is very true. Yes. So I'm going to join you on the floor and read this story with you. It's my last chance to tell you a story, so I want to sit with you. So let's all look at the pictures. It's called Beautiful Hands. What will your beautiful hands do today? Will they plant? What can you plant? What kinds of things can we plant? Tomatoes. What else? Lettas. Celery. Flowers. Treats. Oh, trees, yes. Well, we also can plant ideas for people, give people new ideas. Or what can our hands touch? What kinds of things can we touch? Toys. The ground. Bugs. Yeah, we can touch bugs. How about hearts? We can touch hearts. And our own hands. You're right. What can we lift? What can our hands lift? What kinds of things do we lift? We can lift toys. And our sister. Yeah, we can lift those. Right. Well, we can also lift spirits. When people are feeling sad, we can help lift their spirits. Right? What can our hands stretch? Rubber toys. Slime. Silly putty. Play-doh. Uh-huh. Mines. Let's see. We can stretch our imaginations and think all kinds of things. And how about reach? What can we reach? Okay, at the top of your swing set. Yeah. And there are probably things you shouldn't touch. Right? But how about reaching for love? We can reach out for love and love. Yes. Or peace. Or truth. Or faith. Or hope. Or dreams. All of these things we can reach for. And what will your beautiful hands do tomorrow? That's the question. What will your beautiful hands do tomorrow? And I invite you to talk with your grown up friends. Such a talkative bunch. I invite you to talk about all the things you can do with your beautiful hands. Thank you for spending time with me. Let's go to class. These be seated. Some text today is an excerpt from a sermon by the Reverend Tim House in Rockland, Massachusetts. To me, the heart of our religion is not tolerance. The heart of our religion is engagement. Curious, creative, another, and with the world. Theologian Henry Nelson Wyman called such engagement creative interchange. In his book, The Source of Human Good he asks what that we intentionally engage others who have different ways of making meaning. Different values and truths. Not tolerating them. Engaging them. With the intention of enlarging our understanding of what is valuable and true. In creative interchange he says the needs and interests of others get across to me. Transform my own mind, my own desires and felt needs so as to include theirs. Real value what Wyman calls the good is not in what any of us individuals has created, but in the process of ongoing creation that takes place when we honestly open ourselves to one another. Value is created in what happens between us. Creative interchange is a process of engaging our interdependence in a spiritually transformative way. One's Hands by Karen Solvay Anderson. A couple of weeks ago I was proves in football journals and books. I came across a photograph of a lineman. He was watching the field intently. His body hunched against a heavy sleep. His hands free to the elements. I turned the page of the book and there was a close up of his hands bent and mangled covered with scabs, bruises and scrapes. I was fascinated by his hands. Hands he would inevitably have to soothe. Hands that held violence and pain, but also gentleness. I put the book down and glazed at my own hands and thought about them what they touch and experience and create. I already have my grandmother's hands with index fingers twisted slightly inward. I have bulbous veins winding under the skin that speak to my life typing, playing the piano, soothing hurts, applying bandages caressing and daily work. I like to look at farmers' hands or fisherman's hands laboring hands. Hands that speak to the life of the person. I don't want smooth hands. I want mine to speak to my person to my life's work to a sum of my parts. A place I can look and see generations before and after me of work started or incomplete. I have a colleague who is often present with people when they die. Once I asked her what she did did she have some ritual for the passing of a life? She said she helps wash the body. She washes the face as a symbol of what that person has seen. The hands as a symbol of what that person has done and the feet as a symbol of where that person has been. I love that symbolism. Unlike the lineman who can look at his hands and see what he does most of us can forget what it is we do and who we are connected to but it's all there in our hands. Those we touch or greet in welcome and friendship of creation made possible through writing or painting or playing of conversations retold of tears shed in cradled hands. The shakers have a saying hands to work in hearts to God. They believe that your life's calling your work should be no less than an act of joy an act of work is an act of worship. So I stare at my hands and I whisper Amen. He invited us to practice shared striving and shared thriving so it is not entirely by coincidence that today's service focuses on holding on to one another in good hands. This phrase to be in good hands means to be in the safe, competent care of someone to be cared for with great attention. I need to acknowledge though that this title in good hands is a form that we were all born with and still have fully functioning hands when unfortunately this is not the case and lest we over romanticize the metaphor we must remember that human hands have done some terrible things as history and current events have shown. So I offer the image of helping hands to represent generosity to sentiment and commitment to care for others so we all can do regardless of physical abilities. It also is Mother's Day. For those unfamiliar with the history in 1870 Unitarian Julia Ward Howe appealed for women around the world to unite for peace. Her pacifist reaction to the carnage of the American Civil War and the Franco-Prussian War later became known as the Mother's Day Proclamation. But Howe's efforts to establish a national Mother's Day for peace were not entirely successful. Years later the Mother's Day that we know was initiated by Anna Jarvis to honor her peace activist mother Anne Reeves Jarvis who cared for wounded soldiers on both sides of the Civil War and to honor all mothers because she believed that a mother is the person who has done more for you than anyone in the world this I would say is debatable and I feel a need to acknowledge that while many of us have been blessed not all of us have had positive experiences with either mother figures or motherhood and so this holiday can be a source of both joy and sorrow. This service is not meant to focus on mothers rather in the few weeks before we disperse for the summer I thought we might focus on caring for one another holding on to one another and the rituals that help us do that. Each of us has our own daily rituals of greeting and parting a kiss a hug a wave of the hand a handshake a fist bump we may have rituals to connect with one another weekly such as gathering for worship and our exchange of friendly greetings watching a ball game or walking or biking together having conversations over coffee we come together now and then to mark milestones in our lives and in the lives of those we know and love we honor or celebrate rites of passage for births, birth days comings of age school graduations as we see this weekend weddings anniversaries retirements and deaths these gatherings might involve cleaning the house or decorating homes welcoming guests giving gifts telling stories singing songs playing games or sharing meals all of these experiences we have in common serve to connect us over our lifetimes memories of these times will sustain us we Unitarian Universalists are notoriously independent minded organizing us can be like hurting cats which the staff here I'm sure has to and while many of us have chosen helping professions I think we enjoy fending for ourselves most of the time we've been socialized not to ask for help not to impose on someone when we do need assistance we sometimes celebrate rugged individualism at the expense of community and we can forget how dependent we really are on other people one of the challenges of being part of a large congregation is that we can feel disconnected First Unitarian Society offers a host of programs for new and long time members interest groups support groups classes, choirs ministry teams spiritual practice groups despite these offerings and despite a very engaged group of volunteers and staff members it is still possible to feel cut off from others by the hardships in our lives by the ways we cope with those hardships or even by our own stubborn pride we depend fully on others at the beginning of life as we often do at the end of life but we also need help as we all know in the midst of living we are navigating life's uncertainties disappointments and