differentiation of implicit functions or functions which are defined implicitly so first of all what is the meaning of an implicit function so implicit function is basically where your variables x and y cannot be separated out easily see up till now what happened most of the functions that you got where of the type y is equal to f of x where there was a clear cut or you could easily separate the two variables from each other so the independent variable x and the dependent variable y terms could be separated out from each other so finding dy by dx was easy so this is called explicitly defined function this is called explicitly defined function explicitly defined function so today we are going to talk about implicitly defined function where your x and y separation is very very difficult so I cannot write y as a function of x so it's basically a mixed function of x and y okay equal to zero or equal to constant whatever you want to say okay so this is a implicitly defined function implicitly defined function implicitly defined function to give you some examples for the same let's say I have a function like this I have a function like this sine square y plus cos of x y equal to 1 so this is an example of an implicitly defined function so how do we find dy by dx in this case that is what we are going to learn how to find dy by dx in this case that's what we are going to learn in this particular expression so what I'm going to do here is I'm going to let's say call this term let's say I have this function and I want to find dy by dx okay so we'll call this term as let's say u okay let's say I call this term as a u okay now let's say I differentiate both sides with respect to x I differentiate both sides with respect to x okay now see how this works how this works if you differentiate this with respect to t let's not write x as of now let me just give you a very generic idea if you differentiate this with respect to t du by dt will be please make a note of this this is something which is beyond your school level syllabus it is doh f by doh x into dx by dt plus doh f by doh y into dy by dt basically what you have used here is actually a complete form of a chain rule this is called total differential so if you're differentiating you with respect to t and since this function since this function has x and y as its input you can write this as doh f so let's say you is equal to f x comma y you are differentiating with respect to t so this is how this expression will come out you will study more about this in your undergraduate syllabus early in your second or third semester of J of sorry of engineering college so now what I'm going to do is I'm going to equate this to zero I'm going to equate this to zero and what I'm going to do is because it is a constant I'm equating it to zero because constant derivative with respect to anything will be zero right so derivative of this with respect to t anything would be zero right now what I'm going to do is I'm going to change my t with an x earlier I didn't want to use x because then you would not get a feel of this term it would become a one actually now I want you to realize that this term is also there so what I'll do I'll put my t as an x okay so this expression here would become doh f by doh x into dx by dx this will become doh f by doh y into dy by dx got it okay so what has happened in this case this will become a one so your dy by dx can be written as minus doh f by doh x divided by doh f by doh y okay so everybody please make a note of this this is the expression for dy by dx for a function which has been implicitly defined this is an expression for dy by dx for a function which has been implicitly defined okay now normally what happens for school exams we don't use this formula okay so I'll write it down use it only for competitive exams use it for competitive exams so how does this work let us take this example itself which I had quoted over here so let's say this is your implicitly defined function and I want to find dy by dx okay now in school what do they say they say that okay differentiate this from left to right and make dy by dx the subject of the formula correct so if you do that it becomes sine square y plus cos xy equal to 1 so let's differentiate both sides with respect to x okay so if you differentiate this it becomes 2 sine y cos y dy by dx what will happen to this term this term will become minus sine minus sine xy into derivative of xy derivative of xv x correct am I right and this will be equal to zero so this is your C that you know the term which you wrote as a C on the yeah this part see okay now let us collect the terms having dy by dx in them so you'll see terms here which is 2 sine y cos y correct from here you will get minus x sine xy okay and this term if you take on the other side it becomes y sine xy y sine x okay so dy by dx will become dy by dx will become y sine xy upon 2 sine y cos y minus x sine xy so this is when you don't use the formula okay so the process is you differentiate from left to right take dy by dx terms common non-dy by dx containing terms send it to the other side and make dy by dx the subject of the formula okay so this is how you would be doing it in the school exam okay for competitive exams what do we do is I'll do it in the second way this is your first way second way if you use the formula that we have discussed right now dy by dx is minus dou f by dou x by dou f by dou y see how the process will become super easy you can do it in one step somebody has a question oh sure dou f by dou x dou f by dou x means you are differentiating by keeping only x as the variable y and constants will be treated as constants so let me ask you this question if you differentiate this with respect to partially with respect to x keeping y is a constant what will happen this time will go for zero what will happen to this term this term will become minus sine y sine xy into derivative of x y now remember your y is a constant so y will come out over here correct hold divided by dou f by dou y dou f by dou y means you have to keep x and constant as constants only y will be treated as a variable so this will become to sine y cos y this will become minus sine xy into x that's it done is this the answer that we had got see one step one step is what is required of course minus and minus will become a plus this is what you will do for your j so one minute sure but please do not use the second formula for school exams you will straight away get no credit for it who knows your parents and all maybe called the other day for meeting and ask what she is doing in the car what he or she is doing the school exams second formula you just have to follow this formula adhan if you're asking what is dou f by dou x then let me tell you dou f by dou x is the derivative of f with respect to x keeping y as a constant of course constants will be constant but y will also be treated as a constant so if you differentiate this keeping y as a constant sine y square or sine square y will be zero because it's a constant cos xy derivative will be minus sine xy into now derivative of x y keeping y as a constant will just give you a y it's like something like 2x so 2 will come out right so same way y will come out at the point no no they'll not be giving you more than two variables but this is extendable this formula is extendable Venkat so if that more than two variables this formula is extendable as you can see du by dt will become dou f by dou x dx by dt dou f by dou y dou y by dt dy by dt plus dou f by dou z into dz by dt yes sir what you want to find out you can find out from there oh okay okay this is something which is not there in your this thing school syllabus or for jee also they don't use it but I'm just telling you in case it may be helpful to you so could you please roll right sir yeah why not yes sir right yes sir one second sure it's a dancer thank you sir then yes sir let's take a question on this