Hello everyone let's talk about urban planning again. Today let's look at urban forests and how they are implemented in cities around the world. So urban forests are gaining popularity as a solution for climate adaptation in urban planning but what does it really take to successfully implement a new urban forest project? This is why we have the help of our guest, Victor Munoz-Sans, is from the Delft University of Technology in the Netherlands and we'll be exploring with Victor the planning processes of urban forest projects in three different cities in Almir, in Madrid and Boston. Victor it's a pleasure to have you here. Thank you, same flag question. Before we jump into the article itself and your research what are urban forests? So tell us about the concept. Well, probably in Europe if one talks about urban forest to a layperson one can imagine a park which has lots of trees and that's kind of an urban forest and that indeed is one way of defining urban forest but in terms of a discipline of scientific discipline and urban forestry and urban forest is broader. It really talks about the official definition, talks about all the trees that are part of a city that are in a city and that's what is considered urban forest but with this article of course we also want to question that definition. Of course it's a good context of background for us to start so can you give us a bit of that background about this research in particular so why is this important? Yeah well we know temperatures are rising, biodiversity is being lost and greening has become kind of high in all the political agendas and we also know that green has multiple benefits, carbon capture they mitigate the heat island effect for example green improves public health so trees are important and indeed in Europe the European green deal is already asking cities to kind of to develop urban greening plants by 2030 so it has become a municipal task to develop greening plants. So as this greening and in the end the question of the urban forest becomes mainstream we basically wanted to know well how can we implement this with success and does the focus of our paper how do we make it in work how do we make it actually possible? Precisely that's where I want to follow up and correct me if I'm wrong so your research along with your authors so wants to point out good practices and success cases in urban forest strategies even though strategies like this have been implemented for years correct? Yeah so we are looking I mean we are looking at specifically at how there might be a mismatch between the processes that usually take place in city planning, city government and the principles of urban forest so and that comes to the course with the analyzing of the success of the other past projects. Specifically we looked at very recent projects what we call a new generation of urban forest because we thought there was like there could be keys to understand new trends and new ways of dealing with this question and so in our paper we have three main objectives one was really to understand how a city and different urban stakeholders as from the bottom could make a case for green in a project how to argue for the need of planting more trees. We also were interested how then these actors seek for support for the projects and to overcome there for the possible barriers for implementation and third we want to know what were the models both in terms of organization but also in terms of typology of a forest that were chosen in connection to those the goals of the projects and the different actors that were involved in them. Let's tell us about the findings then. Well in relation to the first aim so in this how different stakeholders make a case for green we indeed see that I mean among all these actors the main concern is environmental questions so the need for more ecosystem services in cities and they point that you know we have to effectively implement this into the city planning to mainstream things into city planning but we also show that there is a poor development of criteria that could allow to assess whether a project is successful or not so we miss that there is not enough benchmarks or in urban plan urban forest plans that could help different strategies and to support the adaptive management of the process how do you evaluate the process in the projects in time and adapting as you constantly assess them and then concerning the second research aim which is how different actors seek support we did notice that in order to kind of go beyond some of the different barriers that happen even within urban planning city municipalities actors seek support from different stakeholders mainly intermediaries from academia for example and also other agendas national international to the legitimized projects and and what was very important going to that is that indeed most of the barriers that were perceived by the different actors we interviewed were not really the ones that we could maybe at first think they are the ones that could hinder the implementation of a project of urban green for example you know the pressures of densification or urban development most of the barriers were internal they were about lack of information silos silo thinking it within the the municipal departments and continue it in time so as governments change election after election priorities change and we know that in order to to build a forest one needs to think in a long-term perspective 40 50 years and then finally in in relation to which models are being favored what we concluded is that it did we need to question the traditional scientific definition urban forest that i told before of the all the trees within a tree or within all the trees within the city because that that idea demands a lot of coordination within a city you know and not all cities have the resources to think in that scale of a normal forest so we need to think of different ways of also bringing green in our cities which could be also coming from the bottom up okay so the findings concerns stakeholders different actors models that are favored over others can or have these findings impact public policies somehow you refer to municipal planners in your article for example so can you elaborate on that yeah i think that what what we think is the strongest point of our results is that we have to consider a broader spectrum of possibilities and a diversity of ways of thinking about urban stories and that's very much depending on the context of environmental and social context so that there is not just one solution and to achieve the desired environmental goals and you know as a tone and also to overcome the different barriers so these results can be always support public policy in anticipating consequences in the media in the medium and long term of favoring a model for example if you