Hi, I'm John Overdeck and thank you, David, for your remarks and thank you, Pete, Marcus, and the band for the wonderful music this evening. It's my pleasure to introduce our main attraction, our two next guests, our 2017 honoree Keith Devlin, Professor Keith Devlin, and the host of NPR's Weekend Edition, Scott Simon. As we already learned from the lyrics of the song, quite a bit about them, I can keep these remarks quite short. I'm particularly excited to introduce Keith because he hails from Stanford, where I was once a student, and Stanford has a wonderful tradition of naming things, having funny acronyms for all kinds of things, so he is the co-founder and executive director of the Human Science and Technologies Advanced Research Institute, the H-STAR Institute. He co-founded the MediaX Research Network, and he's also a senior researcher at the Center for the Study of Language and Information that we used to call Cicely. I don't know if it's still called Cicely, it is. As well as being an author of 33 books and a decent, fine mathematician, he's a great communicator and writer. His passion for communication has led him to be the math guy on NPR since 1995. As you might guess, the name math guy wasn't entirely meant respectfully when he first received it. When he was visiting the local California NPR office, somebody would just write up on the scheduled math guy to mean that it's Keith's turn. Joining Keith on stage will be his partner in mathematical exposition for the masses, Scott Simon. We've already learned that Scott has won a Peabody and an Emmy, and just about everything in between. I got in my notes an absolutely striking list of awards. I don't know anybody else who's won the Presidential End Hunger Award for the coverage on the Ethiopian Civil War and Famine, the James Beard Award for a story called Conflict Cuisine in Gourmet Magazine, and the Barnes and Noble Sports Book of the Year for his book Jackie Robinson and the Integration of Baseball. With that kind of diverse interest and with Keith to speak with, we are in for a treat. Unfettered by the strictures of the FCC and public radio, I am pleased to introduce Keith and Scott for a special live rendition for the math guy unfiltered. Before Scott and I start the unscripted parts, on your table you'll find little versions of this thing called a Newton's cradle. You might want to just sort of grab that without getting the things all tangled up. Just see what happens when you first of all pull one ball out and let it go. Somehow there's a mysterious transfer from one end to the other of the motion. If you are trained in a university or even a high school in classical mechanics, you will say, oh, it's transference of momentum, or maybe you'll say conservation of energy. Unfortunately that's not quite what's going on. Those of us who went through a classical education tend to believe the lies we were taught because 99% of the time those lies work, in fact that's true of mathematics, I mean actually this room is full of quants and quants know that 90% of the time the mathematics is reliable and 10% it's not. In physics it's maybe 99% but we tend to believe it because almost all the time it's true but it's not always true. Mathematics I sometimes describe as providing us with lies that are very useful most of the time because mathematics doesn't really fit the world in its simple form. The world can bite us. To see about the momentum you would think to yourself, okay momentum is to do with mass and velocity so maybe I'll pull two of these out and then presumably it will transfer to two at the other end, indeed it does. So you get two going. So it looks like momentum is sort of being passed from one to the other. Well first of all, can I ask a question? Oh yeah, that's his job. So that's not for every action there's an equal and opposite reaction. You do see a little bit of a reaction because it does bounce back but it's not the classical Newton's. It's another lie that I have swallowed all the years. You're allowed to tell the lies because you're not the math guy but the math guy is supposed to tell it true. That's what we do in the news business anyway. First of all, when you say momentum is transferred is that magic? How is it transferred? First of all let's remember momentum is just a mathematical invention. These things were invented at the time when science was invented in the 16th, 17th century period. These things got invented and they were invented to attach numbers to the world. And so when we say momentum is something else must be going on. What's actually going on is that when one ball hits the other the balls squash. Even though they're steel they squash a bit and when they squash they bounce back out. What's really going on is you've got squashing and bouncing out. You can do that with rubber balls and you can see it more clearly. In the case of steel balls because of steel it doesn't absorb much heat. The energy isn't getting dissipated in heat. If you use steel balls the little bit of squash pulls it out. So what you've really got is you've got a whole wave passing through. In fact you can get two versions of these things. When you buy them some of these Newton's cretals, it's called Newton's cretals. Some of them the balls are actually touching. In that case you can really if you want to analyse it you've really got to analyse it by a wave passing along. So it's a squeezing and bouncing out process that's happening. If there's a slight gap between them then it really is a sequence of nudges. But the model that we have in our mind that momentum is transferred is very useful. We understand things. We send rockets into space using that. It really is useful. You can sort of talk about momentum in baseball and Newton's reaction. You can talk about it but it's not actually true. If you really want to see what's going on, the one you have doesn't do this but the one I've got I can take away that guy and replace it with a big one which is twice the volume and hence twice the mass of the others. So if this was a matter of momentum, if I use that as a driver I should get the same behaviour as using two touching balls coming down. Which is what I did.  