yeah see the meaning of dou f by dou x means see when there is a multi-variate function how does your function get affected or what is the change in the function when there is a slight change in the value of x that is called dou f by dou x so what the change in the function when there's a slight change in the value of y that will be called dou f by dou y so do it when it basically says when there is a multi-variate function if you fiddle with one variable how does the function change right that is called dof by dox okay many of derivatives the change in the output when there is a small and nudge in the input budge by nudge right yeah so yeah that is what is dof by dox similarly dof by doi has a similar meaning etc done one minute sir this is easy just to remind you people who join late will stop the class at 7 15 today in the light of your ut is going on yeah thank you so if you see this you can write this as log x squared plus y squared minus 2 tan inverse y by x equal to 0 so you can treat this as your fx comma y okay equal to a constant constant is 0 but it so dy by dx I'll be using my shortcut formula is minus dof by dox by dof by doi okay so dof by dox you what it will be negative of 2x by x square plus y square okay remember you're treating y as a constant here so this will become minus 2 by 1 plus y square by x square into minus y by x square into minus y by x square okay remember y by x derivative when you're doing you'll treat only x as a variable y is a constant whole divided by whole divided by again derivative with respect to y and treating x as a constant will be this minus into 1 by x correct is this fine so few things that we can do directly over here we can make it as 2 plus 2 y so make it as plus 2 y 2 factor you can remove from everywhere if you take x square up if you take x square as a LCM over here this x square gets cancelled with this x square so you end up getting y upon this okay here also we can do the same if you take x square as the LCM over here okay and take this x square on the top that will get cancelled with this x leaving you with the x on the top okay so the numerator term is minus x plus y denominator term is minus y minus x remember the denominators of both numerator and denominator will stand cancelled so this is nothing but x plus y by x minus y which is your RHS and moving on this is basically a twin set of problem which I'm giving you we'll solve only one of them normally in school exams also it appears the question goes like this if under root of 1 minus x square plus under root of 1 minus y square is a x minus y then prove that then prove that dy by dx is given by under root of 1 minus y square by under root of 1 minus x square okay now the various versions of the same problem the next version that you will probably see is this one the next version that you'll see is this one if under root of 1 minus x to the power 6 plus under root of 1 minus y to the power 6 is a x cube minus y cube then prove that dy by dx is going to be x square under root 1 minus y to the power 6 by y square under root 1 minus x to the power 6 okay if you look at a generic case that's what we're going to do now but I'm just uprising you of the different forms of the same problem that you may get if you see the generic case is 1 minus x to the power 2n plus under root of 1 minus y to the power 2n if it is this then prove that dy by dx becomes x to the power n minus 1 under root of 1 minus y to the power 2n upon y to the power n minus 1 under root of 1 minus x to the power 2n let's do the last one let's do this one so what exactly that a doing here sir is just a constant okay but can we do without it also nothing will happen is just a constant very good to see Matthew T is over for you Monday so yesterday this problem if you try to do with brute force really if you try to use your formula also minus dy by dx upon dy by dy you may not get the result very quickly okay you may have to slog you may have to struggle you may have to spend a considerable amount of time to get the output so what do we suggest over here is using some trick substitutions using some trick substitutions okay so what I'll do is I will take x to the power n as let's say sine of theta and I take y to the power n as this is sine of phi okay so what I'm doing is I'm going to take x to the power n as sine of theta and y to the power n as sine of phi it is just for our simplification purpose it has no role to play in the differentiation it is just to simplify that given implicit function okay so basically what I'm doing I'm converting an implicit function into another implicit function okay so now this term over here this term over here will actually become under root of 1 minus sine square theta this term over here will become under root of 1 minus sine square phi okay and this is nothing but sine theta minus sine phi so this is nothing but cos theta this is nothing but cos phi this is nothing but a sine theta minus sine phi okay let us use our transformation formula that we had studied in our class 11th cos a plus cos v formula that's to cos a plus b by 2 into cos a minus b by 2 and this becomes 2 a cos a plus b by 2 into sine a minus b by 2 okay 2 gets cancelled 2 gets cancelled cos term gets cancelled cos term gets cancelled that will give you cot theta minus phi by 2 as a okay that means theta minus phi by 2 is cot inversing take a 2 on the other side okay so this is where we reach out reach up to now what is theta as per the given expression theta is sine inverse of x to the power n and phi is sine inverse y to the power n and this is just a constant you can call it as a c okay now from here we can use our implicit differentiation formula dy by dx is negative doh f by doh x upon doh f by doh y okay so what will it become so negative doh f by doh x will become what 1 by under root of 1 minus square of this term but at the same time we have to use our chain rule so x to the power n derivative is n x to the power n minus 1 now remember the other term is a term in y so that will simply become a zero you don't have to worry about it okay whole divided by doh f by doh y doh f by doh y this will become a zero this will become negative 1 by under root of oh I'm so sorry I think I missed out a 2 1 minus y to the power 2 n into derivative of y with respect to y is this fine now let's simplify these terms so minus here and a minus here will be taken care and n and n will take care of themselves so you end up getting x to the power n minus 1 under root of 1 minus y to the power 2 n divided by y to the power n minus 1 under root of 1 minus x to the power 2 is that fine is this what you wanted to prove is this our right hand side okay now people who are saying done how did you do it was there any other method or did you find longer time to do it people who said done okay okay so this is a very classic problem normally it comes in school exams also but in school exams you'll probably get a lighter version I think the first one you'll get maximum access the second one you'll get but I took a generic case this is a this is a more generic case general case okay let's take another one let's take another one let me put the poll on for this oh sorry Adweta do you want me to go back to the previous page Adweta sorry I missed your message five four three two one please vote everybody most of you have gone with option B let's check see you don't have to start another difference differentiating the function from the word go you can make some small changes into the function for example let me write it like this okay let me square both the sides let me square both the sides so this will give you x square one plus y is equal to y square one plus x okay so this will give you x square y minus x y square plus x square minus y square equal to 0 okay take x y common over here and take x plus y x minus y common over here so that'll give you two expressions over here one is this and the other is that