crises this is where ritual love and trust come in where being part of a faith community can help us rituals are the symbolic behaviors we perform before, during or after meaningful events the symbolic behaviors we perform before, during or after meaningful events rituals reduce anxiety increase confidence and alleviate grief even for those who don't believe they do the lighting of a flaming chalice a flower communion as we will have next week a water communion weekly worship and fellowship rituals like these remind us of who we are and to whom we belong my father is one of eight siblings so I grew up with lots of aunts, uncles and cousins gathering at the family homestead most years for Thanksgiving was a tremendous joy for me we would play games hike in the woods go swimming relax and have fun the adults would have their adult conversations and the children would play Thanksgiving dinner was pot luck so everyone brought something to share and before we ate we would stand around the table and go around and say one thing for which each of us is thankful maybe you have that custom as well so in my family the meaningful event was the gathering of the clan the symbolic behavior was renewing our bond of kinship with one another by sharing a little about each other about ourselves in a statement of gratitude this was our Thanksgiving ritual which we shared in an environment of love and trust we gather here in covenant with one another we strive to make our spiritual home a safe space because sometimes our living space or our workplace is not safe and we need someone to confide in, rely on or lean on did you know that First Unitarian Society has 22 lay ministers serving as extensions of the senior ministers that is one lay minister for every 63 members which is actually pretty good the lay ministry program was conceptualized in 1985 and put into practice in 1996 so it's been around a while these volunteers are trained to listen recognize needs be sensitive and empathetic guide or prompt spiritual thinking and provide a confidential caring presence to congregants whenever they undergo stressful life challenges how many of you were aware that this program is available to you some but not all there you go if you are in need if you've been laid off or your parents health is failing or your child is in the hospital or jail or your partner has asked for a divorce for example I encourage you to reach out ministers lay ministers and staff members can help you only if they know about your struggles I also invite you to be more fully present with one another and consider doing so a spiritual practice I've mentioned before Martin Buber's I and thou concept of being authentically and deeply committed to connecting with others engaging in relationship when I think of what feeds my spirit the long term commitment of and to my partner the joy of parenting the communion of choral singing the wonder of nature I realize that each of these requires my being in and I thou relationship so I invite you to consider how sharing expressions of love and trust with one another might serve you even sustain you as a spiritual practice by remembering this I thou and embodying it Unitarian Universalist minister Tom Ohentoll once wrote that we come into existence with our fists clenched as babies but when we arrive at death's door our hands are open and during the intervening lifetime we are summoned to progressively unclench our fists and open our hands in love and concern to all who crisscross our path we are summoned called to love and to serve one another by the golden rule parables, fables, stories songs and scriptures ancient and modern from around the world we also are summoned to service by instinct intuition and conscience we know that the benefit of generosity and service can be reciprocal a study from the University of the south in Siwani, Tennessee suggests that performing random acts of kindness boosts our psychological health by activating the release of dopamine the feel-good neurotransmitter in our brains this is referred to as a helper's high so being motivated by generosity can benefit the giver as much as the receiver most of us I think resonate with Marge Percy's poem to be of use we love those who submerge in the task and move in a common rhythm for the satisfaction of a job well done we yearn for work that is real most of us want to help people serve people and serve them well because we know that is work that is real sometimes though we don't know how to help someone or we feel uncomfortable with their pain so much so that we avoid them or give them advice or rush to immediately fix things without really acknowledging the person's experience author Debra Tannen's research has shown that men are more likely to do this than women which is a topic for another day Parker Palmer among others in an article he wrote a year ago counsels us to resist giving advice or trying to fix things but simply instead be present to one another the human soul he says doesn't want to be advised or fixed or saved it simply wants to be witnessed to be seen heard and companion exactly as it is serving as a companion for people as they are does not mean we shouldn't challenge ourselves shouldn't confront our own unconscious beliefs or biases shouldn't strive for personal transformation or even work to dismantle systemic oppression we should but when we deal with one another we need to be mindful that each of us has a story that each of us bears some untold truth about difficulty or suffering and that we need to have it acknowledged witnessed before we consider solutions Karen Anderson in our second reading suggests that our hands reflect those stories our hands have done so much she says have touched lives of so many people they connect us to past and future generations they reflect all the work we've done and left undone in our lives what we do who we are connected to is all there in our hands she says these are the hands that have touched life have reached out and by them we are blessed so many stories so many minds and hearts that need to be gently witnessed this requires that we engage with one another and this is what we hear are all about Tim Haus in our first reading declares that the heart of Unitarian Universalism is not tolerance but engagement we make meaning in our lives through creative interchange she says engagement is more than tolerance it means welcoming other people's differing values and truths by which we may enlarge our own understanding of what is valuable and true real value he says is not in what any of us as individuals is created but in the process of ongoing creation that takes place when we honestly open ourselves to one another value is created in what happens between us what happens between us and what a piece of work we are truly each of us is a living breathing wonder as today's story suggested dreams planting ideas touching hearts lifting spirits stretching imaginations reaching for love peace truth faith hope and dreams we have it within us to create together a circle of love and trust bravery and forgiveness and even dare I say it save the world and when at times our own light goes out let us seek one another so that our own inner flame might be rekindled with all that lies ahead for this congregation a capital campaign a minister and staff transition discussions about sanctuary and white supremacy continuing work of justice in a changing landscape I invite you to look on these in the months to come as opportunities and growth I invite you to attend to one another take the risk to share your needs and to witness the needs of others when we experience hard times difficult transitions forces that threaten to divide us let us engage those rituals that help us hold on to one another in good hands let us love trust and live with integrity our opening words invited us to be vulnerable here human and true and messy or try to be just a little let the real peek out from its usual hiding places that we might meet each other see each other see life for what it is still possible still becoming still trying to be whole again with our help tomorrow so what will your beautiful hands do tomorrow the work and mission of this congregation is sustained by your generosity we will now take our offering which will be shared with the Girl Scouts of Dane County whether each week as a community with joys and sorrows written on our hearts in this place we are love and we forgive and are forgiven we give and receive in return we come together to find strength and common purpose turning our minds and hearts toward one another seeking to bring into our circle of concern all who need our love and support and so today we take this time to send healing thoughts for Gay Eliason to Ken Ragland who is in the hospital with a broken hip and we come to terms with