choose to put down what a bottom up right in the middle model what we show can help to identify anticipate possible and problems or challenges you could have in the future for example you have a top down project it will be interesting to still implement some sort of like a collaborative structure that could reach the gap between the top and the bottom and also and there and that way activate you know local initiatives for example and involving citizens and businesses in the maintenance and future long term management of the of the forest to overcome this question of the long term management for the from the perspective of the municipality on the contrary when a bottom up approach and given that they are mostly based on and rely on on on personal leadership let's say somebody who decides that this should be done as a project that has challenges in terms of long term development because as soon as the leader goes then the project may may wear down so in this kind of approaches bottom up with advice that one is to find ways to make them institutional and to set other frameworks that could take the project beyond these personal initiatives. Of course and for researchers and planners that are working on urban forestry so where they should look now I'm curious to know about future venues for research. Well we feel that there's a need to go in depth to this question of organizational model and mostly because and also to understand how can we better assess the projects in time. So we hear regularly on the news and in both planning and urban design news and that cities are putting out of money planting trees but just after that we don't hear anything about that we don't know how the projects have been assessed as they've been monitored what can we learn from those initiatives. So what we feel is that we need to find ways of monitoring projects in a dynamic way so that one can learn from these practices and these best mistakes you know practices failures and therefore identify possible next practices of urban forestry. Of course we that has to do with some of the limitations we have in the in the study which is that we are focusing only in Europe and also America so we would be will find interesting to find insights from other geographies which could bring other alternative ways of thinking urban forestry and also we are based our analysis on interviews and collective studies so we will need to kind of have a larger sample and a quantity of approach to provide a richer for the waste cities based in the challenge of greening. Of course so there's there are some different methods that can be conducted other geographies and there's apparently an issue with follow-up and monitoring as you were saying. So what further materials can you share with who's listening about this topic? Well there's also a lot of things being written and can worry about about urban forestry if I can be so specific. Last year there was a very interesting conference in the Cal 11 which is called urban forest forestry which is basically the same team that wrote wrote these articles for urban planning, wrote a follow-up with a lot of case study and that so you can find that in my research profile. And also the proceedings of this conference are very interesting research for those interested in urban forestry because you can have an overview of their work that's in the specific type of light or view of the research of urban forest. Additionally dealing with the question of the barriers or the challenges for implementing the climate adaptive and mitigation policies. Recently I closed an article for urban planning which in the issue of planning and urban polarization which deals with also the question of how the public accepts as a participation in climate acceptance projects works, includes as a meeting. So to anticipate kind of how a project may have difficulties in being accepted and how to overcome that in the process. Of course all the materials to our listeners all the materials that Victor just suggested will also be available if you are watching on let's talk about urban planning website below the video of this conversation so you can find there all the recommended materials that Victor just just shared. So Victor let's close this episode with a punchline message, a grand finale. So if you could condense this conversation in two, three sentences what would you say? Well there is a serious need to take greening seriously that's the first thing and to consider that the time matters in addressing the challenge we have now in terms of climate. Time matters in terms of like we have to act urgently and time matters because these solutions take a while to start taking effect so we need to consider not just the act of implementing policies now but actually their long-term life and maintenance and management. So indeed we need to question to imagine new ways of organizing planning and to imagine new ways even of new institutions that could manage such projects in the long term. Thank you Victor. Thanks. If you are for our listeners if you are watching us on YouTube you can find all the resources all the materials of this conversation on the Let's Talk About Urban Planning website and you can also listen to these episodes wherever you get your podcast.


Dawg. Yn gweithio. Yn dweud ym môr coronavirus, am ychydig yn ysgrifennu. Rwy'n gweithio'n gweithio'r hynny, ac mae'n gweithio. Mae'n gweithio'n gweld â'r dweud ym môr coronavirus. Mae'n gweithio'n gweithio, ac mae'n gweithio'n gweithio'n gweithio'n gweithio. Diolch i chi'n gweithio i chi. Efallai yna yn ei bod yn bach ar hi, ond mae nid eu cockpit yn ôl yn maen nhw. Roedd yn ei ei wedi'i cynnig felly fan ynRI, ddim yn ymweld am ar y cwmwyllfa. Roedden nhw'n syn symud hynny. Fe yma'r angen o'ch cyfnod o'ch cyffreddau o'r hynny o gyfle. Rwy'n ôl yn arda llwydda llawedd. Dyna o'r mewn cofio, sy'n drafod o'i'r hollwch am eu stryw, fel yna'r hollwch am bach o'i'r hollwch gan yr holl gafodd, ddiweddar ar gyhoedd, a dweud ag oesol a chynydd, maen nhw ymlaen i ddweud i ddweud. Oni'r dweud i chi'r cymryd yn y bwrdd ac mae hwnna have to try again, I'll use the same message day one, day three, day six. Build your immune systems. Don't get stressed, don't allow your immune systems to drop and try and stay focus and present in the moment to take away any of the anxiety, the depression or any of the fears that you've got. It's absolutely understandable for our human mind, I know I call it my muggle mind, to be able to be kicking in and giving us a fight or flight scenario. The reality of the situation is is you know over a period of time this will pass.  