okay now probably x equal to y yeah probably x equal to y is an extraneous factor that has kept into the system okay so this is an extraneous factor because x need not be equal to y right that means the factor which we are interested here is the other one so this is a this is the one where we are interested in okay this is the one which is important to us this probably has come because of the process of squaring as Venkat is rightly pointing out squaring may change the function may introduce extra roots into the system that's why this factor has come into the system okay and if your y is equal to x then divide by dx is going to be one which is not in your options okay so let's not worry about the other factor so let's worry about this guy x y plus x plus y equal to 0 now this is I mean this is actually an explicit function you can express x in terms of y or y in terms of it but let me tell you the formula which I have discussed with you is equally applicable for a cases where you take a bit of effort to separate x and y okay so what I'll do is I'll treat this as fx comma y okay and directly apply my formula dou f by dou x is minus dou f by dou x by dou f by dou y okay so the derivative of this with respect to x keeping y is a constant will just give you a I think why and this will give you a one okay what about the denominator terms denominator term will give you a one plus x okay now hold on this doesn't match with your options but that doesn't mean you have done it wrongly that doesn't mean you have done it incorrectly probably they want you to replace your y completely in terms of x they want you to replace your y completely in terms of x which anyways we can do it from here so we can say why 1 plus x is negative x so why is negative x by 1 plus x okay so place it over here so it becomes negative 1 plus y 1 plus y upon 1 plus x so if you simplify this that clearly leads to the result minus 1 yeah I'm correct minus 1 upon 1 plus x the whole square which is option number b which is option number b now I'm not saying this is the only way to do the problem you could have done it by other ways or as well okay but just be aware that you don't have to start differentiating in the raw form as given to you in the question is that fine any questions here any concerns sir could you go to the left yeah I know thank you sir okay yes sir in this case it was quite easy to determine which was the extraneous roots but then in other cases how exactly do we do it you have to get it clarified from the options you have to see which option yeah which option is basically meeting your requirements okay yes so could you give an example like that there's no ready made example as of now in my mind but I'll do one thing I'll just send you a few questions based on that okay sir I'm unable to put my question so I'll give you my question let's say yeah this is basically a equation of a family of conic passing through origin okay this is basically an example of a family of conic passing through origin it also includes a pair of straight lines by the way okay then then dy by dx is which of the following option a why did I put a bracket on this option is minus x by y option b minus y by x option c y by x option d x by y I'll put the pole on alright at the counter five I'll stop five four three two one go alright so 25 of you have voted so far and 60% of those people say option c okay now very easy question to you know I mean prima facie looks easy but most of the effort probably may go into the simplification because you can see in the answer there is no a there's no b there's no h so to begin with we'll use the formula dy by dx is minus dou f by dou x by dou f by dou y where this is your f this is your f okay so minus dou f by dou x minus dou f by dou x means you're differentiating only with respect to x as a variable so that will be 2 a x this will be 2 h y dou f by dou y will give you 2 h x plus 2 b y okay the factor of 2 can be removed so it becomes negative a x plus h y by h x plus b y now what to do from here on because my answer is in a different format so what I'm going to do is we all know that we can break we can break this term like this so what you can do is we can break this 2 h x y as h x y and h x y so I'll do a x square plus h x y and the other term I'll take it on the other side so minus b y square h x y okay so if you take x common from here you get a x plus h y and if you take a y common from here you get h x plus b y by the way this is b you may look like an h yeah okay so from here we can directly say a x plus h y by h x plus b y of course you can borrow a negative sign that will become y upon x that will become y upon x so this answer that you have got over here is actually y upon x that's nothing but option number c most of you got this correct 60% of you voted for c very good any questions here any any questions here somebody was asking something yes like I mean we also learned let's assume for time being I know I'm complicating it but just for knowledge sake or whatever like let's assume for some time that these are a pair of straight lines okay sir and then okay and let's assume they have been up also like slope is y by x okay that's most of you would have solved it by that method also that's that's a very excellent method of solving it accident I think okay so trippin also has some question trippin you're saying something trippin oh no trippin is same thing same thing yes okay origin then there'll be two slopes yeah so I'm not saying it is differentiable at the origin as you can see it will become zero by zero for yes okay so it is it is non-differentiable at the origin next question I think this is very straightforward you just have to use your formula and your result should be there but if you go by the school method of you know finding dy by dx for the entire expression I mean differentiating it from left to right well you'll have to waste some time collecting your dy by dx terms okay in another 45 seconds we'll stop one please vote please vote only 12 of you voted three minutes is good enough for time for this kind of problem three minutes okay and the poll maximum people have voted for option a a is the most voted for option okay let's check see again if I were you I'll just use my formula dy by dx is negative dou f by dou x by dou f by dou y okay guys are be careful about this negative sign I've seen people missing out on this negative sign as a result they get a negative of the actual answer okay so negative dou f by dou x now dou f by dou x you have to treat x only as a variable everything else should be treated with a constant okay so this will just be 2x e to the power y okay just 2x e to the power y this will be this will be 2 y is like a constant so you can take out so you have e to the power x plus x e to the power x correct all divided by dou f by dou y dou f by dou y the first term will just give you x square e to the power y second term will give you 2x e to the power x okay just do one small thing divide throughout with e to the power x because this doesn't match with any one of the answers so just divide by e to the power x both numerator and denominator so this will give you negative 2x e to the power y minus x minus 2 y x plus 1 upon this will become x square e to the power y minus x and this will become 2x is there any option which says so oh yeah there you go option number 8 so a is absolutely correct much convenient method if you are using this formula but equally cumbersome or you can say you have to collect all your terms with dy by dx collect all your terms without dy by dx and then make dy by dx a subject of the formula but unfortunately you have to follow the same in school you can't do anything about that sir yes sir sir in school just asking could we use the proof you gave like let's say they gave a question and then we use it yes sir they say if you want to use anything else extra I can prove it then use it yeah they say yeah but I've seen them that they did it to marks for that I don't know why they do that but they don't keep their promise yeah seriously I mean some of my students tried proving it law with our rule and they started in a solving it but they were not on it they're not given marks okay so let's let's not try it if at all there's a problem let's not do it yes sir okay let's take easy most of you are able to do it quickly so this is basically why is equal to under root of sine x plus y so y square is equal to sine x plus y now I would basically do this problem by the use of my implicit differentiation I like it like this okay so dy by dx is negative of doh f by doh x so derivative of this is going to be with respect to x is going to be just negative cross x by doh f by doh y so this will give you 2 y this will give you a 0 this will give you a minus 1 so as you can see you can get the answer in straight away two or three steps good enough so could you show the previous slide for one second previous slide for one second yes sir I can't disconnect it huh so that doesn't mean squaring always gives extraneous roots when it it may give extraneous roots


