Okay great well hopefully you have an idea what's happening in the urban metabolism field how things work you've seen these different projects that are happening in Cape Town and we've discussed some of the issues the challenges that exist so I'll be showing you a project as we've been working on with metabolism of cities to address some of these these challenges so let me first and this this platform is called multiplicity I hope you enjoyed the pun as much as I do so before we go there let me explain a bit about our organization so metabolism of cities Lauren mentioned something that started as a band of people and I think that's quite an accurate description it's very much a project that came up as a passionate side hobby of a number of researchers who all worked in urban metabolism and who felt that it's great to get together and to jointly do things and our main and first project was to put information that we all gathered and we did we all came across on an online open-source website so we were all doing research and we found literature scattered around and we figured let's put a database of what literature exists and how it relates online from there we started growing and adding other tools and sections to the website and we started adding more people as well bit by bit and most people located in different countries so none of us meeting initially and over the years we've met a number of times now but we remain a group of people who get together out of passion to work on urban metabolism and who very much it links back to Sarah's comment we very much care about accessibility of the field itself accessibility of information accessibility be accessibility of data so we try and build generally online tools that help address that so that's the core of our our project we are strong believers in in open source in open access so we bring that in and yeah we think urban metabolism is interesting and has a lot of potential and we also think there are challenges and we try to work in that space and collaboration is key in what we do so that's a quick background on metabolism of cities here you can see we defined a goal for our group for organization so as you can see here to collaborate on system medically improving the sustainability of cities you gotta aim high by creating and sharing urban metabolism knowledge and accelerating its implementation in policy and practice now you can see from that that our event today is very much in line with our goal and it's it that's why it was something that we really wanted to organize and we're very happy that that we could because we feel that these kind of events are our attempt to contribute to this implementation in policy and practice now if you look at this work in urban metabolism it's come up a number of times that data is an issue and there are a number of challenges when you do work in urban metabolism around data so having done an MFA so material flow analysis is one of the accounting approaches to urban metabolism in which you draw up a material balance on any scale but of course in our field on an urban scale and that's needs to be or is often economy wide it's all the materials that go in and out of a city and it's incredibly difficult to get that data that applies to Cape Town but that also applies to Paris or London in no other place is there a system in place to just track everything the borders the boundaries are very porous everybody goes in and out lots of different actors involved etc so that makes that the work of getting data incredibly time consuming and lots of researchers spend most of their time not analyzing or interpreting information they spend it collecting data not necessarily the best use of time but it's what you have to do if you want to do this work at the same time if you look at policy for someone in policy to do something with this you need data first which means you need time which means you need money to just get this going so that's of course a limitation at the same time even if there is data available it doesn't mean you can access it so thinking of material material and energy flows through a city lots of that information sits in private hands or in the hands of government agencies who don't necessarily make it publicly available that is a challenge in itself and yeah it makes life even harder if you know that there's data out there but you just can't get your hands on it the scatteredness is a problem because we look at the system we look at all material flows well who's gonna have all materials in and out of Cape Town or any city no one has that level of data people in the water sector have information on water and then people on the supply side have data on the supply and people on the output side of data and output but that has nothing to do with transport that has nothing to do with food that has nothing to do with construction so it sits in all these different places and you have to go to all these different places to get your data and put it together bit by bit lastly formatting so you'll see some data published in a beautiful government report which is actually some sort of a scant version of a table well you have absolutely no way of getting that into your system other than typing that over you've other people who publish it at least as a spreadsheet but you also see all kinds of other years geo reference data comes in the form of a shape from there are all kinds of different formats in how the data is packaged and also different formats in how data is being published the goal of many organizations is not to provide data for urban metabolism resources researchers they have other goals and they publish based on their needs or their users needs so a wastewater treatment work may publish data on the contaminants in the outflow but they will say percentages they will not report on the total quantity that flows out in that particular material out of their waste water treatment work and you'll see all these different users report in different ways so you have to calculate you have to do all kinds of applications of massaging the data to get it in the right format uncertainty is a whole other issue but you'll see it's difficult this whole data picture is difficult but fundamental we need to get these numbers otherwise we can do our work remember our pyramid so if we look at this pyramid we if you have paid attention you remember that this whole data bit is part of the foundation having systems to record it but as we go up you also see information about the visualizations also important it helps engage people with this make it understandable now remembering this and remembering the difficulty with data we figured okay we need to try and do something that can help address these challenges and given the fact that we do things online and we run a website we figured a potential way of doing this is by developing an online system that helps address some of these these challenges so we came up with this idea the system we called multiplicity which is basically a data driven online dashboard for urban metabolism so the core features of this at number one we talk about this needs to be user friendly this is not for researchers this is not only for researchers we want this to be accessible by anybody who has no clue what urban metabolism is and still feel this is interesting and engaging and useful at the same time we do want it to be something that researchers researchers can engage with so we come from an academic background and we needed academic rigor in our system so even though there's nice images and scrollable maps we also need to be able to go dive into the data and see where does it come from how is it generated what is the reliability etc remember we believe in open source and open access so that applies to our system as well to anything that we develop licensed as open source and we our goal is to make this data more publicly available so community driven we can't do this by ourselves you've seen how much time it takes to gather data well if it takes so much time why don't we try and involve more people spread the work so you must have heard of the idea of crowdsourcing something and we feel we can apply the same methodology to urban metabolism data for some of the things you need to have advanced degrees to understand what's happening but for many things you don't and there are lots of things related to urban metabolism you can see from all the discussions that lots of information and knowledge can sit with people who just know something about something and that's already useful so to give an example when we study urban metabolism i would love to know where wastewater treatment works are located just knowing where they are what their name is where i can find them that's incredibly useful and you can find it on the website of the government but that's the kind of stuff that takes time to put together you can find a company having published a in their annual report data on their local we have a brewery in Cape Town so they publish data on how much beer they brew here in Cape Town that's incredibly