So let's just see if mathematics is really accurate. Is it just a matter of momentum?  The whole thing splays out. You get an interesting pattern which looks, first of all, you could actually look at the size of these things. You could do a gallery there. You could look at how many things do these come out part. It looks like some kind of an arithmetic progression. I'm not sure what it is but it's certainly not the same as the original one. So that simple model we have of momentum being transferred is an effective lie. It's a very useful lie but it's not capturing the world. So those of you in the quantum business, when people go on the radio and I sometimes go on the radio and say, they say, well, you know, they're applying mathematics to a world where you can't quantify things so accurately. You have to be in a world of 95% confidence or 100% confidence. Well, guys, guess what? That's true of the physical world as well. Mathematics doesn't actually apply totally accurately. It's an extremely good approximation. Not only is it a good approximation but thinking about things in terms of momentum actually is useful. So I'm not trying to say forget momentum. It's one of the best models we have. But it is a model and like all models, it's only true when it's true. And when it's not, you have a financial crash. On that happy note... Can I begin by saying something about you? Well, I'm going to anyway. A little bit certainly for me to realize tonight that Keith and I have worked together for 18 years. Mercy. 22 years. Oh, 22 years. You're the math guy. Poetically, right. This is why it's a good team. Thank you. That's so embarrassing. By the way, I was told this was a gala tonight. And I've been flying most of the day from Chicago to try and get here and didn't have a chance to change into my black watch plaid dinner jacket, which I was going to wear because it's, of course, Keith's ceremonial tartan. I had no idea though that dressed this way, just a shrub to fly on an airplane. I would fit right in with a lot of the guests at this gala.  Right, absolutely. Going to stuff some pencils in my pocket. We were not looking for a math guy. This is not the sort of thing that a news organization sits down at least 22 years ago and said we need a math guy. By the way, I think one of the ways in which Keith has been in the advance of a lot of change in the news business is you can make a compelling case for that now. At the moment, he walked into our studios and at first it was just going to be a one-off interview about something. He was so compelling, such a good teacher. He couldn't understand that ridiculous accent, of course, but that was part of the charm. And he began, there was no higher compliment in the news business. He began to help us take a fresh look at the world. And that's why Keith has been on our show for 22 years, because each, several times, a number of times of the year, he's able to find something that nobody else is looking at or he's able to see a circumstance that nobody else is looking at quite like him and other people that he knows. And it makes us take a fresh glimpse at the world, and that sort of contribution is invaluable. So I'm so glad to be here tonight, my friend, to be able to be here. Have you here, Scott? Really, very pleased. No matter how impressed. Tonight, I'm just going to ask him questions. That's all right. Tonight is the first night of the World Series. There's a game we play in this country called, you're familiar with it, right? You can use your hands. Let me put it that way, baseball and a bat. I was startled to learn this week, I didn't know this, that most of the general managers in baseball now are not former players. The pendulum, if you please, has swung over on the data side. They tend to be people who are either from a financial background or some kind of analytical background that they've applied to sports. How do you see this development? First of all, it means that Amazon and Google will probably end up owning all the baseball teams because they have the expertise to do that. But no, it's a couple of reasons why that's happening. One of the reasons that happens is because sufficient elements of randomness in baseball that you can use the behaviour of random sequences to make accurate predictions. In fact, if I remember correctly, there was a study some years ago that we actually talked about on the show where they looked at all of the good players and they sort of compared their performances with random distributions generated on a computer. And there was only one player that didn't fit the possibility of being random and that was Babe Ruth. So, Babe Ruth apparently wasn't random but everybody else could be random. There's a funny story attached to that is that that piece that we