Hey, welcome back everybody. Jeff Frick here with theCUBE. We're at the Innovation in Motion Mapping and Navigation event. It's a cool event here in Milpitas hosted by Western Digital and we're talking about levels of mapping granularity that I never even knew existed, even just in the application for autonomous vehicles. It's been a fascinating conversation, a lot of panels, mainly panels, a couple keynotes, but we're really excited to have Andy Parsons. He's the CEO of Increment P North America. Welcome, Andy. Thank you. So you were in a cool panel, really talking about this debate between maps versus sensors. You know, is it the historic thing in a map that the autonomous vehicle more depends on than sensors that tell it's where it is now or the other way around and nobody in the panel wanted to really commit one way or the other. Well, they're not mutually exclusive. Right. You know, I mean, in order to make an HD map or what we call dynamic maps in Japan, you need sensors to be able to build those maps and to build that 3D vision of the world that the car needs. So it's, as I was saying on the panel, a map is historical information about what's supposed to be there and sensors are telling you what's there right now. Right, right. And hopefully you can correlate them and everything works perfectly, but sometimes you'll see differences between what the sensors see versus what the car thought should be there and that needs to be resolved by some intelligent algorithm. So let's dig down into the vocabulary a little bit because you've made a very specific reference to an HD map. So what exactly does that mean? Well, I think most of us think of a map as kind of a 2D representation of streets and features on the surface of the earth, excuse me, but an HD map is really something way beyond that. It's really, if you can think of it, it's more like what we see when we're looking at the world. When we're driving around, we're seeing features on the road. We're seeing curbs, we're seeing lane markings, we're seeing trees and foliage and other kinds of landmarks that are nearby. An HD map is all of that. It's knowing where you're going but also seeing what's around you as far as the kinds of objects that we take for granted that a human can avoid or pay attention to while you're driving. A map typically doesn't include that, but an HD map, again, what we call a dynamic map because I think it's a bit more specific about what's going on, it's all the things that are going on around the car when you're driving it. Or when the car's driving you. So dynamic by implication means moving, right? Changing within real time. Right, I mean a map to a certain extent is somewhat static. You've got roads that don't change. I mean they're there and they're not gonna just suddenly move to the left or something like that. Although lane markings can move, that's kind of a semi-dynamic thing that can change over time. Potholes can form, those are things that can change over time. You also get to a more extreme version of dynamic when you talk about things like weather and traffic. So for a car to get from point A to point B it needs to have a plan or a route, right? So the map can do that for you. You can say, okay, this is the best way right now to get from point A to point B, taking into account all of these elements like weather, traffic, construction projects, quality of the road, toll roads. A lot of people don't think about that but maybe you don't want to pay a toll. They don't want to bypass them. So all of those things need to be taken into account when you're planning your route. And that's what an HD map does for you. And even things like if you are talking about a vehicle that is a high profile vehicle, then maybe you want to be very careful about not being able to go or not going in a place where that might have some difficulty getting through. Like a low overpass or foliage that's growing into the street. So there's a lot of elements that have to be taken into account for the HD map or a dynamic map. And I would imagine that it even then goes to the things that we can't see with our human eye. Like is there bandwidth? Is there connectivity to feed the systems? Are there other things going on? Is it a restricted zone or whatever that feeds that map as well? Right, I mean there are unexpected things that are always happening on the road. As we all know when you're driving around things happen like maybe a kid's chasing a ball into the street. There's no way a map can know about that, obviously. So you need to be able to react to that and do it safely. So an interesting question came out of the panel when you were on the panel, something somebody asked, can you describe what everything's gonna look like in 10 to 20 years? I don't know if it was, you were one of the other panelists that said no. And anyone who tells you they can, they're talking science fiction. But I thought you very quickly though segwayed into conversation about what's happening in Tokyo for the 2020 Olympics. And I thought it was interesting because it shows that the evolution into this space is not gonna be smooth, it's gonna be lumpy and partially driven by specific activities or specific events or specific things that will move the ball forward a certain amount. So I wonder if you can kinda share with the audience what's happening in Tokyo for 2020 around autonomous vehicles. Sure, the Japanese government in Japan has effectively defined the Tokyo Olympics as kind of a milestone where we need to be able to have what we call level three autonomous vehicles up and running in the areas immediately surrounding the Olympics. So the objective is to be able to get cars that are able to navigate and get around properly by the time the 2020 Olympics happen in Japan. In order to do that, they've formed a coalition of companies where Japanese automotive OEMs, the car manufacturers, as well as the map producers, the map providers like my company, Increment P. And we're basically trying to work together to figure out a way to make this an ecosystem, if you will, for high definition mapping and navigation such that it will work in Japan by anybody who wants to use it. It's like coming up with a standard, if you will, for use in Japan. So it's a very ambitious goal to do that in only three years.  