the loss that we have been feeling for a number of months here at FUS it's been a rough spring today we recognize the family and friends of Anne Nelson whose memorial will be May 18th at 3pm at Oakwood friends and family of Lillian Redding family and friends of Gabriel Rochester for whom many of these flowers are here today thinking of her husband Tim and children Ava and Milo and our thoughts are sent as well to Joan and Becky Burns and family in the death of Carol Burns Stotz Carol was the eldest of Joan and Bill Burns and she passed unexpectedly on May 2nd so I invite you now into a space of quiet and peace to ground yourself in noticing your contact with the chair and floor by sitting up straight by becoming aware of your breathing look at your hands they've been through a lot these hands they have strengths, scars beauty I invite you to remember that it is your hands that do the work of love in the world these hands may hold another's hands these hands may type emails to politicians sign cards of consolation and congratulations these hands may patiently teach quilt beautiful works or write words urgent peace these hands may bathe children, feed elders nurse the ill work the earth or organize communities these hands clasp in prayer open in release grasp in solidarity clench in righteous anger these hands are hands your hands God's hands our hands a great mystery of flesh and intention a great potential of embodied love I invite you into a moment of silence to reflect on all that our hands have done and on those cares of our community and so we remember all the joys and sorrows too tender to share but that live in the fullness of our hearts may we remember that we are part of a web of life that makes us one with all humanity one with all the universe may we be grateful for the miracle of life that we share and the hope that gives us the power to care to remember and to love if you would please rise in body or in spirit for our closing hymn number 1021 please remain standing and take hands with the person next to you belongs to a person whose heart is sometimes tender whose skin is sometimes thin whose eyes sometimes fill with tears and whose laughter is a beautiful sound the hand that you hold belongs to a person who is seeking wholeness and trusts that you are doing the same as you leave this sanctuary may your hearts remain open may your voices stay strong and may your hands remain outstretched Blessed be please be seated before the postulate


Today, we are going to consider approximation by polynomials of a continuous function. If the function is sufficiently differentiable, then we can look at the Taylor series expansion and truncate the Taylor series and obtain an approximating polynomial. For Taylor series expansion, we need function to be differentiable and we will need to know the value of the derivatives. So, we will look at Bunstein approximation using for a continuous function using Bunstein polynomials and we will see what are the advantages and disadvantages of this approximation. Then we will consider best approximation and lastly we will look at polynomial interpolation for a given function f which is continuous on a closed and bounded interval a b. So, our setting is going to be c a b which is vector space of real valued continuous functions defined on interval a b and the infinity norm or the maximum norm of a continuous function is maximum of modulus of f x when x varies over interval a b. A polynomial p n x in the power form it is written as a 0 plus a 1 x plus a n x raise to n where a 0 a 1 a n these are real numbers. So, we are going to look at real valued functions and our polynomials are going to be real polynomials. So, the coefficients a 0 a 1 a n they belong to R and x varies over R. If coefficient of x raise to n a n is not equal to 0 then that will be a polynomial of exact degree n and if n is equal to 0 then the degree of the polynomial is going to be strictly less than n. The derivative of p n is given by a 1 plus 2 a 2 x plus n a n x raise to n minus 1. So, thus to store the information about p n we need to store the coefficients a 0 a 1 a n in computer. For the derivative the coefficients a 1 a 2 a n they come into picture and if you look at indefinite integral then it is given by a 0 x plus a 1 x square by 2 plus a n x raise to n plus 1 by n plus 1 plus there will be constant of integration. So, once again what we need is only the coefficients and the derivative of a polynomial is a polynomial of 1 degree less and indefinite integral is again a polynomial with 1 degree higher. So, now the first thing we are going to look at is Taylor series expansion. We had already looked at the Taylor's theorem. So, our function f will be defined on closed interval a b and we assume that the first n derivatives and function f they are continuous on closed interval a b. We assume that nth derivative is differentiable on open interval a b that means our function n plus first derivative exists on open interval a b. Under these conditions what we get is f of x is given by f of c plus f dash c into x minus c plus f n c upon n factorial x minus c raise to n plus this is the remainder term the expression in blue that is a polynomial. Our point c is fixed x is very small. So, we are considering over interval a b and the remainder that is a function of x. The disadvantage of this polynomial approximation is it is going to be a good approximation in neighborhood of c. What we are interested in is approximation of our function by a polynomial on whole of the interval a b. So, this is not ideal approximation of our function f and another thing is that we need to know the derivative values. Now, many times in practice what is known is value of the function, but wanting to know the derivative values that is not possible for all functions but then we have got the well known Weierstrass theorem. So, this Weierstrass theorem tells us that if you have a continuous function defined on closed interval a b then there exists a sequence of polynomials p n which approximates your f in the infinity norm that means you have got a sequence of polynomials p n converging to f uniformly. Now, in numerical analysis just the statement of the type there exists is not of much use. What we should be able to do is construct it like given a function f whether we can construct the polynomial sequence of polynomials p n which converges to f uniformly. Now, fortunately it is possible to write down the explicit expression for p n which approximates the given function f and what we need for writing this is value of the function not the derivative the derivatives they do not come into picture. In fact, our function f may not be differentiable function is continuous on closed interval a b then there exists a sequence of polynomials p n converging to f in the uniform norm. So, here is the Weierstrass theorem f is a real valued function which is continuous then there exists a sequence of polynomials p n x such that norm of f minus p n infinity this is tending to 0 as n tends to infinity. So, in contrast to the Taylor's theorem where we had approximation in the neighborhood of c. So, here you have approximation over whole of interval a b. Now, this sequence p n. So, there are various proofs of this theorem, but we also have a constructive proof and in the constructive proof the polynomials p n they are given by Bernstein polynomials. So, for the sake of simplicity we will take interval a b to be interval 0 to 1 afterwards by a fine change of variable one can take care of the case when the function f is defined on interval a b. So, our function f is defined on interval 0 1 it takes real values and it is continuous. So, Bernstein polynomial it is given by I denote it by b n f at x to be summation k going from 0 to n n factorial divided by k factorial n minus k factorial n minus k factorial factorial value of our function f at point k by n x raise to k 1 minus x raise to n minus k. So, this is going to be a polynomial in x these are some real numbers f of k by n that is going to be fixed and what we have is x raise to k 1 minus x raise to n minus k. So, it will be a polynomial of degree less than or equal to n whether it is of degree n or not that will depend on our function f. Now, this is going to be our p n x and norm of f minus p n infinity norm will tend to 0 as n tends to infinity. We are going to calculate the Bernstein polynomial for three special functions and those functions are f x is constant function 1 f x is equal to x and f x is equal to x square. For these three functions we will show that norm of f minus b n f it is infinity norm tends to 0 and n tends to infinity. Then this map which associates your function f to this Bernstein polynomial we will see that it is a linear map and it is positive. That means, if f is bigger than or equal