useful but it just sits in this hidden away report that just requires someone to go there and get that out so we feel we need to harness people's enthusiasm interest in this field and say well anybody can help because we need so many pieces of the puzzle that don't need advanced training and whatever training you need we're here to help so replicable and scalable if we think of cities and if we think of urban metabolism work we often encounter when we try to do comparisons between cities we encounter problems that cities are so different that it's very difficult to compare and urban metabolism in itself is already trying to make that easier because of indicators and because of things like that but basically the standard situation if you try to just compare cities you you're talking about you're trying to compare apples and forklifts what we hope to achieve is that we can compare apples and bananas that would be great so then we're already getting somewhere that makes things better and in this whole field there is no such thing as perfect but at least if we advance things we get to the next level of understanding that's already better than where we are now so replicable and scalable we want to roll this out not for one city but for a number of cities and in principle we've set up this online platform that allows people to put data in for Cape Town just as well as Oslo and Mexico City and Bogota doesn't matter the idea is that any city can do this and of course there are local nuances to take into account but we are there to adjust to the local nuances so that at least we can see what's happening in different cities you have the same data set and you can see it across the board so we've been working on this for about a year now it came to our minds a bit more than a year ago and we started working on it as our group of people as an unfunded endeavor just to try and get this out it's still in a prototype version and we still feel there's so much more to do but you got to start somewhere and you got with these kind of things we can tell people about it but the best way is to show people what we have in mind so that people see what's happening and they can envision better what our final goal is so let us go and see some of these things so i know the screen may be difficult to see but you can check things out later as this is also online i need just help from iris finding a right button on this computer as it has a french keyboard can you just do control plus a couple times to make the okay so in our dashboard we have started working with capetown because well i'm located here and you've seen there's a number of researchers in the city making data on capetown relatively abundant um so we figured let's start loading some of the data that we have on there let's start seeing how things look for capetown and then we can take it from there now this system is in our website in the city's tab where you can then see an initial dashboard that shows a little bit of the city okay here we see the other cities that exist in the system as well you can see we're working on a number of other cities that we have lined up with the project we're doing starting at the end of the year and we've also been working with the city of The Hague in the Netherlands where we've also set up a Kickstarter the first initial round of data so if we look at the platform it starts out by a simple homepage that shows a photo a map and in which we describe or list a number of sectors that are relevant to urban metabolism many sectors nearly all but not all many sectors are relevant to urban metabolism because nearly every sector uses resources in one way or another so our ultimate goal is that on this website you can find information on any sector that you're interested in and you can find all information on that sector from an urban metabolism point of view so what kind of information well you can see on the left one thing is data sets so that kind of you know spreadsheets and whatever information we can find on that sector on people in that sector on flows that go through but that's not the only thing so mapping things out wonderful useful we want maps on there we also figure out we want to figure out what research has been done and catalog for each city what journal articles are there what reports are there they have people done theses on the on the city etc remember we want this to be accessible to anybody and just photographic appeal helps a lot and all these places i mean i don't know if you've been here into these these different the ways what the treatment works over there it's great to see that and how many people know what a waste water treatment plant looks like even though every day they depend on it so just putting up photos of the infrastructure that goes with this is incredibly interesting at the same time if you think of different cities i would love to see apparently in brussel they incinerate all their waste i have no idea what an incinerator looks like so if i could go on a platform and just even if all i do is browse photos i think we already bring people a bit closer to the idea of resources and the infrastructure that's behind it and how things move through a city okay so i'll show one of the sectors that we've set up in capetown that we started setting up so let me go up so now we're in the water section so what we do is we try and describe how does water move through and what is related to the water industry in this case capetown so we didn't have to describe the process or the life cycle of water the number of ways of doing it but one of the first things we do is we try and describe the different bits and pieces that are related to that process so water is first in the sketchment area it gets collected in dams it goes to water treatment works it goes to reservoirs it gets pumped around in the city it is being consumed it then goes to waste water treatment works it goes out in marine outfalls and you also have your treated effluent now all of these things have bits of text and information in them so people can click on any of these things and then read up so if i'm a user and i say i want to learn what's what's the deal with water i go through there maybe you can click on one um you go through that section and all you do is you just click and you say okay let me read up on that short paragraph in capetown we were very lucky that with the water crisis some great documentation was developed by the city on exactly this on explaining what is this infrastructure and all the processes around water so that's the top part then as you go down you'll see that the next section shows some of the data sets that we have on water um this is something that will take time to populate but we start by just putting in what have what have we found so far and you can see here maybe you can do preview of the other one so there's an interesting data set on on oh sorry in the preview um on the wastewater effluent it's in the open data portal of the city of capetown and it has data on what has been going through each of in terms of volume through each of these wastewater treatment works this is the kind of information that's incredibly interesting for a research or point of view but if you don't know where to go it takes time it takes effort to find this data and when you download it it's difficult to process it and here with a couple clicks you can just explore you can see which is the biggest wastewater treatment work how has volumes changed over time so that's the kind of stuff that we try to show people in a more visualized visually appealing way while also allowing people to dig deeper so for a data set if you go down a bit in the form of up if you look at this this profile you can see where it came from what was the original source you can see some data quality indicators you can go to a table and get all the the numbers you can download data and you can see if you want to replicate this data set what are the steps i need to take and sometimes the steps are go and download sometimes the step are follow these 20 procedures to transform that into what you need but we need to share it with people so that we can continue to update the information and others can see how it came about so there's a number of as you can see number of different ways of visualizing data and all that's needed is that a user uploads a data set in a particular format because remember we want this to be crowdsourced you don't need to have a a degree in programming or visualization to get this so all a user does is they format data in a particular order column a is the time frame column b is the place column c is the volume something like that copy from a spreadsheet into a field click next review their data describe how they got it and the system does this the system creates this overview shows this creates the the different ways of visualizing data if you georeference it it can put it on a map no we'll show it there yeah okay so um that's the data set part and there's there's much more to it but just to give you a brief overview we also very much