Well, it's been an ongoing effort for a while. It was formed last year, this company. But everybody has a seat at the table and we're all doing the best we can to try to make it all come together in time for the Olympics. So I'm just curious. You talked about everybody coming to the table, often competitors, trying to make this thing work. We got a unified goal. But I'm just curious to get your take on how the world is changing in terms of standards. We see in the traditional software space open sources become a very big thing and derived a lot of what formerly would be maybe managed by some type of a standards body. And standards bodies are less important, kind of in establishing these things. What's been the experience as the move towards full autonomy over time and trying to define some of these communication standards, especially as you mentioned, you want to share different types of data for the benefit of all. Is it going to be government regulated? Is it going to be we're coming together around a particular project? How do you see that kind of shaking out? Well, it's like pretty much any other standard, if you will. It's oftentimes comprised of the input from a variety of private organizations, private companies that agree to get together and make a contribution for what ultimately becomes a standard. The trick is how you can do that in such a way that you can still manage to run your business at the same time. So standards are great. But if you give everything away, then you don't have a business anymore. So the challenge is how do you make a contribution to a standard and yet at the same time, allow your company to differentiate itself and be able to provide solutions in the commercial marketplace. So this is always difficult. I mean, if you go back to the world I came from, which is the home entertainment industry, for example, like DVD and Blu-ray, there are a lot of companies that made contributions to those standards. And they all have competing visions of what should happen. And so we're all trying to make these contributions in such a way that you ultimately end up with something that everybody can look at and say, I'm going to build to that spec. So it's something that's always going to be a challenge when you're talking about trying to standardize something. But I think in Japan it's unique in the sense that there's some government guidance that's happening to help encourage the companies that are all doing this on a commercial basis to try to come together as best they can to put together a coherent solution. Right. And I apologize. I didn't give you an opportunity to tell us about what is increment P. And before I'm sure there's a lot of people that like to know more. Well, increment P is a company that's been around for more than 20 years. And what we do and what we have been doing is building up a very rich database of the country of Japan for mapping purposes. And again, a map a lot of people think of as a two-dimensional thing, but even in what we call normal or standard definition maps, there's many layers to that. So you've got terrain, you've got road networks, you've got points of interest, and many other things like railways and public transportation. All of those are different layers that sort of build on top of them. So we built up this very rich database of the country of Japan and we licensed that technology to a number of other companies that use it. Could be for in-car navigation. It could be for a map application on your smartphone. And then we are continually maintaining that because in Japan in particular, the quality has to be very high. Most of the time when you're using a map, you want it to make sure it takes you where it's supposed to take you. If it sends you to the wrong place. It's gonna be a good thing. If it sends you to the wrong place, your confidence level in that map goes way down at least for a while. So it's got to be high quality. So we also put a huge amount of energy into making sure that the quality is as high as we can possibly make it. And that's a pretty labor-intensive and fairly difficult proposition to do. And that's something that's kind of our specialty. Right, right. So I'm just curious too to get your point of view which has come up again in the sessions in terms of using third-party data, whether that's community, users, maybe an independent, different government data set. How do you balance kind of the value of getting that much more contribution into your system versus your maniacal focus on accuracy? And how has that kind of been working out and evolving over time? Well, there are many, many data sources we have to use. We of course have our own data that we collect through our own survey vehicles that are literally 365 days a year driving around all the roads in Japan to get updated information. But we also get data from other sources like, for example, traffic is provided by a single provider in Japan that is something that we license ourselves and then sub-license to other people. So, and then points of interest, those are things that we maintain but we also can get those sources from other places as well. Right. So it's really trying to put all that together into one single category of products that are useful and dependable. Right. Again, nobody wants an inaccurate map. Right, right. Well, and then before I let you go, I'm just curious, obviously maps are important to get us from point A to point B, but as you license this data, I wonder if you can share any kind of interesting or surprising or unexpected stories of people using your data for a specific application that we may have never ever thought of. I think it really just comes down to, in our business, we're, as I've said many times, we're focused on making accuracy kind of our top priority. We do have a mechanism to give people to report to us when they think there's an error. And this is the other thing that when you talk about crowdsourcing, sometimes people will tell you something that's wrong. They'll say, you made a mistake. This is not correct. Then when you go verify it, it turns out that actually it's not incorrect. It is correct. And that person was making a mistake. So one of the challenges you have is how do you do verification on reports that are coming back to you or you could say it's a kind of an early form of crowdsourcing when people are reporting errors. You need to make sure that the accuracy is paramount. So it's interesting. Sometimes you'll think that you've got some dire problem.  And that's a fairly labor-intensive process and humans still have to be very involved with that. Right. All right, Andy, well, I'm gonna leave it at that and I want to wish you well in two and a half years in counting. We'll be watching Tokyo in 2020. Thanks again for stopping by. All right, Andy Parson. I'm Jeff Frick. You're watching theCUBE. We're at the Autotech Council Innovation and Motion Mapping and Navigation Conference here at Western Digital. Thanks for watching. We'll see you next time.


This is a really special environment to be in. It really, I would say, inspires you. Going to a women's college, you have that Mount Holyoke network that extends for the rest of your life. I just feel so supported by this community and I feel like we all want to see each other succeed. Students here are very involved in a lot of different activities. So there's always an opportunity to meet people who are passionate about the same things that you are. Mount Holyoke doesn't only build you as a student but also in the steps after graduation. I've used my internship experiences and found out that I really do want to go into a certain field after graduation, whether that's tech or I want to go to law school. I'm graduating feeling really confident about my prospects in the job market and where I'm going to live after graduation. The community here is built on diversity, this very multifaceted, dynamic, constantly changing and evolving community. Just by engaging with people from different parts of the world, with different cultures, you learn so much about their traditions, how things work in their countries, their language. And it's like traveling the world but you're still in one place and you're meeting all these wonderful people, it's pretty great. I now have a lot more confidence with not only my voice but my opinions. I feel like I have the education to support them in a meaningful way.