Give yourself the best chance of fighting it or avoiding it. So that's the coronavirus information put out there for you and it's a difficult time. We're going into self isolation here now locking it down, not for any reason because any of us have got the coronavirus. I just think we're at that stage now where over the next coming two or three four weeks, we're going to stay inside. That's my wife, my daughter and my son-in-law. The sun is coming up over south Wales which is just coming into my face here, which is beautiful and today on day three we're going to talk about meditation and the brain. Okay, because day one you're all excited and you get involved in it. Yeah this is great mindfulness.  A'r gwaith, galw'n gweithio'r pethau. Yn y gweithio'r ffaith ar gyfer, pan dweud, dyna gwneud hynny, ma yw ddim yn fawr gwertho? Gweithio'n fawr gweithio'r ffaith? Mae'n fawr gwneud, mewn ei wneud i gynhyrd ymateb gwaith ac i mi wneud yn oed yn dweud y mae gwellio'r ffaith ar gyfer ac, i wneud, yn oed i gweithio kaeth yw'r ffaith. If you're having positive emotions or you have a positive outlook on life, there's higher activity in the left of the frontal cortex, which is the front of your brain here on the left, okay? And when people see higher activity as a ratio from the left to right, it normally says that people are more optimistic, they're more creative, they're more joy, they have a bit more vitality in their lives. There are other people who have higher activity in their right of the frontal cortex, which can show signs of depression, anxiety, worry and stress, okay? Now what happens there is, is that some people, you know, the ratio between left and right and right and left can vary based on the experiences you're having around you. So, you know, if you win the lottery, your left hand side is going to be going crazy. If you're really worried about the coronavirus, the right hand side is going to be going crazy. And the idea here is, through mindfulness, is that we build higher activity in the left than we do the right. And there was a study of a Tibetan meditator and when they looked at his brain, they could see there was a huge difference from the left to right ratio. Now the question here is obviously, was he born happy? Is he naturally a happy person and the mindfulness has only activated and amplified it or did he actually amplify it, cultivate it and build a new thought process from the left ratio to the right through mindfulness? And there's so many studies out there, but there was a random assessment of 45 people, which, you know, 45 people, it's not 45,000 people, but it's 45 people, there's much bigger studies. And when they undertook mindfulness for four months, they saw significant differences from the left to right ratio. Significant differences. Now, this is huge for me. We've only on day three, so we've got a little while to go yet, or maybe you've been doing mindfulness for years and you feel it. But what they found, which was really, really interesting, is that individual people have what's called a happiness set point. And what does that mean?  So some people are naturally more happier than others because of this happiness set point. So, as I said, if you win the lottery, they're going to have a blip of happiness, but 12 months later, you're going to come back to your happiness set point. If the opposite happens, if you have a traumatic experience or something really fundamental changes in your life, maybe like my dad, for instance, passing, that we'll have a drop in that happiness set point, but at some point, whether it's 12 months, we'll start to get back to that happiness set point. So we're born with it. A new research shows that we can change, the change in our exterior, okay, win in the lottery, grief passing traumatic experiences has effect on your happiness, but it comes back to a set point eventually. But what's been found is, is if we change the interior, it has an effect forever and ever on men. It's been scientifically now proven that we can change our basic levels and base levels of happiness through changing the interior landscape through mindfulness and loving kindness. It's been proven, not in three days maybe, but it's been proven we can shift our levels of happiness. Happiness can be grown by changing the structure of our mind through mindfulness practice. Now it's called neuroplasticity, I think they never say this word, plasticity. I still can't say umbrella either, I just did. Neuroplasticity, okay, which is repeated experiences shape our brain. There is a great study of London taxi drivers and when they analyze their brain, the part where they had to have navigation, okay, is much higher than anybody else. And the reason for that is that they have to learn over 23,000 different routes and roads to be able to pass their test and every day they live it.  It's called cortical thickening, okay. We actually thicken the cortex of our brain and it takes place through practice. So in the earlier ones where we said, what we practice become stronger is absolutely correct.  How can we change that? Well, we know that mindfulness can now dig a new road. We can look at the cortal pathway, we can build new roads, we can shift down a different highway. And everything we do through this mindfulness course, we strengthen that pathway, we create a new mindful super highway of happiness that allows us to create positive habits. But more importantly, it pushes our happiness set point to a position where we become happier, where we have less anxiety, where we have less stress, where we have less depression, where we have more compassion, where we have more empathy. Mindfulness has been statistically scientifically proven. I've got to put my teeth in this morning, OK? And that is there for us to be able to use. So we're on day three. Day one was great day two. You're going to get to a point where you might think, oh, I've got time. Stay with it, OK? Stay with it. It is proven you can improve your happiness by altering your interior landscape through neuroplasticity. I still can't say it. Plasticity, right? By cultivating mindfulness. What you practice becomes stronger. When you sit now on this loving kindness meditation that we recorded on Monday, when you sit with it today, you will be strengthening your cortisol thickening and creating a new highway. It takes time to make a motorway, OK? A super highway, a freeway, wherever you are in the world. Be patient with it. Sit with it. Conduct it and see the difference. I hope this makes sense. Day four will be tomorrow. Thank you very much for joining me. There's one last thing I'd like to say. And that is this. We all need at this moment in time a little bit of help, support, love, and importantly, talking. So if you're sat at home today feeling a bit bored, phone someone. Have a chat with someone. Get on to Skype, get on to Zoom, get on to a video conference. There's no need to be lonely now because we have lots of people around us. I'm here. Hello, how are you? OK, send me a message below. But talk to each other. It's been, again, scientifically proven that if somebody is having anxiety or whatever, it reduces greatly by talking to another person. So reach out to somebody today. Accept the call from somebody today. But importantly, let's keep talking. We are stronger together. I hope this makes sense. Enjoy the meditation next. And I'll see you in a few moments when you've completed it. OK?  