to 0 implies our polynomial also is going to be bigger than or equal to 0 and then we will appeal to what is known as Karoukin theorem. So, let us first calculate the Bernstein polynomials for special functions 1 x x square and show convergence for these three special functions. So, we have b n f at x to be summation k going from 0 to n n factorial by k factorial n minus k factorial f of k by n x raise to k 1 minus x raise to n minus k. If our function f is constant function f x is equal to 1 then b n f at x will be summation k going from 0 to n n c k that is n factorial upon k factorial n minus k factorial x raise to k 1 minus x raise to n minus k which is nothing but binomial series expansion of x plus 1 minus x whole thing raise to n. So, this is going to be equal to 1. So, when f x is equal to 1 b n f x also is equal to 1 for n bigger than or equal to 0 and hence norm of f minus b n f it is infinity norm is going to be 0. So, we have got convergence next let us look at function f x is equal to x. So, when you consider f x is equal to x then f of k by n is going to be k by n and hence you will get b n f at x to be summation k going from 0 to n n factorial by k factorial n minus k factorial now f of k by n is k by n x raise to k 1 minus x raise to n minus k. Because of the presence of this k the term k is equal to 0 will not contribute and hence I can write this as summation k going from 1 to n n factorial and you have got n here so 1 n will get cancelled and you are left with n minus 1 factorial k and k factorial. So, that will give us k minus 1 factorial n minus k factorial x raise to k 1 minus x raise to n minus k. Now, what we are going to do is we are going to take 1 x common then you have here summation k goes from 1 to n. So, let me substitute l is equal to k minus 1 when k varies from 1 to n l will vary from 0 to n minus 1 n minus 1 factorial divided by k minus 1 is l.  So, it is same I have taken 1 x out. So, what I am left with is x raise to k minus 1 so that will be x raise to l and then 1 minus x raise to n minus 1 minus l. So, this is equal to x raise to n minus 1 and now you have summation l goes from 0 to n minus 1 n minus 1 c l x raise to l 1 minus x raise to n minus 1 minus l. So, this is going to be x plus 1 minus x raise to n minus 1 the binomial series expansion. So, it is going to be equal to x this will be for n minus n bigger than or equal to 1 and when I look at b 0 f at x when you have got n is equal to 0 then that is going to be 0. So, for n bigger than or equal to 1 the approximating polynomial b n f x is equal to x. So, once again norm of b n f minus f it is infinity norm will be 0 for n bigger than or equal to 1. What we want is norm of b n f minus f it should tend to 0 as n tend to infinity. So, except for the first term you have got a constant sequence and now let us look at the function f x is equal to x square. So, we had for f x is equal to x the Bunstein polynomial was function itself and then we had for f x is equal to x again we had Bunstein polynomial to be equal to function itself for n is equal to 1 onwards. Now, when you look at f x is equal to x square then it will not be. So, but still what we are interested in is norm of f minus b n f whether it tends to 0 for f x is equal to x square and then that is going to be true. We will be using again the same idea write down the Bunstein polynomial for function f x is equal to x square and then try to manipulate the terms and get an expression for Bunstein polynomial for function x square. So, here is our Bunstein polynomial general form we are looking at f x is equal to x square. So, b n f at x for this function will be summation k going from 0 to n n factorial by k factorial n minus k factorial f of k by n is going to be k square by n square x raise to k 1 minus x raise to n minus k. So, that is the Bunstein polynomial for function f x is equal to x square. Now, presence of k will make the contribution of the term k is equal to 0 to be 0. So, we have got summation k going from 1 to n n factorial and n square. So, cancel 1 n. So, you have got n minus 1 factorial k factorial and k square so cancel 1 k. So, it will be k minus 1 factorial n minus k factorial 1 k by n got cancelled. So, you are left with k divided by n and now this k let me write as k minus 1 plus 1 x raise to k 1 minus x raise to n minus k which is going to be equal to.  So, this will get cancelled and you will have summation I am looking at first the term k minus 1 by n and then I will look at the term 1 by n because of k minus 1 the contribution from k is equal to 1 is going to be 0. So, it will be summation k going from 2 to n n minus 1 factorial divided by k minus 2 factorial n minus k factorial and then you have got 1 by n because this k minus 1 got cancelled with k minus 1 factorial x raise to k 1 minus x raise to n minus k. So, this is 1 term other term will be plus 1 by n summation k going from 1 to n n minus 1 factorial k plus 1 by n summation k going from 1 to n n minus 1 factorial by k minus 1 factorial n minus k factorial and then you have got x raise to k 1 minus x raise to n minus k. Here in the second term I can take x common and then as before you have summation k goes from 1 to n n minus 1 c k minus 1 here it will be x raise to k minus 1 here it is 1 minus x raise to n minus k. So, when I take x out what I have got is this summation is nothing but binomial series expansion of x plus 1 minus x raise to n minus 1 and hence it is going to be equal to 1. Look at the first term here I have got n minus 1 factorial. So, that I will write as n minus 2 factorial into n minus 1 and take out this n minus 1 and n out. So, let us do it and see what expression we get for f x is equal to x square. So, we have got for f x is equal to x square b n f at x is equal to from the first one I have got n minus 1 by n and let me take out x square common. So, here I am taking x square common I have summation k going from 2 to n n minus 2 factorial k minus 2 factorial n minus k factorial x raise to k minus 2 and 1 minus x raise to n minus k. That is this term and this term as I said earlier it will be 1 by n x and then x plus 1 minus x raise to n minus 1. So, this is for f x is equal to x square. So, now I get b n f at x to be n minus 1 by n x square it is this term plus 1 by n x which is going to be equal to n by n and x square. So, it will be x square and then plus 1 by n x minus 1 by n x square so it will be plus 1 by n x into 1 minus x our f x is x square. So, when I look at modulus of f x minus b n f at x this is going to be 1 by n modulus of x into 1 minus x and hence norm of f minus b n f its infinity norm is going to be maximum of this when x belongs to 0 to 1 the maximum will occur when x is equal to half. So, it is going to be 1 by 4 n. So, this tends to 0 as n tends to infinity. So, now I am going to take this as 1 by n minus 1 for the 3 functions f x is equal to 1 x and x square we have seen that we have got convergence of f minus b n f its infinity norm to 0. So, here is the result when you have got f x is equal to 1 then we saw that b n f at x that is equal to 1 for n bigger than or equal to 0. So, norm of f minus b n f infinity norm is equal to 0 for all n if you look at f x is equal to x then the Bernstein polynomial is equal to x for n bigger than or equal to 1 and for 0 when n is equal to 0 b 0 f x is equal to 0. So, the norm of f minus b n f infinity norm is equal to 0 for n bigger than or equal to 1 for the function f x is equal to x square you do not have b n f to be equal to f eventually, but what you have is f minus b n f its infinity norm is going to be 1 by 4 n and this tends to 0 as n tends to infinity. So, we are interested in the convergence of f minus b n f its norm to 0 for all continuous functions for the 3 functions 1 x and x square we have proved that the norm tends to 0. So, now what we are going to show is that the map which takes f to b n f that map is a linear map and if f x is bigger than or equal to 0 then b n f the approximating polynomial also will be bigger than or equal to 0 for all x belonging to 0 to 1. And then we will appeal to Karaukin theorem for deducing convergence for all continuous function. So, once again let us look at the Bernstein polynomial it is b n f at x is equal to summation n c k value of f at k by n x raise to k 1 minus x raise to n minus k k going from 0 to n.  So, f and g are 2 functions defined on interval 0 to 1 continuous function. So, f and g are 2 functions defined on interval 0 to 1 continuous function alpha is the real number this will be summation k goes from 0 to n n c k alpha f plus g it is values at k by n x raise to k 1 minus x raise to n minus k.  