We've all had our eyes on Amber heard while she's at the center of the decade's most publicized domestic trial against ex-husband Johnny Depp. Whether you're a fan of the actress or not, Amber has the world's most perfect face. Well at least that's what some plastic surgeons concluded when analyzing her features with computer mapping technology. But has Amber always looked the same since her debut in Hollywood or has she gone under the knife like many other stars? Well in this video, we're gonna find out. Hey guys, it's Cara the Vampire Slayer, welcome back to Famous Entertainment. This series is titled Before and After and today we're gonna check out Amber Heard and her transformation over the years including whether she got any work done or not. Don't forget to like, subscribe and follow me on Instagram to chat. Now let's get into this video. Amber Heard is an American actress known for roles in films such as Never Back Down, The Rum Diary and Replaying Mira in Aquaman and its upcoming sequel. Her marriage to actor Johnny Depp from 2015 to 2017 and more notably their divorce has drawn global media attention after Heard claimed Depp was abusive during their relationship. In 2019, Depp sued Heard for defamation over an article she wrote on sexual and domestic abuse in The Washington Post and in 2020 she countersued him. The Depp vs Heard trial began in Virginia in April 2022. But enough about that. Like I was saying, some plastic surgeons claim that the 36 year old actress actually has the world's most perfect face. While using computer mapping technology on Amber's features, Dr. Julian De Silva, a London based plastic surgeon, concluded this. He explained, The golden ratio was a mathematical equation devised by the Greeks in an attempt to measure beauty. The premise behind this is that the closer the ratios of a face, body or room are to the number 1.62, the more beautiful it becomes. And Amber, well, after measuring her across 12 key markers for the nose, lips, eyes, forehead, chin and facial symmetry and shape, he found that compared to some other celebrity faces, Amber was the closest to the ancient Greek principles for physical perfection. He continued, According to the research, the most beautiful female celebrity is Johnny Depp's former wife Amber Heard who was found to be 91.8% accurate to the Greek golden ratio of beauty. So did Amber have any help in making her face so perfect? Looking at photos of Amber when she was only 19 in 2005, she was lightening her hair to be blonde. At this time, her nose was a bit more prominent than it is now and her mouth looks different. Her upper lip was a lot thinner than the lower. Besides that, in a photo two years later, we can also see Amber has a bit of an overbite causes her front teeth to protrude over her lower lip, something that no longer is an issue for her. In 2007, Amber's top lip began to get slightly fuller so the changes may have slowly started happening. Then, a year later, at age 22 on a red carpet appearance, it seemed that she may have started her dental work since her teeth looked whiter and possibly more pushed back. Soon after this, Amber's look would change and she rocked a dark brown hair to switch things up. Given her thin brows and intense hair color here, it seems in her lower face she's lost a lot of fat compared to 2007, which might not be all natural. Another moment we can see facial changes in Amber while she had the dark hair was in 2009 at a gala event while rocking a rare updo. Her lower cheeks seemed more hollow and her nose was evidently wider than it is these days. Additionally, Amber's upper eyelids were starting to look a bit heavier with some more skin. By 2010, Amber was uplawned again and we can note that this year and the following year, she had definite improvements made to her smile, including very white teeth. Also in 2011, the actress's nose looked more sculpted. However, it seems that around 2012 and the time that Amber started to date Johnny, she may have undergone further work on her facial features. Her teeth now appeared to be a better size for her mouth, her top lip was more balanced and plumper than it used to be, and even her eyelids may have been fixed, giving her a more open-eyed look. By 2013, you can really see how much Amber's smile had transformed compared to 2005 or 2007. With an open-mouthed smile, we can see how much the size, shape, and position of her teeth had changed. By 2015, it became more evident that Amber was dabbling with lip fillers at least in the top lip because there was a dramatic increase in volume over the years. Especially if you take a look at the outer corners of her lips, which used to disappear. Soon after, we could also note changes to her eyelids, nose, and perhaps even Amber's chin. By age 30, it seems that Amber only got prettier as the years went on. In 2017, we saw Amber at the Unite for Humanity gala and her top lip was fuller, her teeth were less protruding, her brows were higher up, and her nose looked more petite. About a year later, photos of the actress also showed more of an angular look to her face that seemed different, likely because of cheek filler or implants in that area. In 2019, Amber rocked her plumpest lips today, while her cheeks seemed to become more prominent, maybe continuing with the treatment that she was doing before. In 2020, while it had appeared, Amber had some subtle enhancements over the years. It looked like she still had a natural quality about her face, and whatever injections that she was getting had settled a bit instead of taking things overboard. All in all, while looking at Amber's beauty evolution and her appearance now, it does seem that she has gotten some work done, but nothing too dramatic. Or at least it looks very subtle on the surface. Rhino-plasty or a nose job is the most common celebrity surgery, so it's no surprise that even someone with as nice a nose as Amber would get some tweaks done. Her nose was already straight, but it does look like she got some slight refinement to the tip and the area beside the nostrils for a narrower, more petite shape. It's even more noticeable when we look at Amber's face from a side angle. We can also assume that Amber had one of the most under-the-radar procedures out there done, which is called a blepharopus. This is a little trick that some celebrities use to change their face, such as Jennifer Lawrence. And it's a procedure where a surgeon removes some excess eyelid skin on the top lid usually to open up the eyes and stop any drooping near the brows.  First off, we can see the actress has gotten at least some lip filler over the years to increase the size and volume, especially in the top lip, which was much thinner in her younger years. Also, Amber's front teeth no longer overlap her bottom lip, so she likely used something like an Invisalign to improve that, and maybe bleaching and or veneers to perfect the color and shape of her teeth.  But this procedure would have given her that more chiseled look to her lower cheeks that she's had in recent years. Another probable area where Amber could have had work done is her chin. It seems that the shape had changed since her earlier years, which may have been in the form of filler or a chin implant to reshape the area. Finally, Amber's cheeks have changed over the years and are noticeably higher than before. What's most likely is that she got cheek fillers and slowly built them up over time for subtle changes. While Amber may have had a handful of procedures done, and even if they aren't too noticeable, we can detect some changes when we take a closer look. Whatever the actress had done or didn't have done, the outcome is pretty positive. According to considerations, Amber's current bright white smile and stunning face shape, we can't deny that the actress looks great. Alright, well now we've taken a look at the beauty transformation of Amber heard from over the years. What are your thoughts on her before and after? Do you think that she had all of those procedures done, or is it all natural? Be sure to drop me a comment down below about Amber's beauty evolution and your thoughts. Thanks for tuning in to this video, follow me on Instagram to chat, and I'll see you all in another one. Bye!