into this mapping thing and to put things on maps so as long as we have information about where a waste water or water infrastructure is located we can have the system generate a map of that so if you go further down so here you see the different pieces of infrastructure that exist in the water sector at least that we have identified um and as you go down depending on our internet um you'll see a map pop up um that displays all those different pieces of waste water infrastructure so you can see where remember we saw earlier in the same press in this in the same block we saw the map of all the waste water treatment works that is a an image that's great to see but it's difficult to get hold of it now once someone has uploaded the information here anybody can get hold of this information you can browse it maybe you can just if you see on the right hand side it's the different types of infrastructure you can you can click and filter and what you can do is you can click on any of them you'll see their profile and maybe just open the random one so here we can see this is clip hugo so waste water treatment works yeah um and actually i've opened the one for the cape flash because that's where we're at so as we go up this is each of the piece of infrastructure has an individual profile so this shows where we are right now this cape flats waste water treatment works someone up well in this case it was me but it could be anybody someone uploads data on the gps points what this is what the name is and then the system creates this profile so it puts it on the map it shows a satellite image the moment someone else or the same person uploads a data set and says this is related to this place the system pulls it together and says okay that's linked to this waste water treatment work so the same data set we saw before of all the ways where the treatment works it's one upload but it's then separated out and filtered out so people see it in different ways um yeah and if someone takes a photo it can be uploaded there and we can browse and see what's happening at that place so that gives you an idea if you go back to the the water page oh sorry go to waste so it gives you an idea of what's happening in the water sector this is replicable to any sector so we've done the same with waste so same idea describe the sector get some text on there people can orient themselves um put data sets on there that are available on what's happening in waste if you go further down show what's the infrastructure that exists around waste we have one landfill right across from here you can see it on the map um you can open it you can see what the volumes are that are in there oh and where you haven't shown at the same time we pull through those documents that have been uploaded that are related to waste to water to whatever it is so yeah i think this is enough to give you an idea i invite you to check out the website and see what's happening still a prototype we're still working on it but it gives you the idea of where we're going now our goal our end goal isn't to just have a place to centralize data it's useful and it's it we think it helps a lot of people and we also think it's something that will motivate others to contribute who now not contributing to contributing to this conversation to make it so much more accessible to view it or to contribute now great what do we want in the end remember we want to move ahead in policy and practice we feel that this is a useful tool for policymakers to visualize and to to see things but what we really want is that we can create tools that help decision making and if you think of this kind of data there are a number of tools that you can invent that help people and that allow for a much easier way of running a model and to give a very simple example let's say there is data on what infrastructure is available say on waste on wastewater so we have data on what the technology is at each of these wastewater treatment works we heard it earlier and we can have it in the system well if we have data on what each of the technologies implies in terms of environmental impact of course there's a different impact if you use this or that or that we put in the system what technology is being used and we have in the system what the impact is of each of them then we know what is your current impact of the technology that you have add to that a database of what other technologies are there and what could that mean add to that information about what the cost is and what you could do is you press a button and you say show me what we could do what we could upgrade or change in the city and what the impact is of that so if we swap out even say landfill if we if we talk about incineration we have examples of cities who do incineration that has a cost that has an impact that needs a certain structure but if we have tools like this then we can have those tools help us understand what options we have and how do they compare in terms of cost in terms of impact in terms of whatever we can can think of we're still very early on in that conversation but we have a colleague of ours who does work on islands who's with whom we are developing metabolism of islands who has experience engaging with government and a lot of ideas on these models and that's the kind of stuff we want to work towards it's going to take time but this is step one or step zero and we have step two to six in our head but we don't know if that really makes sense and we really hope today to also get ideas from the other side of the table so if you are a policy maker if you're a decision maker could be a city level could also be at a corporate level companies are also decision makers so we'd love to learn what are tools that could help you if you think of having data sets and infrastructure information and whatever you can envision we are keen to build but we are not sure what we should be building so you let us know you brainstorm with us and we think that's the way we can try and co-create something that helps improve our system but helps most importantly your decision making like I said it's early on in the conversation but for us that's really a key thing we hope to get out of here to at least start that kind of conversation and keep us in mind we are a collaborative platform we are we try to get funding here and there wherever we can but whenever we have the option we just do things it's not about funding it's about getting things done and the cool thing is if you do it for one city it becomes available to another city so let's push so Brussels wants lots of things and then Cape Town can get them as well yeah so I think that's an overview of the platform that we have visit it check it out give us constructive criticism and let us know what you think and what you need so and start uploading yes exactly if you have data give it to us put it in here there is there are videos that show you how to do it there's all kinds of different things you can put on there from you know even if you're not an expert on this we want to know if you see something that's interesting in terms of your metabolism just put a photo on there that's already very interesting very useful so there's lots you can do and contribute okay thank you


Well, we're here in Kamloops at the new Brocklehurst Gemstone Care Center, which is a 130-bed home for long-term residential care. So seniors throughout the Interior Health Authority around Kamloops can come to this beautiful center and live a wonderful life. There's 125 publicly funded beds here, which really looks after the wait lists that we've had in the region and takes it down to almost nothing. Well, for the families, certainly Kamloops is the center of a very large geographic area. With a community like this, the extra beds will enable people to be able to still be close to their grandparents and their parents as they continue to age in place this way and are able to really keep that family connection going. We are focused on making sure joyful living occurs all the way through and finding those moments of joy for people. It's something very special at whatever age we are. Well, one of the things that is really impressive is the fact that it is so light. There are, because of the design of the building, there are windows almost everywhere. So you have a great feeling of openness and light, which I think is important. Well, for me, it's like working in a resort. It's a beautiful building inside and out. And one of the residents when I was cleaning his room, I asked how he was liking it and he goes, well, all we're missing is the casinos down below on the first floor and it would be like living in Vegas, he said. So yes, it's a real honor to be here and it's a lot of fun. This is a facility that is very modern. It's got all the comforts of home, a really caring staff and a beautiful, beautiful building close to the river here in Kamloops at a beautiful setting. And it shows the commitment from Interior Health and the province of British Columbia to caring for our seniors, who of course have cared for us for so many years.