So, let us take alpha common and split the sums. So, this will be alpha times summation k going from 0 to n n c k value of f at k by n x raise to k 1 minus x raise to n minus k plus summation k going from 0 to n n c k value of g at k by n x raise to k 1 minus x raise to n minus k. So, this is alpha times Bernstein polynomial of f at k by n x raise to k at x plus Bernstein polynomial of g at x.  Suppose, our function f is greater than or equal to 0 that means, f x is bigger than or equal to 0 for all x belonging to 0 to 1. If that is the case so, f x bigger than or equal to 0 that will imply that f of k by n is bigger than or equal to 0 for k is equal to 0 1 up to n. If I fix x 0 in the interval 0 to 1, then b n f at x 0 will be summation k going from 0 to n n c k x 0 raise to k 1 minus x raise to 1 minus x 0 raise to n minus k multiplied by f of k by n minus k. This coefficient it is a constant I am fixing x 0. So, let me call it beta k this beta k is going to be bigger than or equal to 0 because our x 0 is in the interval 0 to 1. If beta k is bigger than or equal to 0, then our b n f at x 0 is greater than or equal to 0 and f is going to be bigger than or equal to 0 and thus f bigger than or equal to 0 implies b n f is bigger than or equal to 0. So, now, we have got b n to be a map from c 0 1 to c 0 1 such that b n of alpha f plus g is equal to alpha times b n f plus b n g when alpha varies over real numbers and f g belong to c 0 1 not 0 1 it should be c 0 1 and f bigger than or equal to 0 implies b n f to be bigger than or equal to 0. So, that is a positive map and here is Karoukin theorem which says that if you have p n to be a map from c 0 1 to c 0 1 which is a positive linear map and if norm of p n f minus f infinity norm tends to 0 for 3 functions 1 x x square then norm of p n f minus f it is infinity norm tends to 0 for each f belonging to c 0 1. So, it is a very interesting theorem that we want continuity or we want convergence for all continuous functions. Now, this convergence will be assured provided you can show convergence for 3 simple functions just the function constant function 1 function x and then function x square but of course, our map should be linear and it should be a positive map. So, we have already proved the convergence for these 3 functions and which will tell us that you have the convergence of Bernstein polynomials for each f belonging to c 0 1. So, now, we have got we have a constructive way given a function defined on interval 0 to 1 which is continuous I know how to write down its Bernstein polynomial it is just that look at the value of f of k by n and then multiply by n c k x raise to k 1 minus x raise to n minus k and sum it over. Now, mathematically it is a very elegant proof and nice constructive proof but then one is interested also knowing how fast it is going to converge because when you are use going to use computer then you are allowed only finite number of steps. In mathematics it is fine what I want is as n tends to infinity. Now, this is one of the disadvantage of this polynomial approximation that the convergence is going to be very slow. So, here is the fact that we had seen that for function f x is equal to x square the error is going to be 1 upon 4 n. So, suppose I want the error to be less than 10 raise to minus 6. So, this will be assured provided 1 upon 4 n is less than 10 raise to minus 6 that means my n should be bigger than 25 into 10 raise to 4. So, that means n should be bigger than this number. So, the convergence is very slow. Our function f is so simple it is f x is equal to x square and then I want the polynomial approximation and I want to write it down. So, I want the error to be less than 10 raise to minus 6. So, it is really a modest requirement and then I need to choose n to be so big such a ridiculously high number as such when I am looking at function f x is equal to x square. The polynomial approximation is going to be just take it itself is a polynomial, but then this Bernstein polynomial it tells us a way that given a function f construct it in this fashion. So, what it does not do is it does not reproduce polynomials. If the function is polynomial then the best approximation is just take the polynomial itself as your approximation. So, the two disadvantages of this Bernstein polynomial approximation is that the convergence is very slow and it does not reproduce polynomial. It means if the function is polynomial then eventually my polynomial approximation should be polynomial itself when I have got f x is equal to x square. If I want its approximation by constant polynomial then of course there will be some error for linear also there will be error, but when I want approximation for x square by quadratic polynomials then it should be a quadratic polynomial and it does not happen with this recipe. So, why not look at the best approximation that means I fix a degree say degree n is equal to 10 and then among the polynomials of degree less than or equal to 10 I choose the polynomial p n star which has got the least error because after all what I am interested in is the error in the maximum norm. So, among the polynomials of degree less than or equal to n if I can choose the polynomial which has got the least error then I am doing my best and when I do such a thing then already I know that Bernstein polynomials they converge. So, I will have convergence also and I will have convergence as fast as it is possible, but then the question of construction comes that when you are doing numerical analysis we do not want just existence we should be able to construct the error should converge to 0 at a reasonably fast rate. So, you have the best approximation. So, there exists a unique polynomial p n star of degree less than or equal to n such that norm of f minus p n star is less than or equal to minimum of norm of f minus p n. So, you are taking minimum over all polynomials of degree less than or equal to n.  So, convergence is assured, but here the computation of p n star that needs an iteration technique and that is known as the second algorithm of Remy's. So, the problem comes with the construction that not just there exists. Now, whether this best approximation whether it will reproduce the polynomials. Yes, if my f is itself a polynomial of degree n then the minimum error will be when I choose p n star is equal to f. So, it reproduces polynomials it gives you convergence as fast as it is possible, but then the computation needs iteration technique that means you do not have a formula for writing p n star in terms of f. So, we have considered Taylor series expansion in which case what we needed was the derivative values and also the Taylor formula is in terms of value of the function and the derivative at a fixed point c. We are interested in the error over the whole interval a b. The Taylor series will be efficient in the neighborhood of c, but as you go away from c the error will increase and then the knowledge of derivative needed. So, that is why we looked at Bernstein polynomials which gives you approximation to all continuous functions, but the convergence is very slow. So, you look at the best approximation in case of best approximation you have to resort to iteration techniques. So, the problem is with construction. So, now that brings us to interpolation polynomials.  Suppose I give you one point x 0 and find a constant polynomial which matches with function value at that point x 0 and it is nothing, it is just the constant polynomial. If I look at two points x 0 and x 1 look at the value of f x 0 and f x 1, then I am going to join by straight line I will get a linear polynomial. And in general if I have got n plus 1 distinct points in interval a b, then what we are going to show is that there exists a unique polynomial of degree less than or equal to n such that p n x j is equal to f of x j for j is equal to 0, 1 up to n. Now, once again you have to worry about there exists, but here one can write down explicitly the polynomial. So, this is fine, then you are going to have a unique polynomial of degree less than or equal to n. So, if your function f is itself a polynomial of degree n, then your p n is going to be equal to f, because if the function is a polynomial of degree less than or equal to n. And if I choose p n is equal to f, then it will interpolate my given function not only at this n plus 1 distinct points, but at all points. So, it is going to reproduce the polynomials like if f x is equal to x square, if I choose three points, then the interpolating polynomial that means you are fitting a parabola and your interpolating polynomial will be same as the function. So, no problem with the construction, it reproduces the polynomial polynomial. So, it would not happen like in case of Bernstein polynomials f x is equal to x square, when your Bernstein polynomial it give you something, there was error, whereas here when I will consider three or more points my p n is going to be equal to f. So, it is going