I'm going to put a nice image up. And I'm going to sound a bell for us to start. And all I want you to do now is focus on your breath. Breathing in and breathing out. Breathing in and breathing out. And just focus on where your breath is. Where do you feel it most? Is it through your nostrils? Is it on your t-shirt? Is it on your jumper? Where do you feel? Do you feel your belly moving in and out? Breathing in and breathing out. And to start this, we're going to offer loving kindness to ourselves. And I want you to focus on the intention of these words. And I want you to continually repeat these words. And if you have a thought that comes in, you welcome it, you love it, you let it go, and you come back to the mantra. So as we're breathing in, I want you to say these words. Breathing in, may I be safe? Breathing out. Breathing in, may I be happy? Breathing out. Breathing in, may I be healthy? Breathing out. Breathing in, may I live with ease? Breathing out. Repeat the mantra. May I be safe? May I be happy? May I be healthy? May I live with ease? Continue that mantra as you're breathing in and you're breathing out. And if you have any thoughts that come in, you welcome them, you love them, you let them go, and you come back to your mantra. May I be safe? May I be happy? May I be healthy? May I live with ease? Breathing in and breathing out. May I be safe? May I be happy? May I be healthy? May I live with ease? Welcome this beautiful love into your life, to be kind to yourself, to connect in with yourself, to love yourself, to heal yourself. May I be safe? May I be happy? May I be healthy? May I live with ease? May I be safe? May I be happy? May I be healthy? May I live with ease?  And I want you to picture that person in your mind. And on this occasion as you're breathing in, you say simply, May you be safe? May you be happy? May you be healthy? May you live with ease?  See them and set the intention of may you be safe? May you be happy? May you be healthy? May you live with ease? Breathing in and breathing out. And don't forget if you have a thought, just let it welcome in, hold it, don't put any judgement on it, accept it, acknowledge it, let it go and come back to your mantra for this person that you love. May you be safe? May you be happy? May you be healthy? May you live with ease? Breathing in and breathing out. May you be safe? May you be happy? May you be healthy? And may you live with ease? Now the focus, we're going to focus on someone you know who's having a difficult time at the moment, maybe someone who's ill, and we're going to offer them kindness. So if there's somebody you know who is in self-isolation or isn't feeling very well at the moment, and we need to place their intention in your mind with them in your mind, and you say again, may you be safe? May you be happy? May you be healthy? And may you live with ease? Breathing in and breathing out. This is somebody who you know who's having a difficult time at the moment, who may be ill. Set the intention of that person, put them in your mind's eye, and as you do you say to yourself, may you be safe? May you be happy? May you be healthy? And may you live with ease? And if you find your attention or your mind wanders, don't worry. Just love it, let it go, and return back to your phrases, your mantra. May you be safe? May you be happy? May you be healthy? And may you live with ease? These mantras are now your anchor. May you be safe? May you be happy? May you be healthy? And may you live with ease? Now what I'd like you to do is choose someone in your life that you might have difficulty with, or have some tension with, or have had an issue with, or something that hasn't sat right between the both of you for a little bit of time. This one can be difficult, but we set the intention and we put them in our mind's eye, and we say to them in our mantra, breathing in and breathing out, may you be safe? May you be happy? May you be healthy? And may you live with ease? And again, if something, a thought comes in, I want you to allow it in, accept it, acknowledge it, put no judgment, love it, and let it go, and come back to your anchor, your mantra. For the person who you may have had difficulty with, or have had an argument with, there's some tension between you. Set the attention, put them in your mind's eye, and say may you be safe? May you be happy? May you be healthy? And may you live with ease? And if at any point you find that difficult, then you can just direct it back to yourself. May I be safe? May I be happy? May I be healthy? May I live with ease? Now I'd like you at this moment to direct your loving kindness, your attention, and your intention to all forms of life, people, animals, all beings, those people who need it most all over the world at this very difficult time. And I want you to say for them, may all beings be safe? May all beings be happy? May all beings be healthy? May all beings live with ease?  May all beings be safe? May all beings be happy? May all beings be healthy? May all beings live with ease? Breathing in and breathing out. May all beings be happy? May all beings be healthy? May all beings live with ease? May all beings be safe? May all beings be happy? Al iddyn nhw'n siwr hefyd... Al iddyn nhw'n siwr hefyd. Fawr byddwch, a'r gwaith. Rwy'n gweithio eich lawr i ddeithasol, iddyn nhw, gallwch gynyddiad ddaw hanes ar yMynd a gallwch nhw'n sut y seng ymlaen a'r ddweud y byddwch i fy JES o'r cyflwyondeb yr ychydig a'n gyflwynt ar y rymig a phwy'r gwneud i ymwy. Felly yn hoffi chi'n meddwl ychydig i'r gwy Shouldn. Do ddim ychydig ymwneud! Fel hynny, Lleodd yn cael y gwneud i'n achos gŵr i'r gweithio, bod ni'nID i'n gofnod, rwy'n dda'r gyfer yr ystyrwyd yn ei gwrdd siarad yn ystod, rwy'n gweithio. Rwy'n gweithio'n gweithio. Rwy'n gweithio. Rydyn ni'n gweithio rydyn ni'n meddwl i'ch gweithio'n glynodwch rwy'n gweithio'r i wych byddwch iawn i ddweud o gaelog ddiogel ac i ddwaith ac yn dod o'r hwn. Felly, rydyn ni'n gwbod i'r ddef nhw'n gweithio. Rhyw ddim yn ystyried, sidio ei ystafello, dd境wch profi o ddysgol, yw'r ddwych yn amlwg, ac mae'r drinffodd ar hyd wrth ddweud o'i cyfnodd, a'i ddwych i ddwych ar gyfer a ddwych, rydyn ni'n mynd am ymgylch a'r ddwych a'u ddwych ond o'r llwyddiad, mae'n cael ei wneud o'r virus. Ond hefyd, rydyn ni'n gweithio'r gweithio'r llwyddiad. Rydyn ni'n gweithio'r llwyddiad a'r llwyddiad, rydyn ni'n gweithio'r llwyddiad. Rydyn ni'n gweithio'r llwyddiad a'r llwyddiad.