At Audiology Associates of Prestonsburg, you can live your life the way you want and find the freedom of better hearing. With 40 years of being in the hearing care industry, you'll experience patient care that is specific to you with exceptional follow-up care that ensures your hearing and balance needs are being met. Audiology Associates at 1428 Northlake Drive in Prestonsburg. World Immunization Week celebrated the last week of April, works to highlight the importance of vaccination to prevent deadly diseases. Millions of children missed out on vaccines during the pandemic, leading to an outbreak of measles and other vaccine-preventable diseases. We spoke to Public Health Director Tammy Riley on the importance of vaccination and vaccine safety. Many probably are aware that Kentucky's recently had three documented cases of measles, for example, and that's something that we've normally not seen in recent years. So herd immunity for most diseases varies depending on the disease, but for most it would be around 95 to 96 percent participation in vaccines to achieve herd immunity. So when that occurs, then the whole population is safe. And that's how we eliminate diseases, such as polio. The World Health Organization estimates that childhood vaccines alone save over four million lives every year. Well, vaccines are available for a wide variety of serious diseases. And over the years, vaccines have prevented serious, deadly diseases and saved millions of lives. But the fact of the matter is, vaccines, the recommendations vary widely depending on an individual's age, their health, where they live, their travel habits, their occupation. So there's a wide variety of factors that influence vaccine recommendations. But the fact remains that vaccines are widely available, that they prevent serious and deadly disease, and that it's documented. But that we are seeing in recent years an emerging, a reemerging situation of diseases that we normally haven't seen in years, like the measles situation recently in Kentucky and across the nation. For more information on vaccine safety, immunization schedules and more, go to PikeCountyHealth.com. Reporting for Mountaintop News, I'm Brianna Robinson.


So, what we will do today is we will look at, I show you a demo, right, we will use hat functions and what we have seen so far is the intervals are important, the number of intervals that we have or the size of the interval is important for orthogonality and the fact that the intervals do not overlap is important for orthogonality, right, the size of the intervals so far did not matter, right, we took them as equal sizes, right, that is what we basically had. The number of intervals increase the accuracy with which we were able to represent the straight line for box function, if you think back to the example of the box functions, how well we were able to represent the straight line, it got better, the error reduces as the number of intervals that we use increases. So we will try to see if there is a relationship between the function that we are able to, that we want to represent, right, and the number of intervals, is that fine and I made use hat functions to do the demo, so this is a, so there will be a combination of things that we are going to get out of this demonstration. I am going to use a scripting language called Python, a particular version of it called IPython, just so that what comes on the screen does not sound strange to you. I am going to use two packages, I would suggest that you find for yourself some package with which you are able to plot, right, plot graphs and check them out on the screen, there are lots of packages out there. I am going to use a package called grace plot, which I will, this is as I said so that you understand as I pull up those packages what it means, what I am doing. And to represent arrays, I am going to use a package called numpy, numpy I guess for numeric Python, but anyway numpy, which is all that we basically need, right. So the objective is to see that if I take a function say for example sin x and I use 10 intervals on which to represent sin x, how well do I do it? And then we will change the function sin x, sin 2x, sin 3x, sin 4x to see whether it gets better, gets worse, you expect that maybe it gets better, gets worse, anything before we start? If I use 10 intervals and I try to use, I try to represent sin x as opposed or sin 2x or sin 3x, sin nx, you expect the representation to get better, representation to get worse, you expect the representation to get worse, fine, you expect the representation to get worse. And the reason why I pick trig function, actually numerous reasons why I pick sin x, one of them of course is that it is possible for me to scale it, I can change, there is an inherent length scale associated with sin x and by changing going to sin 2x, sin 3x, sin 4x, I am changing those length scales, okay. So let me start it off, the first time around I will set it up, maybe subsequent classes I would not go through as much of a setup, I am going through the full setup this time so that you see what I am doing, okay. So I am going to run I Python and I am going to set up my grace plot, initially it is going to be very intense, I will change the background color, I create the plot, it is going to be an intense white color, so I will quickly resize it and I have set up something that will give it a more not so bright, not so bright color, not so bright color, okay. Now what I will do is I will now get my numpy in, from numpy import star, you can go learn Python if you want to use this programming language, most of the stuff that I am doing I am not going to do anything unusual, I will create an array, so numpy has in it various things including definitions of pi, sin and so on, right. So I will create a row vector, peculiar way by which you do it going from a row vector whose values go from 2, 0 to 2 pi and I want 11 points that is 10 intervals, so if you say what have I managed to do, that is what I have managed to do, so I have x going from 0.628 so on to 26.28 which is 2 pi, right close to 2 pi. Now it is very simple, I create y which is sin of x and since my plotting utility is going to connect these points by piecewise straight lines which means that inherently underneath I am doing hat functions, I will just use it to plot, okay. Now if I were to use qubits, quadratics I have to be a little more careful, I cannot just use these straight lines, I have to be a little more careful, I have to actually plot the interpolating polynomial, right but in this case by coincidence it happens that it is going to join them by straight lines so I am using hat functions, okay. So I say g dot plot x, y and I get that, what do you say, are you happy with the function? You have to be a bit critical, is there any problem with this function? First of all it does not seem to reach the maximum, right, it does not reach 1, there is a small gap, there is a small gap, there is a small gap here, why does it not reach 1? Because I am not sampling the function at that point so I am using the word sampling here, I am not sampling the function at that point, I am not sampling the function at the peak, okay which is the