to reproduce the polynomial. So, now comes the question of convergence. So, suppose I am choosing interpolation points. So, for a constant polynomial I will need one point, in order to fit a linear polynomial I will need two points, in order to fit a parabola p 2 I will need three points. And in order to fit a polynomial of degree less than or equal to n I will need n plus 1 distinct points. So, every time you have to specify the points and then we can fit a polynomial. So, suppose I choose this set of points by some rule for example, take the equidistant points. Then the question is whether for every f belonging to C A B this sequence of interpolating polynomials f minus p n norm infinity tends to 0 as n tends to infinity. Unfortunately, this is not true. See your interpolating polynomials they are going to depend on your interpolation points. So, if I take equidistant points like first take the midpoint of interval A B, then take the end points, then divide it into three equal parts and take two equal parts and take the end points and so on. So, then one feels that maybe this is not the best way of choosing interpolation points do it some other way. But what one can show is no matter how you choose these points there will always exist a continuous function for which this error will not tend to 0. For some specific functions it will tend to 0, but whether for each f. So, I choose the interpolation points I fit a polynomial then norm of f minus p n infinity will not tend to 0 for each f belonging to C A B. So, there is always a trade off that sometimes for the interpolation points it was nice you can construct, then you will have the polynomials they will be reproduced, but then the problem is with the convergence. Now, still the polynomial interpolation they are going to be used and the reason is that even though interpolating polynomials have no convergence as such one should not work with high degree polynomials. When we talked about the good algorithm then one of the criteria was stability and I had said I will comment about it little later. So, here is what happens when you deal with high degree polynomial, here is a polynomial of degree 7 it has got roots 1 2 3 4 5 6 7 now if you expand it is going to be x raise to 7 minus 28 x raise to 6 etcetera. Now, if I change this minus 28 to minus 28.002. So, if I change this minus 28 to minus 28 I am making a very small change, but then the roots which were 5 and 6 they become complex and that becomes minus 5.459 plus or minus 0.540 i. So, a small change then it makes a lot of difference in the result and that is the stability problem. So, then the what we are going to do is we are going to use low degree polynomials we will try to choose our interpolation points in a way best we can and then we can go to piecewise polynomials. So, the topic of our next lecture is going to be polynomial interpolation. Thank you.


just kind of checking where it's coming on moving forward um there's no other further questions or comments on this agenda item um we shall move on to 5.06 which is the use of force incident report with that i give the floor back to the chiefs thank you uh chair and mosh um january uh from january first 2021 to january 31st 2021 there were a total number of six use of force incidents within those six incidents there was a total use of force of nine uh nine uses of force in those the percent of uh sorry and during that time period we responded to 1250 incidents so the use of force incidents two versus the total incidence is about 0.4 percent 0.5 percent uh the gender uh we had two white males uh three black males although note one black male had four uses of force and two separate incidents and there were two white females and to note that one white female had two use of forces in one incident the first incident um was an arrest on a misdemeanor on lawful trespass the there was a complainant who called our dispatch and the officers were dispatched to it the subject had been treated and cleared at the hospital three times and now was refusing to leave the property and was attempting to re-enter the hospital but was being prevented hospital security officers were advised that the subject is an aggressive to leo and past interactions the subject walked towards the officers when they arrived and shoved an officer up against the wall um she later uh after being handcuffed she fell down onto the ground and refused to stand up and walked to the cruiser so she had to be picked up by the officers and placed in the back seat of the cruiser the white female age 39 she's five foot five hundred and sixty five pounds there were two officers that used force the first officer used verbal direction empty hand controls and picked the subject off the ground officer two used verbal direction empty hand controls and assisted and picked the subject uh up from the ground the arrestee was not injured and the officers were not injured the next one was an arrest on a felony aggravated assault violation of court conditions of release this also had a complainant who called our dispatch and the officers officers were dispatched to it the subject threatened to assault the caller the complainant with a three and a half foot long metal pipe on scene subject was compliant uh while at the police department for processing an officer was walking behind the subject into the holding area the officer accidentally stepped on the heel of the subject's shoe the subject turned around towards the officer and the officer placed his hand out and pushed the subject back out of his personal space subject did not fall it was a black male age 26 510 180 pounds one officer officer one used verbal direction and empty hand controls the arrestee was not injured and the officer was not injured the next was um well do you want me to go over it as the one you guys have all watched the video on um it was a incident on grand street or sorry um george street do you want me to walk through that or do you all feel that you've uh and well briefed on this one you personally i am okay but i'm asking here if you don't mind try i would actually like to just hear how that would be described relative to the other ones just to get some sense of of um comparison sure sounds good this is a rest on a felony uh assault uh interference with law enforcement officer and vandalism these are the charges from the actual valkour incident there's a complaint new called dispatch and officers were dispatched to the area subject had been observed breaking into a motor vehicle confronted and been confronted by the caller followed subject no police arrived on scene subject refused to stop for officer and attempted to leave the area while screaming but that's just wearing at the officer the officer called for another officer and continued to prevent subject from leaving second officer arrived on scene and they attempted to place subject in the handcuffs subject began to fight with officers and all of them ended up on the ground the subject grabbed one officer by the throat and punched the second officer in the head subject and one officer were able to make it back up to their feet the second officer attempted to deploy a taser while on the ground but at malfunction the darts did not deploy only arc that just means electricity arc back between back and forth between them that second officer was able to get to their feet they deployed their taser a second time causing the subject to lock up subject fell onto the stairs porch area of a residence third officer arrived on scene and assisted in handcuffing the subject the black male age 19 five foot seven 195 pounds there are three officers involved in the use of force officer one verbal direction empty hand controls handcuff control techniques officer two verbal direction empty hand controls the controlled electronic weapon malfunction and then a controlled electronic weapon was used and the darts were deployed officer three used verbal direction other force used is the pulled subjects are out from underneath the subject while handcuffing officer one had injured his throat officer two had injured your hand the uh sorry scroll down to the next page your st had cuts on his hand the next one was an arrest on a felony burglary complaint there was two complaints the first complaint called dispatch and officers were dispatched to it a subject demanded entrance into complainants residence and made threats to resident if he was not allowed in the complaint did not know subject and did not allow subject into residence subject then attempted to break through basement window but was unsuccessful residents younger daughter called 911 a second time stating the male had some type of weapon was breaking into the garage the first arriving officer observed uh the side door of the garage to be open with the lights on and could