Hello and welcome to this Australia Papua New Guinea Network special event here at the Lowy Institute. My name is Jonathan Pike and I'm director of the Pacific Islands program here at the Lowy Institute. I'm coming to you today from the lands of the Gadigal people of the Eora Nation and I would like to pay my respects to their elders past and present. I'd also like to thank the supporters of the Australia Papua New Guinea Network project for their ongoing support, the Australian Department of Foreign Affairs and Trade and our event sponsors Bank South Pacific and Coca-Cola Amatul. We're pleased to be able to host this event today to talk about the biggest issue in the P&G-Australia relationship at the moment, the global pandemic of COVID-19. Papua New Guinea has been particularly hard hit by the Delta variant of COVID-19 in recent weeks. An explosion of case numbers in the Highlands and the nation's capital has sent alarm bells ringing with the true picture of caseloads throughout the country remaining murky at best. To discuss the latest on COVID-19 in P&G, along with a number of other pressing issues for Australia's nearest neighbour, I'm pleased to be joined this afternoon by Gary Juffer, governor of the Eora province in Papua New Guinea. Governor Juffer was elected as governor and a member of P&G's parliament in 2012 and is leader of the People's Movement for Change Party. A prominent and outspoken member of the opposition through the Peter O'Neill years of government, Mr Juffer joined the Marape government in 2019. Being a member of the government hasn't stopped Governor Juffer from being an outspoken representative, asking the tough questions on politics and governance, leadership and the environment. Gary Juffer, thank you for joining me. Now Governor, we've caught you in a very unique position in Glasgow as part of P&G's official delegation to the COP26 negotiations. I do want to come back to climate change and what's going on on the ground in Glasgow in a moment, but instead want to start with a snapshot of how things are going back home. How is the current COVID situation on the ground going? What are the biggest challenges facing authorities in responding to this crisis? Well, firstly, the usual challenges that are facing all governments worldwide. For example, the capacity, you know, we just simply don't have enough health workers in all the electorates, in all the provinces, in all the districts, in all the hospitals, district hospitals, aid posts and health centres. So that's the first. Secondly is the information management or the management of the information space. I feel, and I have to admit and I will probably be disciplined for this or review for it, but the fact of the matter is, we have not managed the information space as well as we should have, you know, and had we done that, then, you know, you would be, you would be ahead of the crowd in terms of basically informing them about what is going on. Today's world where information is so readily available, people can access information, misinformation and disinformation in an instant, you know, it's at their fingertips.  And promote it with in great haste, and also the elements of influence in society such as churches and community leaders, you know, youth leaders and tribal leaders. They need to have been engaged and this did not happen as quickly as we ought to have, you know, and had we done that then you know I think we would have seen greater vaccination rates. But there's huge hostility to vaccines, and primarily because people are suspicious because they don't know what's going on.  And it does seem like this information is moving even faster than the virus is through through Papua New Guinea at the moment, and many, many people in P&G even don't even believe that the virus is real but you yourself had your own battle with COVID earlier this year what would be your message to those that still don't believe in this virus. Well, it's very real. I mean, it is, it is absolutely real. I, it was the worst experience I've ever had, you know, and I've never felt like, you know, this could be it. And I'm saying this now, but at that time it was, I felt that you know and I'm, I'm a fairly healthy person I manage my diet I exercise and I keep a, you know, relatively reasonable fitness regime. And so, if it affected me that way I thought man it really affects a lot of other people differently I'm sure those who answer well, you know, but I can tell them it's very real. And, you know, the, what, what a lot of people need to consider is the fact that they've got to be there for those who they are responsible for, you know, for myself. I, after getting it I went and got vaccinated after the tyrants was given by the doctors. And the reason for that was because I wanted to demonstrate that this is the responsible thing for you to do so that you can be there for your children for your people those who you love and care about and are responsible for they need you out. So, if that's my message to the people do that. And it's very real I mean four million plus deaths around the world is nothing to snooze at that's, that's no joke.    And so what will be your message to what are you, what conversations you're having with with your government with with Prime Minister Marape about what more the png government can be doing to help turn this misinformation around turn this vaccine hesitancy around. I've said this many times I'm sounding like a broken record myself and a few others have been saying this manage the information space. This is a team of people that are going to be sitting on social media countering this misinformation this information just responding to it, everything and anything just hammering it home this is an information war that we're fighting and we are losing.  The message I've been giving and also, you know, we've got to also understand that as much as possible, maintain health services because the other diseases aren't going to take a break, you know, malaria and pneumonia and TV and all these guys aren't going to say you know what covid they're going to take a break for a little while. No, they won't they, they're going to keep marching. So you've got to treat those as well. So to do that the district hospitals and eight post health centers need to be fully equipped.  They've been seriously taken care of you know if anything this is a huge wake up call. And also it's a, it's an opportunity to review our different national department of health, which has been deteriorated, and has been decimated by corruption for so many years that it is basically not a responsible functioning entity as it ought to be, providing to the expectations of our people. Well, the health system is an issue in an area that you've been so vocal on for such a long period of time and you're in your career as a politician and in Papua New Guinea. What do you think would be the first steps in turning around the country's health system and cleaning up misappropriation of funding and corruption you've been speaking about what would be the first steps. The first steps would be to look at the public accounts committee reports that were tabled, and they were not just the final report, but reports that were tabled along the way during the inquiry, and we shown the spotlight on the procurement committee and the processes in the report that we identified the individuals, you know, and nothing has been done about this. You know, nothing has been done about those individuals we identified the companies that were involved in corrupt dealings and nothing has been done in fact many of these companies are still operating, still engaged, still providing, barely providing I would say goods and services and they're still there you know. That's what happened about this, I would say that the emergency task force of some sort has to be set up. It probably shouldn't be the Department of Health overseeing this the Department of Justice should be in charge, and we should bring good credible people in and say we need to deal with this. Now meanwhile the procurement system under the Public Sector Reform Committee, Special Parliamentary Committee on Public Sector Reforms which I chair, I have said we will need to create a space where donor agency development partners can come with together and establish a procurement system that can actually work in function for the people of Papua New Guinea because right now National Department of Health has totally failed the people of Papua New Guinea. Not only has it failed them but it's failing to correct the, you know, the situation that it is in, even when we have highlighted that situation to them and delivered a set of recommendations as to how they can improve. It seems like very tangible first steps there governor. Now the COVID crisis is developing in the shadow of a looming election scheduled for the middle of next year with campaigning already in full swing up in Papua New Guinea. What are you talking about? The logistics of undertaking, the logistics of undertaking. 