I'm Zor. Welcome to Unisor Education. Today's topic will be functions. There are different definitions, certain principles which I would like to explain today, very briefly, because it's a really very American class subject. Okay, what is a function? Philosophically, generally speaking, function is certain dependency between something and something. So, to be a little bit more precise, let's imagine that you have two sets, absolutely abstract sets, one sets, one set and another set. We will call this one domain and this one we will call codermain. Let's imagine that you have certain rule which, given an element, any element from the domain set, you will be able, by applying this rule, come to a certain element of a codermain set. Well, basically this is a definition of a function, function of dependency. So, this is a definition of a function which is defined on a set of elements which we call domain and it takes the values in the set which we call codermain. Not necessarily all the values will fill out completely the codermain. So, maybe all the images of elements of the domain will actually be a subset of the codermain set, in which case we will call this subset of all the images of the elements of the domain, we usually call it at range. So, the function, if we want to define it a little bit more precisely, should be defined as the combination of one set, which we call domain, from which we will take the elements which are called arguments. So, these are elements of the domain which we call arguments. And then we have to have another set which we will call codermain and then we have a rule which allows us for any element from the domain for any argument to be able to find the corresponding value, this is called a value, an element in the codermain. And a range is a set of all the different values which this particular function which we can call f whatever, which this particular function can take. So, this is not really very mathematically rigorous definition but still kind of an explanation which allows to understand what the function actually is. Usually in school people are talking about functions which are defined on the set of all the numbers. So, the domain is usually the numbers and the values, the codermain is also usually the numbers but in theory function can be any. Let me give you just a very simple example.  Well, what is the function in this particular case is a set of all flowers and the codermain is a set of all colors. Well, obviously not necessarily the color of all the flowers you see somewhere in the forest or wherever you go would fill up completely all the spectrum of all the colors. So, the range can be a little bit smaller. For instance, you see only red flowers. Then the range will be all the variations of red and the blue ones are not really in the range. So, the codermain is actually all the colors like blue and red and yellow and whatever but the range is actually only different shades of red. For instance, it can be different. Okay, what other functions can be defined in this particular set? Well, we can have a different function which is defined in the same domain which means elements of this domain. Flowers will still be arguments but the function itself will be let's say a size of a flower. Whatever the size might be, I mean bigger, smaller, measured in millimeters, centimeters, inches, doesn't matter. So, in this case again domain is a set of flowers and what is our codermain? Well, that's basically all the different numbers. If you measure in let's say centimeters, then from tiny flowers of fractions of centimeters to maybe gigantic flowers, I think the biggest flower can be like at a meter, I don't know, two meters. No more than that probably. So, this will be the range. So, the codermain will be all the numbers but the range, all the real sizes which flowers can really have is definitely smaller than this. Let's say it's numbers from zero to 200 centimeters. There are actually some other functions which might be defined on the set of flowers like shape of leaves and all the kind of shapes can be that all the different kinds of shapes can definitely be defined here and where this flower is growing, for instance, the continent or the temperature this flower is usually comfortable with. So, there are many different characteristics which we can talk about as functional dependencies between the flower and something. And now we are approaching a very interesting point. If I will tell you only that the color of a flower is let's say red, does it define the flower itself? No, there are many different flowers which are red. Okay, what if I will tell you? Okay, the color is red and the size is such and such and the shape of leaves is such and such and the temperature and the continent and the form of roots and whatever other different components I can talk about each one of them is a separate function. This is one function, this is another function, this is a third function, etc. All the different characteristics each one of them representing a function defined on the set of flowers. Does it really define completely what was there? Well, as you see we have to really go from value to the argument of the function or maybe value of many different functions defined on the same argument and try to determine what was the argument which produced these values. So, basically as you see we have a different concept here. We have a concept of inverse function. So, what is this? Again, if you have certain domain these are elements and you have certain co-domain these are elements and you have some kind of correspondence from each element you go and you find the value of this argument among the different elements of the co-domain. If you can define absolutely deterministically knowing the value you can actually go back and find the argument. Well, what is this? This is another function actually which has co-domain as an argument and domain as the value. So, one function is defined from here to there this is argument this is value and another function is from the value back to the argument. Is it always possible to find this other function which is called by the way inverse? Well, not necessarily. If I have something like edges wrong here then yes it's obvious that for each element of the co-domain I can find by following this line backwards I can find what the argument was and that would be the value of the inverse function but what if situation is like this what if the same element of the co-domain the same value can be derived by applying the function to two different elements of the domain. In our example with a flower what if there are two different flowers which have exactly the same color as you see color does not define flower completely. Okay, so inverse function is not always defined properly or I mean it's not defined at all in certain cases but in some cases it is actually defined obviously in school mathematics we usually deal with functions which are which which have domain and numeric or all the different numbers and the co-domain and range of the function always the numbers and among these numerical functions there are many which really have something like one-to-one correspondence between the arguments and the values so you can always go backwards from the