reason why anything else? It looks reasonably like sin x, it has a reasonable semblance to sin x, okay, it has a reasonable semblance to sin x, maybe if we try, excuse me, higher number of grid points or whatever it is, maybe, right. So what about sin 2x, shall we try sin 2x? Sin of 2 star x, I do not have to keep typing this every time but anyway, what now? So is it any better, it is worse, it is worse, it really does not look like, I do not know, maybe the peaks also do not seem that great, the peaks also do not seem that great but actually if you look at it, it will turn out that the point where it crosses the 0s, the 0 crossings so to speak, these are the 0 crossings will turn out to be quite good, okay. On this graph of course, it is not that easy to find out but you can actually evaluate it and check it out and you will see that the 0 crossings are actually fine, okay, the point at which the function crosses the 0 is fine and the number of 0 crossings is fine, the number of peaks and valleys is fine, the number of peaks and troughs is fine, right. The function values seem okay, they are not great and with 10 intervals, this is not looking that good, so we will see what happens if I go to 3x, okay. So tell me what we have, I see a lot of smiles, tell me what we have, right. So here obviously that minima is gone, the minima and the maxima here are gone, okay and has it caught at least the frequency, does it seem as though it has caught the frequency is the number of 0 crossings that I am going to have, right divided by 2 in this case, so it is like this, so here to here, I can count, I can count, it looks like the frequency information has been picked up properly, the 0 crossings seem to have been picked up properly, amplitudes are totally off, okay. So now I am doing sin 3x with 10 intervals, okay, so there is clearly some relationship between the number of intervals that you are going to pick and say the underlying frequency or length scale that you are trying to represent, okay, so that is one of the things that we want out of this. Let us go ahead, I mean we will just try 4x and see what it does, sin 4x, now this is like nothing, I do not know, you tell me, I mean what do we have, do we have anything at all, do we have any, is the frequency information correct, is the number of 0 crossings correct, how many times does it cross 0, there is 1 here, 2 here, 3 here, 4 here, 5 here, 6 here, 7 here, 8 here, 9, 4x, that looks about right, okay. So in some sense the frequency content is there, right and the function representation, that is if I were to show you this, you would be hard pressed to say that oh it is actually sin of anything, sin of 4x or something of that sort, it is difficult to say that it is sin 4x, okay. Let us try the next one, what do you think will happen here, it has been getting bad, how bad can it get, how bad can it get, this is I should actually, this is, you have to look at the scale before you look at this, right, that is 10 power – 15, okay, 10 power – 15, just for the fun of it, just so that you can see the kind of stuff that you may have to do when you are running or exploring these programs, I will reset the scale here to – 1, plus 1 and that is what you get. So what happened, we sampled the functions exactly at the 0 crossings, now this is a bizarre situation, we picked up the 0 crossings exactly but we have nothing, right, we do not, the functions just totally disappear, we have not picked up the peaks anywhere, so we really have nothing, is that fine. So let us go on, we go on and see what we get, after all we have 10 intervals, 11 grid points, 11 points, 10 intervals, so what do you expect for 6, going to be the same as 4 is the prediction that I get, indeed it is the same as 4 or similar to 4, is it the same as 4, this is a sign change, this is a sign change, so you would expect and I think if all of you are familiar with this, I do not have to do all of, I do not have to go on, so if I go to 7, 7x, okay and we see that the pattern is repeated, the pattern is going to repeat, the pattern is going to repeat, this is what I expect, I go quickly to 8x and I plot that, right, 8x is getting better but it is not 8x, right, 8x is getting better but it is not 8x, 9x, what do we expect for 9, yes, a bit strange, it is clearly the sign is flipped and 10, 10 will be, 10 should be a straight line and of course I mean, I do not, I am not going to rescale it again but 10 should be a straight line but we will see if we get the same, actually we get the same value, it should be a straight line but remember we are dealing with numerics and yeah it is not quite the same but yes the scale is still 10 power – 15, so if I make it – 1 to plus 1, it will just be a straight line, the value is essentially 0, it is within machine epsilon, value is essentially 0, okay. So and if we go on what do you expect to happen, if I were to repeat this process what do you expect to happen, it is going to keep repeating, it is going to keep repeating, so the first lesson that we get out of here is if I have 10 intervals, if I have 11 grid points, there is a largest frequency that I will be able to represent with those grid points, right. If I have 11 grid points and I am using hat function, there is a largest frequency that I am going to be able to represent using those grid points, is that fine, does that make sense, okay, fine. Now we can try a different set of grid points just to see if we can infer anything else, there is a largest frequency, we had 10 intervals and the largest frequency was what, it was 4, so it looked like maybe right now you would say n by 2 – 1 or something of that sort. So let us pick some other number, let us say we take x to have going to 2 pi, how many shall we take, 21, you want to take 21, take 21, 21 grid points is about 20 intervals, it is 20 intervals, right. So we go through this business again y equals sign of x, check this out, 21 grid points is definitely better than, right, 21 grid points is definitely an improvement on, 11 grid points, 20 intervals is definitely an improvement on, right, 11, it looks like though it is pi, pi by 2, you know you are not going to actually sample it exactly at pi by 2 but we are quite close, we cannot make out that, so it looks like it is pick the peak well, the function representation is quite clean, okay. So this is one thing that we should bear in mind now, a lesson that we take from here that if you are representing a function, good idea sometimes either an engineering way to look at it would be to double the number of grid points to see whether there is an improvement, any improvement, this is something to bear in mind, right, it is something to bear in mind. I will, we will come back and visit this, that suggestion at the end of the semester, okay. So yeah, the function definitely looks a lot better, what about, what do you expect will have to happen to 2 star, 2 times 2x, this I would expect, this should be the same as sinx on the 10, right, so now it is very clear, this should be the same as sinx on the 10, right. So if I now go ahead, if I now go ahead, we will do a few of these and then I will sort of jump ahead after that, so if I do 3x, 3x of course