hear glass being broken with the knowledge that the officer received from dispatch by the weapon possibly being present the officer drew his firearm as the officer approached the garage he observed a male inside behind a vehicle the officer could not see the subject's hands this put him in fear for this that the subject may could brandish a weapon the officer displayed his firearm at the subject held the subject at gunpoint while at gunpoint the officer ordered the subject to show his hands the subject complied the officer ordered the subject to slowly walk out of the garage and face the garage door once outside the officer ordered him to place his hands behind his back the subject was being compliant with the orders and they were able to place them in handcuffs with no further incident it was a black male age 19 five foot seven 195 pounds there was one officer who used force he used verbal commands and his firearm was pointed the arrestee had no injury and the officer had no injury um the next one is an agency assist disorderly conduct uh there was a complaint who called dispatch and that officer was dispatched to the scene the officer dispatched to a subject attempting to drive his vehicle that was in the process of being repossessed off the back of a flatbed tow truck upon arrival subject got out of the vehicle and got off flatbed began yelling and screaming at the officer and the tow truck operator running out into the traffic and causing a disturbance the subject attempted to climb back onto the flat bed the officer used his hands on the subject's shoulders to prevent him from doing so the subject stopped it stopped his attempts and paced around until the vehicle was removed by the tow truck and then left the area was a white male age 51 five foot eight 175 pounds there was one officer unseen one officer used force he used verbal commands and the empty hand techniques the subject was not injured and the officer was not injured the next one is an arrest for a felony aggravated assault interference with a law enforcement officer um and disorderly conduct excuse me this was officer initiated the officer was working downtown he observed two subjects a male and a female fighting and intervened to stop the assaults when the officer identified the officer identified himself and told them to stop the officer then tried to detain the male subject who violently resisted and attempted to strike the officer the officer and subject ended up on the sidewalk and the officer was able to control the male the female subject then approached the officer on the ground yelling and screaming at him fearing it was going to be assaulted he pointed his controlled electronic weapon at her in order to back up the female subject complied officers arrived on scene were able to take the male subject into custody uh it was a white male age 29 five foot 11165 pounds white female age 23 five foot three 130 pounds uh there were two officers that used force the male uh officer one used uh on the male subject verbal commands empty hand techniques roughly four of them and then he also on the female subject used verbal commands and pointed his controlled electronic weapon or taser was pointed at her uh officer number two who arrived on scene used his body weight on the legs of the male subject that uh allowed the other the officer wanted to put him in handcuffs officer one had an injury abrasion to his right and left knee male subject had an abrasion on his hand and that was it for the month thank you for that chief um any commissioners have any questions or comments in regards to that incident report not seeing any hands um uh thank you DC LeBrette um next agenda item will be is uh 6.01 um commendations so Shannon take it away um so I do have just two to share with you one uh is a a telephone call um from the mother of uh someone that was having a mental health issue that the officers dealt with she called to express her gratitude and professionalism and compassion um the officer showed in responding to her son uh she said she also wanted to support the entire police department for the work that is done in the community and the second one um is regarding uh the incident that occurred on the ice outside of the echo center um on uh February 14th um with the officer's assistance uh with the Burlington fire department um the communication specialists that were directing the officers and um that the officers were able to help with a rope that was tied to um a vehicle to help with the person that was on the ice and also um that the radio communication was clear and the officers demonstrated a high level of lifesaving um techniques and were able to use quick decision making although unfortunately um the gentleman did not make it and that's all uh thank you um next up is agenda item 7.1 commissioner updates or comments any commissioner any questions or comments or sorry comments or anything that was not adjusting them that was not addressing the meeting tonight please um uh letting me know I'm not seeing any um moving on to oh sorry uh Dr. Dotson go ahead and perhaps in light of the conversation and the topics that are discussed here it might just be helpful to share and experience that uh myself for the chief and an officer and a citizen had there was a individual who had a um a traffic um not even a traffic stop as an accident uh occurred in front of their house uh the new north end um they had an interaction with an officer they had some questions about that uh that person on their own chose not to pursue uh a formal avenue uh they reached out they were actually in touch with their city councilor who was Ali Jang Ali uh turned them my way um so the person reached out to me and said that what their interest was was just to be able to have a conversation with the officer they uh thought that someone down the in the interaction that they had questions about and before they jumped to conclusions about what was going down they uh wanted to have an opportunity to speak about it I talked to the chief about it uh also DC the Breck helped out we identified the person had not uh gotten or remembered the officer's name so we were able through the data systems to um go back and figure out the timing and location and who the officer was uh reached out to the officer um and then there was a conversation um at the once again the request of the community member um it was uh myself the um chief and this uh community member and the officer and we had a conversation and uh according to the testimony of the individual from the community uh she left feeling good about it she had no issues she didn't want to see more video she seemed to uh feel um satisfied and pleased that she was able to about to go about it in that way um seems also relevant to the uh conversation um that it was a white woman but her boyfriend and the part of the um interaction involved a BIPOC male the owner of the car and so um it just seems like uh that's the kind of story that doesn't necessarily get out there but it's something that I had access to and it seemed relevant to some of the things we're talking about tonight thank you for that all right um now to agenda item 8.01 next meeting agenda items um uh I plan on reaching out to BPLA um just to invite uh one of them into our next meeting to um explain and expand on how some of the things that uh Commissioner Green uh talked about with regards to bias training and things like that and um I also have here um I guess what the next step looks like uh in terms of council monitor kind of things like that and um the two table did end items is there anything else that uh Commissioner would like to see on the next agenda item sorry the next big agenda as always please uh feel free to reach out message me and I'll pass that along to Shannon for the next meeting um and um I motion that we go to sorry before I do that um yes so um I motion that we go to executive session to discuss disciplinary matters and citizen complaints I couldn't hear you but I believe that was the second the second yeah um all in favor uh we have four so that's technically still a quorum uh we something had to step out to take care of a personal issue so yes all in favor I move it going to executive session to discuss citizen citizen complaints and um I'm so sorry like citizen complaints in disciplinary matters um raise your hand to say hi hi hi hi passes unanimously um for everyone that's uh tuning in from from the public at the conclusion of executive session our meeting will be done and it will be adjourned at that moment so there'll be no more deliberative actions taken afterwards uh the time right now is 906 um I everyone should have had the zoom link mail to them for executive session and we shall reconvene at let's say nine minutes at 915 everyone else in the public thanks for joining us and our next meeting will be tentatively scheduled for the 23rd of March which is the fourth Tuesday hope to see you then thank you bye