value back to the function to find what was exactly the inverse function well typical example is if function is described as a formula when this is an argument and this is the value and you have to just multiply the argument by itself three times to the power of three then obviously there is a inverse function which basically does the same thing if you take number two this function converts two to eight and this function converts eight back to two all right so maybe I'm jumping a little bit ahead I didn't really talk about powers and the roots etc but in general I'm sure people who went through normal school mathematics now what I'm talking about here so in this particular case function y equals x to the third degree is reversible or inversible it has an inverse function and this is the one but let's consider a different function what if I will have a square here this is a typical example of a case like this because as you understand two goes to four and minus two square also is four there is no such thing as a function which will convert one number four into two two and minus two well conditionally you can say something like y is equal plus minus square root of x but this is not a function these are actually two functions one function which returns the positive number and another returns a negative number so this is just a conditional expression it's not really the function itself okay so as you see functions can be reversible or reversible but inverse function does exist and that might not be the case now let's consider again a function which is described in this particular picture and we will introduce two new concepts the concept of restriction and expansion or extension what is a restriction and extension it's actually a very simple thing let's consider you have a function which represented exactly by this picture so there are four different elements in the domain and no more just these four and you have three different elements which actually constitute the co-domain now if the function is defined like this that's our initial function now let me define a different function function which is basically derived from this one by restricting the arguments only to a subset of the domain so the function now a new function is defined on the subset of the main and obviously the values will be also members of the co-domain elements of the co-domain but it will be probably a smaller range all right so this is a restriction so we restrict it we restrict our function to a subset of the domain so a new function now why is it a new function because we have a completely different domain and completely different co-domain because the subset now of the original domain which is a different set is a domain of a new function so if it has a new domain it's a new function but this new function definitely is related to the original because the domain is a subset now what if we have the other way around so instead of restricting our function to a subset of a domain let's assume that initially we have only the function which is defined on the subset and the values are here if it's defined initially original function and now we have complemented this function we have expanded it we expand we expand the function towards a bigger domain so if function was defined here and these are the values but now they are considering a new function which is defined from the bigger set which is a superset of our original set and obviously a new range will be a superset of our original range now this function is called extension just a terminology no video actually about it all right what else about the functions quite frankly if function is not numerical which most of functions which we are studying in mathematics the function is not numerical there's not really too much we can talk about it's more like philosophy and and other subjects in mathematics we usually are studying numerical functions functions which are defined somewhere among the numbers and take value among the numbers so what are the ways we can represent this numerical function well actually there are many different ways and one of them I have already just exemplified like y equals x where or something like this this is the formula so one thing is to represent numerical function is formula here's an interesting point formula is something which allows to describe the function quite generally actually if you have a formula y equals to f square well does it define the function well yes and no in general we can say yes because well just take any number square it and you will get another number so from argument you get to the value of this function using certain rules of algebra and more precisely you really have to talk about domain and and and co-domain or range of the function what can be squared if I will tell you that let's consider the function which is defined only on a set of natural numbers one two three four five etc so if domain is natural numbers what will be the range well range also will be natural numbers but all of them no not all of them obviously the range is something like one four nine 16 etc squares so if you have the main one two three four etc then your range will be one four nine 16 etc so basically you have to specify not only the formula but also what's the domain range is not really necessary to specify because you can always find out what's the range by substituting these numbers into the formula so this function which is defined on a set of natural numbers is different from the function which is described by the same formula but which is defined on a set of let's say rational numbers why because different domain if it's a different domain it's a different function function is the combination of three things the domain the co-domain and the rule which corresponds to the main point or element to the to the co-domain however we usually omit this type of discussion in in our everyday mathematical life because it's probably not really very important it's not very interesting formula is really sufficient because you can always think about the function which is defined only on numerical but only on natural numbers as a restriction which means defined on the smaller subset of another function which is defined on the set of rational numbers and that in turn is a restriction of another function which is expressed by the same formula where domain is the set of all real numbers and if you're really sophisticated and you know what the complex numbers are you can always say that the function defined by this formula on a domain of real numbers is a restriction of another function which is exactly the same expressed by the same formula but defined on a set of complex numbers every new function which we are said which we are talking about is an expansion of the previous one so we can always say okay this function is defined in the biggest domain we can possibly can and in reality it's either complex numbers or if you're okay with real numbers then that's the real numbers which is the biggest possible domain people really are interested in so there is no really need to talk about the main itself because it's always restriction or expansion of another function which is basically the same thing the same formula so the formula is one thing which we can talk