has no equivalent there, right, so it seems to have picked the peak in some places, it has picked the peak in some places, it has lost the peak in some places, the 0 crossings are not there or I mean are good, otherwise yeah, you could say it is sin, I mean we can live with it, but I want you to bear this in mind every time, so if you are going to, if you are actually trying to represent a trigonometric function, right, it looks like, if you are going to represent sinx, it looks like if you want to capture the function well, right, it looks like you need at least 20 grid points, 21 grid points, 20 intervals, right, employing hat function, you need at least 20 intervals for a given wavelength, that is one lesson that you get out of this, you need at least 20 intervals per wavelength. So if I am talking and you say I want to model this in some fashion, right and you want to model it so that you pick up that signal, you actually pick up the pressure variations, then you need at least 20, you could live with 10, but 10 does not look that good, right, 10 does not look that good, okay.  So you can get, you can get, it is very clear that beyond a certain point the functions start looking very strange and they do not really look like sin x, sin 7x that you want, okay. That is the only reason why I am sort of bothering you with this, it also looks like, if you look carefully now, it looks like on top of this, if I look at the envelope over this, there is a sign kind of something like a sign that is riding on top, okay. So I am sorry about that, there is something like a sign riding on top of that, okay. Let us try 8x, 8x should be something that we have picked up earlier, 8x is something that we picked earlier, 8x corresponds to 4x that we had, however, of course the wave number is different, the frequency is different, normally when you do it in space, so when I am doing these, when I am, when we are talking about some of these kinds of problems you will see me use wave number and frequency in a sort of interchangeable fashion, okay, but normally when it is in space we say wave number when it is in time we say frequency, okay. So the wave number is of course different, but it looks very similar to 4x if you look at it on the 0 to pi interval and what happens to 9, why do I want to try 9, 9 should be the bad one, 9, yeah and it has that envelope again has that variation, so there is something riding on it, a sign x riding on it literally, okay. And what do you expect will happen with 10, 10 should be 0, each one has its own, it is 0 but each one because of the nature of the representation of the number itself on the computer, right, it should be 0 but each one is a little different and the magnitudes you can see that the maximum and minimum magnitudes do not necessarily see it to be the same and we get the impression that as the value as you go further and further down it is getting noisier and noisier that is it is closer, it is smaller value here, closer to the 0 value here and much worse at higher angles, okay, these are all observations that we can make just purely from looking at what we have got so far, right. So what do you expect will happen to 11, 11 will repeat, so it looks over, so this is for this reason 10 would be called a folding frequency, it looks as though the figures fold over, right, fold over in the sense that 9 is the same as 11 except for a sign, right, 8 is the same as 12 except for the sign, so it is as though we have folded, folded it over, right, we have folded it over that frequency and anything else that we can say, so if I give you n intervals or n plus 1 grid points, right, if I give you n plus 1 grid points then n by 2 seems to be the largest, n by 2 in fact seems to be the troublesome frequency, it is 0 there, so let me try a different set of grid points again, let us try something, we have for both the ones that we tried were odd, 21 grid points, 10 intervals, the number of intervals is even, why do not we see what happens when we change, okay, so the first question is do you expect any change, do you expect change, x equals you expect change, okay, how many shall I do, I do 20 or let us keep it at 10, that way we do not have to, of course the trouble with 10 is it would not look that good but it does not matter, that is not bad, the one improvement of 9 intervals over 10 intervals is that we pick the peak better, okay, it is not draw major conclusions from that, right, I mean it is possibly just happenstance that, it may just be chance that that is happen, the function does not look bad, right, the function does not look bad, what about 2x, again like last time we pick the peaks better, this peak is off, that peak is off, 3x, this is a nice sawtooth, we have picked the amplitude, right, we picked the frequency, right, the 0 crossings are right, it just is not sign, that is only problem, right, it got all that information, of course if you talk to your electrical engineering friends, they will tell you to convolve it with some sync function or something of that sort extract out the signal but it does not matter, right, for us as far as we are concerned we are not going to do complicated things like that, I just want the sample points and I just want to use linear interpolants to represent the function and it bothers me that this is what I am getting, okay, it bothers me that this is what I am getting, okay, 4x, give me a guess, 4x, what do you expect, do you expect it to be 0, okay, so it does make a difference whether it is even or odd, so the n by 2 where n is the number of intervals, the n by 2 seems to be significant, right, so at n by 2 if you have, if n is even, n by 2 is going to give me, pick up that 0 or put the other way, if you know you start off by saying that this is my frequency or wave number or whatever, you may make your choice as to what is the number of sample points that you are going to take, right, so you may take at least 2n sample points, 2n plus 1 sample points, right, you may sample the function at 2n plus 1 points at least, okay, is that fine, okay, so what do we have, what is the, so we have, what are the other conclusions that we can draw from this, pardon me, so frequency information is it lost here, frequency information is still there, the peaks are gone, right, so up till that point, so up till the point, so let me, so one of the big conclusions we can draw is that there is a largest frequency that can be represented, okay, there is a largest frequency that can be represented, so if you give me, I will call it a mesh or a grid, right, if you take an interval a, b you break it up into equal number, right, for now because that is all we have been doing, we cannot draw any other conclusion, equal number of intervals, right, n equal number of intervals, then n by 2 is sort of the highest frequency that can be represented, okay, I turn it the other way around for a given mesh, for a given mesh, whether something is high frequency or low frequency depends on the mesh, am I making sense, so if I were to pick a very large number, say 101 intervals, if I were to pick a very large number, right, so it comes out, sine of x comes out extremely smooth but like, I mean it is pretty