Hi, I'm Jane. I'm the Editor-in-Chief here at Food Unfolded and today I'll be addressing the question Does the fair trade system really work? Some of you have asked us to investigate if the fair trade system is really fair or if it's just greenwashing Greenwashing is when a company or organization spends more time and money Marketing themselves as being ethically driven rather than pursuing ethical practices. This is not the case for fair trade The system is actually focused on making a real impact on the ground with producers and laborers from low-income regions Studies have shown that the fair trade system does indeed improve the lives of its producers and their surrounding community They have higher bargaining power and trade negotiations They receive higher shares of revenue because of the minimum price model and the fair trade premium has had a Significant effect on the social welfare of the community as a whole But it's also important to point out that benefits for fair trade producers and laborers will vary depending on product and region A critique will be exploring in just a moment now Whether or not companies are using the fair trade label to market themselves as being ethically driven is Another question companies can't just slap on the fair trade label They actually have to be paying the fair trade price in order to even be allowed to use that label at the beginning of the Trade chain producers are already paid the fair trade price by the first buyer These costs are just passed along the trade chain While some companies might pass on these fair trade prices onto us the consumer Others absorb the costs and take a lower profit margin So whether or not it's just a PR move for companies to use the fair trade system It's still making a difference for producers and laborers in the fair trade chain But the fair trade system does have its limitations here are three ways in which it's limited Fair trade focuses on long-term financial sustainability of its producers and laborers But long-term financial sustainability cannot be achieved without considering environmental sustainability Fair trade international says it themselves Farmers are on the front line of climate change facing higher temperatures droughts floods Extreme weather crop failure and more however while fair trade standards encourage Sustainable agricultural practices. It is not mandatory though It's important to consider that sustainable agricultural practices like organic farming Can have higher costs the benefit of being part of the fair trade system though Is that producers earn more and can reinvest some of their revenue into more sustainable practices in another? Another environmental issue with fair trade farming is a cluster of monocultures present in regions as small-scale farmers are producing the same crop To tackle this fair trade has been working together with farmers to diversify their land crops and products To protect their producers from further losses due to issues of climate change fair trade is also exploring the concept of climate risk insurance One study in the Costa Rican coffee sector showed that unskilled workers like coffee pickers and laborers Did not see an increase in their wages under the fair trade certification system to be clear under fair trade All laborers are guaranteed the legal minimum wage of their region However, there is often a gap between minimum wage and living wage Meaning the cost of having a decent house decent food access to medical care Etc. So then how does paying minimum wage line up with fair trades aim to empower the most vulnerable people in the trade chain? It's very delicate from an ethical standpoint because the farmer himself is already poor when you have this very small Farm where maybe for a few weeks a year they will hire someone to help them and these guys are already poor It's difficult to go and tell them. Hey, you have to pay them a wage at you and not getting But it's an issue and that's something we are trying also to regulate and to check and so on But it's very delicate and it's very complex because first you have to make sure that a small farmer is getting enough Pays so that he can then afford to pay his workers correctly because if you only protect the worker You will chase the farm into poverty. So it's it is a very difficult issue to tackle So what has been done to improve conditions for fair trade workers on fair trade certified plantation and estates Fair trade premiums are 100% Reserved for its workers the workers manage a joint committee that receives the premiums and there they can collectively Agree to use part of that money as supplemental wages Additional to their minimum wage. They can also use the premium in other ways Like buying coats or paying tuition fees for their kids The most important aspect of the fair trade system is that it leaves the decision solely up to the workers not plantation or state owners Allowing them to invest these additional funds into what they feel best fits their needs Fair trades auditing system has also come under fire as child labor was allegedly found on a fair trade cocoa plantation in the ivory coast By an investigative journalist team from Dan watch in 2020 It's unclear whether the lack of auditing was due to travel restrictions under COVID-19 or systematic flaws But whenever you hear one case it really makes you wonder how many more cases like this exist How does fair trade truly ensure that its producers are not using any illegal forms of labor? Fair trade audits are usually announced in advance, but by announcing their audit schedule How can they be sure that there are no opportunities for cover-ups fair trade has criteria to identify high-risk Situations and in areas that have shown to have a higher risk of illegal labor Certifiers do conduct unannounced audits The general spirit is one of trust and it's also sometimes a Western point of view to say we are going to check these guys Because they are the bad guys. Well, they're not necessarily the bad guys. I mean, they're just people like you and me I mean because these people the children work in a farm they would rather send them to school, but they can't afford it and They need them to work on the farm because they cannot afford to hire someone for a few weeks for the harvest That's why I think a whole certification and auditing process is only part of what we're doing and it's a means to an end It's something that that make things better, but it's not gonna solve everything And that's why we also want to tackle the root cause you can limit yourself to audits and rules You know, it's forbidden. You cannot have child labor. That's it It helps to have rules because at least things are clear It helps to check because to create a little bit of pressure But if you at the same time don't do anything to make sure that these people are paid better prices You're not going to tackle the root cause of child labor Fair trades approach is one of balancing strictness and encouraging development Of course if major certification requirements have been breached Certifiers will suspend or decertify fair trade producer organizations or traders But fair trades overall focus is to give their producers an opportunity to address problems and improve over time Fair trade is not a perfect system, but it's a good start Today the fair trade system is comprised of over 1.7 million farmers and workers from 72 different regions and territories And has over 35,000 fair trade certified products It might seem like a lot, but fair trade producers actually account for less than 1 percent of the 570 million smallholder farms worldwide This means that over 99 percent of smallholder farms do not work under a trade model that guarantees fairer prices Or the respect of basic working rights One thing is clear fair trade has impact The movement has been gaining traction with more consumers and retailers Committing to buying more fairly traded goods Since 2014 fair trade farmers and workers have received over half a billion euros In fair trade premiums the fair trade movement has brought light to the pervading and de-brooded problems of the trading system today And while it may not be the solution for change. It's certainly one of many


Landslides represent major natural hazards, which cause every year significant loss of lives and damages to buildings, properties, and lifelines. In the last decades, a significant increase in landslide frequency took place, in concomitants to climate change and expansion of urbanized areas. Remote sensing techniques represent a powerful tool for landslide investigation. This article was authored by Marcos Keone, Laura Langone, Valentina Malillo, and others.