about as a representation of a numerical function well what other forms well obviously there is a very simple form a table you can say that this is the main this is the main this is called the main and you can say that the main is numbers 1.5 in which case the function is 3 the main can be 4.2 in which the function is minus 2 and element is 7.57 and the value of the main is 0.66 okay is this the function yes absolutely the main is a set of three numbers 1.5 4.2 and 7.57 and the range which belongs to the common main of all the different numbers the range is also only three numbers which is 3 minus 2 and 0.66 yes this is also a function obviously the representation of a function in this particular way is definitely restricted because you can't put like infinite number of arguments getting a certain number of values of this function however for certain purely practical tasks this is the only way we can represent the function let's assume that you are measuring temperature in the city of new york during 1957 so you will have 365 days and for each day you have certain temperature which you can write as sorry as a temperature here and the date itself will be on the left so here we will have a set of dates and here we will have a set of temperatures the function is defined for every day we have a temperature that's a function it's represented by by a table and by the way it's not even a numerical function in the full sense of this word because the definition the domain is not a number it's a date however the the value is a number so it's half of the numerical function if you wish the domain is not and the code domain is not that is a number but anyway table is a respectable way to define the function and in practical examples it's really quite common let's go back to mathematics from the practical life mathematicians don't really master's practice they usually try to stay within the framework of their theory okay what other ways to represent numerical function in this case well you probably have heard about the graph graph is also a representation of a functional dependency let's talk about certain function for instance this function well I'm sure everybody remembers Parapola so this is a graphical representation of this function how can we use this particular graph very simply if you have number one here you go up perpendicularly and then go to the left and you found one if you go here two you go up and then left and you find four so for every number if you want to know what's the square of this number what's the value of this function you actually find a point on the x-axis which corresponds to the argument then go up perpendicularly go left and find what's the value on the y-axis so this is the rule for every point on the x-axis which represents a number obviously you can find using this procedure what's the value of the function which means the function is defined that's it all right so that's all about the graphical representation which we will definitely spend a little bit more time some other day just drawing different graphs for different functions so you have at least these three different representations which mathematicians might use for describing this functional dependency between domain and co-domain between argument and value here is another interesting way to consider functions which is maybe a little bit higher in the mathematical level than usually people learn in school however it's a very simple concept so just try to explain it here so let's say you have certain domain here and you have a function which for every argument gives you some kind of value of this function so this is the main and this is co-domain all right great now let's consider a different function let's call this function f let's consider different function for instance we have another function which has this set which is a co-domain for the first function f as a domain and then goes further into a different so this now becomes a domain for function g and this becomes co-domain for it so from this element of the original domain oh let's forget about the domains and co-domains let's call it a b and c that would be easier so there is a function f which is defined as a function from a to b so every element of a has certain image certain value in b and then there is a function g which further makes the correspondence between every element of the set b to a set to an element of set c so what it actually means is that for every element a which belongs to the set a i can actually find element b in the set b using the function f and from this element b using the function g i can go to an element c so we are consecutively can apply functions b and then g to an element a to get to the c what it actually defines is this because for every element a i have an exact rule how to get to a set c to element c what's the rule well first you apply a function f and then i will use a plus in the circle as a combination then as a continuation or a combination with the function f you should apply a function g now if everything every a b and c are let's say sets of real numbers then we can talk about something like this c is equal to function g applied on the result of function f applied on a that's what it is so we take the a first we use the function f to get to this but both of them are actually numerical so everything from number to number and then use this number as an argument apply function g to get to get to the c now this is called the combination of these two functions called the composition in theory it it is actually very much close to multiplication because you're consecutively applying one after another is you're multiplying one number by another and then by another so if you consider a set of all the different functions which are defined all of them are defined let's say on a set of real numbers then you can talk about this multiplication of the functions it's a little bit higher concept and it relates to functional analysis etc which people are usually learning in colleges but in theory you understand that there is something which is a composition of two functions which really gives you another function so it's kind of an operation between two functions two functions can be really operated upon like two numbers can be multiplied and the third number you will get as a result two functions if they're properly defined like one of them is having the range where the second one has a domain if they're properly defined then you can consider the operation of composition in a way similar to multiplication of numbers so that's another concept which I wanted to to talk about so basically that completes the general introduction to what functions actually are and what we will do in in algebra for instance that's what people do in algebra they're learning different functions and how they are really represented by using formulas graphs what kind of functions exist etc so you can say that maybe the whole algebra is a subject is studying the different functions well that's it for today thank you very much