obvious that I can actually represent say 250, I am sorry, say 25, 25x, sine of 25x, am I making sense, it is not a great representation, right, it is not a great representation but on 10 intervals I cannot pick it up at all, right, so it is obvious, right now what we have seen it is obvious on 10 intervals like but that is not, I want you to pick, I want you to, so when we talk, we talk about high frequencies and low frequencies, as we go along you will see that we will keep talking about the high wave numbers, low wave numbers, high frequencies, low frequencies, so the point that I want you to get out of this is, when you say a frequency is high, right, in our world when you are performing some computation and you are trying to determine a function, when you say a frequency is high, the frequency is high with reference to the underlying grid or the underlying mesh that you have used, you understand, so what is high frequency that is wave number 4, right, on a grid of intervals 10 is a relatively low frequency on a grid of 100 intervals, in fact we pick it extremely well, you understand, so the mesh will pick up frequencies, wave functions which are low with reference to the mesh, it will pick it up well and those that are high with reference to the mesh, it is not going to pick up as well, okay, okay, is that fine? So when I say, when we say high wave number, low wave number, when you are talking with respect to the, when you are talking with respect to a physical problem, you may have actually an understanding as to what is a high wave number, what is a low wave number, but when we are talking with respect to computation, when we say high wave number, low wave number, you have to ask for clarification saying what is a grid size, otherwise there is no sense unless I give you that other piece of information, what is high and low does not make sense, right, high and low or comparative terms, high with respect to what, low with respect to what, right, that information has to be given, is that fine, okay, are there any questions, okay, so I think I would suggest that you possibly can go on to trying out different functions, my suggestion is try out different functions with, right, with this, with hat functions as such, try out representing different functions with hat functions, I would also suggest that maybe you can try the quadratics and qubits, okay, and you may be surprised, so I will let you try quadratics and qubits yourself, then I will come back and maybe we will do a little demo again with that, is that fine, okay, are there any questions, okay, so there what I will do is I will stop my demo and I will go back to my regular, I will go back to my regular session of what we were doing earlier, what we had was we in the last class, right, is that fine, in the last class, is that fine, in the last class we saw that if we had or okay, why do not I continue first with, I will just say something a little something about this hat functions and so on and then maybe we will connect up to the last class, I just want to make a point here, so if I have a hat function, that is xi-1, that is xi, that is xi-1, that is x coordinate, I take one hat function, is that fine, I take one hat function and the question that I have is, we saw that we are able to represent the function reasonably well when we have sufficient number of intervals, in the last class we also saw that we may want to use differential, we may want to use Taylor series or something of that sort to represent derivatives, so then the question crops of how well can I represent derivatives here using hat function, okay, so we have looked at three classes of functions so far, we have looked at box functions, right, we have looked at box functions, we have looked at hat functions, okay, so representing derivatives with box functions, if you represent the function itself with the box function, then the box function is a constant, right, it is a constant on that interval, the derivative is always 0 in that interval, that does not work, okay, what about hat functions, hat functions, there is some hope, right, there is some hope but the only point is that the slope is a constant, the slope is a constant in the interval xi, xi-1, xi, okay, so the derivative you can represent the derivative and in fact if I were to take the derivative of the hat function that is centered at x, then what do I have, if I were to take the derivative of this hat function, let us restrict our interest to only xi-1 to xi, right, elsewhere, only in the support, elsewhere the function value is 0, so the derivative n prime, so if I have, let me, let me, let us say if you have f of x is fi ni summation over i, that is the representation, f sub i is the function f at xi at the grid points or the nodes, grid points or nodes, these are terms that you will see used for these points, grid points or nodes, so the derivative, if I represent the derivative as f prime x is in fact summation over i fi ni of x derivative, okay and as a consequence I am asking the question what is the derivative of, what is the derivative of n prime of some hat function n prime of i, okay, remember the function ni of x is 0 for x less than xi-1, ni of x is alpha of x for x, x in xi-1 to xi, 2 xi and 1-alpha of x for x in xi to xi-1 and 0 for x greater than or equal to xi-1, is that okay, so the derivative, the derivative of course as I said I am not really interested in these because these are 0 anyway but it is important they are 0, so n prime of i is 0 for x less than xi-1 equals alpha prime of x, you remember what alpha was, maybe I should remind you what alpha is, alpha of x is x-xi, x-1, xi-1, okay and I really need a subscript here, so if I make this i, either i and i-1 or whatever, I really need a subscript there, okay, so what is this going to turn out to be, 1 over xi-xi-1 and we call that h, right, h is like xi-xi-1 and otherwise it is minus 1 over h, sorry, h is like xi-xi-1, this is for x in xi-1, xi and this is for x in xi-xi-1 and it is 0 otherwise, so what does this turn out to be, what does the graph of this function turn out to be, the graph of that function, what is the graph of that function, is 1 over h on i-1 and i and minus 1 over h on i, i plus 1, okay, so given the relationship between this and the hat function, it looks like a box function but it is not quite the box function, right, it looks like a box function, it is not quite the box function, so we use the box function for convenience of this class for what we are doing and so on, right, we use the box function for that reason, so if you look here, we could take this interval, we could take another interval, they are orthogonal because the intervals were not overlapping, that was all very nice, right, for that purpose it was nice but it is very likely that often we will use the hat function, right, to do a representation because we are going to be solving differential equations and if you are going to be solving differential equations, it is nice to note that the derivative follows some function that we are using to represent other functions, right, so this may be preferable to the box function, it belongs to, it is named after h, it is called the h function, okay, it