2017 sorry that was the moment and election is over they already campaign you already back at it. I mean the logistics of the logistics of undertaking an election in Papua New Guinea are challenging in the best of circumstances. But what impact do you think COVID will have on the nation's ability to conduct elections next year. It's actually challenging many of which we can only predict, you know, since we've never had COVID before situation like COVID, but it will, you know, I'm, I'm nervous about these elections to be honest. I don't know how we're going to manage them in such a way that we can deliver the results that the people deserve, you know, although it is said that people get to, you know, members they deserve because of how they both but that's another discussion. You know, in regards to the logistics it's a nightmare. What happens is after every election, they should be an immediate review of the previous election. And they should be an effort to correct all the mistakes and put in place, you know, risk management strategies to ensure that these mistakes aren't repeated. That never happens. That never happens until the last moment, you know, the electoral commission is completely left alone and no one checks on them as to what they're really doing and that these reviews don't take place as far as I know. And, you know, I mean, every year the election every election year the costs just escalate to a ridiculous amount and we're staring at a massive, you know, budget for the elections and it's, it's at a time when it's almost like blackmail you know if you don't pay this that guess what we might not deliver the elections that type of, you know, attitude is what I sense. It's worrying and, you know, I think if we haven't learned from the last elections then we're in for a rough time. There'll be more violence, more deaths and more elections that are going to be very suspect, I would say. Well, let's all hope that that situation turns around quickly and a good time before the elections do take place. I mean, as you know, as you and I both know all elections in Papua New Guinea are local at the end of the day it's all about local issues but what do you think will be the bigger national issues that are going to help shape this this election. Well, the people will look at how COVID was handled for sure. They will look at also commitments and promises made and whether those commitments and promises were kept or not but again that will be the those who are, you know, somewhat educated. The masses will still vote along tribal lines, clan lines and that's, again, you know, always the danger. That's usually the big chunk of the voting population, how they think, you know, how barreling has already been in, you know, in process and progress since 2017. And, you know, there'll also be in the usual hotspots, the highlands, certain parts of the highlands that will be the, you know, possibility of ballot box hijacking and intimidation and you know, et cetera, et cetera, violence that goes with it, all of that might happen. But, you know, we can be optimistic, we can hope for the best but expect the worst. Well, I'm personally going to keep hoping for the best. Governor, I now want to shift this conversation from from talk about one crisis to another and in many ways a more intimidating crisis that of climate change. You've recently arrived in Glasgow for the COP26 climate change negotiations. How do you see climate change affecting your people and affecting Papua New Guinea going forward. I think it has been a victim of climate change for a long time and in that one we are seeing the disappearance of our coastal areas, our atolls.  The Catron Islands affected the island to such a degree that you can't plant crops, you can't live there anymore. Then you've got issues such as the flooding that's taking place in, you know, river areas where rivers are. And these floods are no longer the normal types of floods that people were accustomed to and could predict and could manage. They're different, you know, they're very devastating, they're destructive. And then you've got king tides you've got we've seen firsthand a lot of the effects of climate change you've seen global temperatures rising in highland areas. So mosquitoes are now making their way up bringing with them all the diseases that they carry, you know, and the people in those areas are not immune to this disease they don't have. They haven't developed obviously, natural immunity yet so they're prone to, there's a lot of deaths taking place in these areas from these diseases brought up by these mosquitoes who can now migrate up that way. And, you know, I mean, it's happening, you can sense that there's something going on going wrong. The problem with that is the illegal longing that keeps taking place of no one should do anything about that. We should also note here the governor's strong track record as a conservationist and advocate for pngs forests and wildlife. To hear more on this topic the governor will be talking on a panel in Glasgow next week on November 8, called exposing the timber mafia. Governor, do you want to give us a little hint at what you'll be talking about next week in that event. I want to highlight the fact that many of these companies are not companies at all.  That's what they are, you know, and I'll keep saying that they are, they take me to court. In fact, they do take me to court. In fact, they found a very clever way to use our court systems to tie us up. So I've been tied up in court with all types of cases that they've thrown at me.  Millions and millions of dollars worth of logs are leaving our shores. And if you dare to question them, first and foremost, the government systems that were established to protect png interests, and specifically in this space forestry. They're not going to step up and help you they'll turn a blind eye they'll look again away. In fact, not only that they'll fight against you, they'll side with the cartels. That's because the cartels have basically compromised all these organizations, they own these organizations, these organizations no longer work properly they work for them. You know, and what's frightening is that these cartels are diversifying into other business, they're going into property they're going into banking they're going into all types of businesses. You know, I would say it's like a covert recolonization program that they've undertaken and successfully successfully and the greatest success is the fact that not many people understand this or know this or care about. Well, Governor, it sounds like that event next week is one not to be missed I look forward to being able to tune in myself. I've covered a lot of ground here today from COVID to the election to climate change. I wanted to end on a bit of a more personal note, Governor, if you will. What message would you have for a young Papua New Guinea and that might be tuning into this event and wondering how they can shape their country in these uncertain times. Your country is so worth saving. It is a beautiful country. It is blessed with the most amazing abundant resources that is the envy of countries worldwide. And it's so worth fighting for. And you have good people on the good people that at all levels at the village level, you know, at the educated level at the youth level at the elder level that's good people but what needs to happen is this good country can't keep standing up alone they all need to come together and they all need to say right this country is so worth fighting for, and they need to develop strategies as to how they're going to do this. And they've got to start thinking about politics they've got to start thinking about the difference between politicians and leaders and work on getting leaders into parliament there's a distinct difference a politician is all about delivering classrooms and roads and bridges. That's to him the minimum so he can get reelected it's not thinking about the future too much a leader will think about the future, the future, where there can be prosperity security and opportunities for everybody and a safe secure natural environment to work in. Governor Jufa thank you so much for joining me today. Thank you very much Jonathan you have a great day. Thank you to Governor Jufa for joining us and remember that we publish a daily news summary at our os png network website to help you keep tabs on the latest from Papua New Guinea. Thanks again to our event sponsors BSP and Coca Cola amateur for their ongoing support for the os png network and to the Department of Foreign Affairs and trade for his ongoing financial support. We'll be back soon with another os png network event. I look forward to seeing you then.


Prior to embarking on my journey towards becoming a nurse, I worked for nearly 10 years as a high school science and math teacher in rural Maine. I know that sounds a little crazy. I left a decade-long career working with teenagers to jump headlong into the nursing profession amidst an unprecedented public health crisis. I prefer to think that solving great problems requires great effort. And I want nothing more from my life than to be part of the solution. The opportunity to be a part of a diverse educational community provides me with perspective that I can't get from a textbook, that I can't find on the internet, and that I can't truly understand unless I have the opportunity to really be with people. To solve the problems of the world, we must first address the issues that are immediately present around us. The place-based education provided by the Wisconsin Technical College System helps students do just that. By learning to take care of the world through understanding where they live and working to take action and make a difference in their own backyards and communities. That, my friends, is where large-scale change and societal progression starts. And the Technical College System provides a space where everyone has equitable access to a seat at the table. Regardless of, you know, the number of zeros in their bank account, their SAT scores, their family backgrounds, their lived experience, everyone can find a seat at that table, and that's the place where, you know, true meaningful education can happen. At the end of the day, that's the type of education that I want and that everyone deserves. Thank you.