So before I tear this down I wanted to do a follow-up video on the HA failover physical layer video I did yesterday, so I still have this set up It's another new day and there was a lot of comments on the video and you know I'm happy that the video is popular but a lot of people some people misunderstood this to think this was a deployment I so I wanted to make a follow-up video and talk about some of the Hardware that was used in an actual deployment that we did and it certainly wasn't this we to my now Well, we've never installed any I can't say there's not some of the some cheap switches floating around data centers I've seen some stuff But I will say that this is not what we would recommend to deploy because some people like oh Yeah, because the power supply on this little neck here could fail and that could be a fail point You're not wrong at all This was more for demonstration purposes now the actual install that we did that I cannot show you photos of because if well most of the data centers Just don't let you wander around with cameras And they don't really care for you doing that if you've ever been in a data center or a colo Where this is because it's not owned by the person putting it in it's a colo system Which means other people are there and they don't want you wandering around taking pictures of everything So what was the actual deployment we use to XG? 7100 now the 7100s are really nice solid performers They can do up to 10 gig and you can buy those and they bought the whole kit as the HA kit all configured So we have some great two PF senses solid one you rack mounts also We got two edge switches one you rack mount each and with the edge switches because yes any one switch is a single point of failure You're not just a power supply in a switch because someone mentioned dual power supply switches Yeah, you can get those as well We went with the dual switch method and by doing the dual switch method each server has multiple network interfaces on it So we've configured each server To have a network interface that copies to Connects to one switch and then the other network interface connects to the other switch and the switches are set up to talk to each other So if either switch fails, they have that redundancy there and this goes on through all the servers All have redundant connections. The only things that don't have is just the PDUs. They were using there's a couple things like the hardware distribution units That are only quit connected to switch zero versus connected to switch zero and switch one But this creates redundant links and basically you can unplug any switch and it system keeps working You can unplug either one of the PF sense XG7100 and the system keeps working They even have the feeds dual feeds coming in to the system and the way that works is we actually took the edge switch And we parsed out a couple of the other network ports on it to be dedicated to the WAN side And then the other sides are all dedicated to the LAN side And you do create segments of the switch and lock it down So this is the WAN side over here and these certain ports and an arrest are used for this and yes I'm aware of setting it to edge port so it gets a connection faster, which ones are set to edge Etc etc. So this is not a real world deployment for those that seem to do it So I figured I make this video so I can just reply with a video To make that pretty easy, but it is cool that you can use any small network Netgate device even these if I had two of these I would end the video with a couple of These basic ones even to get this done you can do fail over with You know less than $200 ones because it's a feature of PF sense Not a feature that is limited to or only allowed to work on certain models But I figured let's do this quick follow-up video to talk about that before I broke it all down This is not the ideal situation. You would have redundant switches You would have you know, obviously the redundant parts here But everything is much redundancy as you can because ideally if something fails in a data center You shouldn't lose connectivity to it. They're the only thing that even for this client I mean they have rate arrays. They have storage arrays But when it does come down to they still only have a couple servers in there So they're still building out redundancy that still needs to be done it's part of the future upgrade path we're doing as this client expands their business and The high availability is a hugely important aspect to them. That's why we went with you know all the equipment that we did But like so this is a follow-up to that. I Don't really have anything more to say But if there's enough comments on this and maybe there's some other video that spins off of this Let me know I do listen and read to read all the comments And we'll come up with ideas or if you want to participate in the forums and have a discussion about this Absolutely, this is a great topic. You know hardware redundancy HA failure, especially when you're dealing with out of remote sites They're critical to having your infrastructure and making sure it all stays up and running. All right, thanks Thanks for watching if you like this video give it a thumbs up If you want to subscribe to this channel to see more content hit that subscribe button and the bell icon And maybe youtube will send you a notice when we post If you want to hire us for a project that you've seen or discussed in this video head over to laurancesystems.com Where we offer both business it services and consulting services and are excited to help you with whatever project You want to throw at us? Also, if you want to carry on the discussion further head over to forums at laurancesystems.com where we can keep the conversation going And if you want to help the channel out in other ways we offer affiliate links below Which offer discounts for you and a small cut for us that does help fund this channel And once again, thanks again for watching this video and see you next time


Hello. The Scottish Parliament Information Centre is the Parliament's internal research service. Our subject specialist researchers provide impartial, factual, accurate and timely information and analysis to support MSPs in undertaking parliamentary business. Our key issues for session 6 briefing sets out three overarching themes likely to be of interest to parliamentarians over the next five years. The first theme looks at how devolution has changed as a result of the UK leaving the European Union. It considers who has the power to make or change laws in areas previously governed by the EU. It also discusses how, where these powers reside, impacts on the work of the Scottish Parliament. The second significant area of interest will be the recovery from the Covid-19 pandemic. Now that the vaccine rollout is underway, the political focus is likely to shift to the recovery and dealing with the aftermath of the pandemic. We explore this in more depth from the indirect health impact to the social and economic effects. The third big theme is the environment and climate. We're living through a climate and ecological crisis, with implications for individuals, society, the environment and our economy. Whilst there is a renewed focus on these issues, how we all choose to act will be a key theme for Scottish Parliament's scrutiny through session 6. You can get analysis of these issues and many others by clicking on the link embedded in this video. We look forward to hearing from you throughout session 6.