is called the h function, so it is clear that the derivative is a constant but it can be represented, what do we do at the node i, what is the derivative at the node i, there is a jump, is there a way I can estimate it, is there a way I can estimate the derivative at the node i, any suggestions, why do not I take these, so if I have a function that is varying in this fashion, it has a constant slope in this interval, constant slope in that interval, I know the slope on either side of this value, why not just take the average as a value here, maybe, maybe it works, let us see what we get if we do that, just for the fun of it, right, let us see what we get if we do that, so if this is fi-1, fi, fi-1, right, what is the slope of the first line segment here, what is the slope of that line segment there, fi-1, fi-fi-1 divided by h and what is the slope of the second one, fi-1-fi by h, I add the 2, I take the average and this seems to tell me that an estimate of the derivative f' at i is fi-1, fi-1 by 2h, sounds good, okay, so from this function it is possible almost everywhere that we will be able to get it, whether this is an acceptable value or not is something that we will have to figure out, okay, fine, right, so before we leave these h functions, let us get back here, before we leave these h functions, I just want to point out something, I want you to try out something, before I get back to Taylor series and so on, I want you to try out something, on the interval 0, 1 you consider a function that is a constant, just drawn this vertical line, it should be a light vertical line, consider a function that is constant and then consider, so I will call this h0, right and I will call something h1 of x, so up to 0.5 it has value 1, from 0.5 to 1 it has value of –1, so you try to find h0, h0, h1, h1, h0, h1, is that fine, okay, so far just to keep it familiar I have been calling these dot products, we should really call them inner products, okay, they are called inner products, you can continue to use the word dot product if you want, inner product, because we use terms inner product, scalar product, dot product, right, dot product, so you can and I call this h1 and I call that h0 because if you think about it, if you look at what is happening on the interval 0 to 0.5, it looks like h0 that has just been squeezed down to 0 to 0.5, so I can again define a small function there which is positive negative, right, I mean you can, so you can see that there is a hierarchy of functions that we can build on, if I define an h2, I would actually normally put subscripts also but it does not matter, so I can have something that is 1 up to 0.25, right and –1 up to 0.25 and actually I can define 2 of these functions, I can define a h2 subscript 1 that is here and I can actually translate that function to the right, right, this function just like, so it looks like we are getting a low frequency, high frequency, higher frequency, right, just like sin x sin 2 x sin 3 x, it looks like we are doing something like that, something very similar to that, is that fine, is that okay, right, so I just wanted you to be familiar, I just wanted you to get that idea, right now what we want is we will just tie up with the derivatives representation of function, representation of derivatives that is basically what we want because we want to solve differential equation, right. So in the last class, we saw that if I have, if I give you f at x that we could use Taylor series, prime indicates differentiation times delta x plus f double prime that we could use Taylor series to figure out what was the nature of the error that we are making, if we were to approximate f at x plus delta x as f of x, right, so in the last class we basically said well if you have a function value at some point, easiest thing for us to do is assume the same function value at a neighboring point, right, at an adjacent point or whatever. So that would be an approximation f of x plus delta x is approximately like f of x and what would be the error that we would make in that approximation, well we have an infinite series here and this infinite series we have basically truncated the infinite series in making this approximation, we have created what is called a truncation error, this is a repeat of what I did last time and the truncation error is typically indicated by the leading term. So the order of the truncation error is f prime of x times delta x and because the delta x has an exponent 1, right, this error is said to be first order. So basically as delta x goes to 0, as delta x goes to 0, this error goes to 0 in a linear fashion, okay, is that fine? What about a higher order representation, how can I get a higher order representation? I say f of x plus delta x equals f of x plus delta x times f prime of x plus higher order terms, delta x squared by 2 f double prime of x, I keep that plus higher order terms. It is clear that if I have the derivative which we saw when we did, when we used cubic, so if you have the derivative information it is possible for us to get higher order representation, right. So there is a connection between what we are doing there and what we are doing here but the trouble is what if we do not have the derivative, then we are in trouble, right. So if you have the derivative information, it is actually possible for you to look at what is, so if you say that this is my function value right now, this was my function value yesterday, right. So my bank account has so much money, yesterday my bank account has so much money, today then I can maybe predict what my bank account is going to get if I know the rate at which that value is changing, right. That is basically what it says, is that fine, okay. This also gives me a different clue. If I look at this, if I look at this equation, this equation tells me, right. So right now we have been looking at saying that what happens at x plus delta x, this equation tells me, so if I have f prime and if I have f of x, this equation basically tells me that I can predict f of x plus delta x, right in a little more, say earlier I was saying approximate but now I am saying, I have changed it around, I can actually predict. So if from a differential equation you are able to get the derivative value, you could actually predict what is going to happen at x plus delta x given the value at x and the derivative at x, fine, okay. That is one way to look at it. This is the argument we have been using so far. The other thing that you can get from this equation and once you have written this equation is you can write it as a way by which you can approximate f prime of x.  That is the other possibility that we can actually evaluate the derivative based on the values at x and x plus delta x given that now you know f of x plus delta x, is that fine, okay. So we will see in the next class how to go about doing this, right and we will see what is the error. So we will now go about representing derivatives, how do we represent derivatives, how do we estimate derivatives, how do we represent derivatives, approximate derivatives and how do you find the error in, right, that we are making in that representation, fine, okay. I will see you in the next class.