recorded actually found its way onto a CD that the NPR released called Driveway Mormons in Sports. And it was about highlights of NPR of things about sports. And the thing we did about this baseball is on there. So I achieved this sort of status of being someone who knew about baseball. And in fact, I was down visiting some mathematicians in Georgia and they took me to the first ever baseball game I'd been to with the Atlanta Braves. I went there into Turner Field and it became clear to my hosts within five minutes that I had not a clue what this game was about at which point they had to ask me how come you're recognised as an authority on baseball? All I can say is, well, I know about the mathematics and that's what got me through but I don't know anything about baseball. I hope that the people who are in charge of teams that get there because they're quants do know something about baseball because otherwise the teams might be in trouble. But yeah, you can make predictions pretty accurately based on statistics. There's a case that will be heard for the Supreme Court I want to ask you about because the mathematics has a role perhaps in its resolution and I'm intrigued by it because it seems to be applying a perspective that to my knowledge hasn't been applied before and of course that's the case that the Court will be taking about gerrymandering which determines our democracy. So the cases before the Supreme Court now comes from this case in Wisconsin which is about the state. For years, whichever parties in power will try to gerrymander things. They will change the district team map so that you can and it's known as packing and cracking you want to make sure that the opposition party can win as many districts as possible so that they can take control which means you want the opposition party to be concentrated in a small number of districts. So you want all of the opposition votes to pile up to give way in excess of needing in maybe three or four regions and then you want nine or ten regions where they just win. So the trick is to win a lot of regions with just slightly more than 50% and then lose two or three regions and whichever parties in power has done that and it tends to wash out and in fact the Supreme Court when it has ruled have said these things will ever wash out, it doesn't matter. The difference in the one now is that in Wisconsin the party that took control was the Republicans and it's only an anti-Republican story because there happened to be that time in the cycle. At the very moment when the ability to run very, very sophisticated algorithms available, the computing power, the technique and the know-how, saw the Republicans in Wisconsin were able to do for voting what Volkswagen did for emissions tests build a system that looks absolutely fair but is guaranteed to keep them in office for the next 10, 15, 20 years unless there's major changes in the demographics. And so that use of computers and sophisticated mathematics is new and clearly is a clear and present danger to democracy. There's two ways to fight it. One is to go back to the idea like we do in California of having an independent group that sort of does the districting or use fight-fire with fire. If algorithms are the threat fight them with algorithms and in fact some mathematicians at Duke University recently came up with what seems to me as a plausible way of detecting whether the redistricting map is in fact the result of one of these long random searches, Markov chain searches to find these things. Because districting maps that have that property that they look fair but really aren't they're outliers. That doesn't happen most of the time. And so you need a test to see if a map is an outlier. If it is, you should throw it out. There was a recent article I think in the New York Times by Justice Kennedy expressing some skepticism about using maths in the court. Unfortunately, maths is in the court because it's what's behind the Wisconsin redistricting map. I should, we'll point out in the interest of balance in our home state of Illinois, the Democrats have been accused of doing the same thing. Of course Chicago is famous for doing strange things as you keep teaching me. Exactly. Not just algorithms but resurrection. When my mother died a few years ago my favorite I shouldn't say my favorite I received a condolence note from Senator Durbin of Illinois who had met my mother on a couple of vacations and said nice things and then he said and I hope I will continue to win her vote in the 42nd award for many years to come and that's considered just a compliment in Chicago. Um when you what's over the horizon that makes you worry at night? Oh good lord. Um well these days it's just looking in the Washington Post every morning so that's as we say is not normal right so which is a term of art in mathematics. Um you know I'm actually a pretty optimistic person I actually don't tend to get depressed I think you know I've lived through enough cycles to know that the when you think there's doom inevitable something comes along and changes you know people are adaptable they change the way we're creative we're resourceful I just think whatever happens whether it's outsourcing whether it's artificial intelligence taking away jobs well that's why I asked the question are we up to competing with artificial intelligence or do I phrase that badly is the notion to use it not compete I you know there's no doubt about it that artificial intelligence get as it becomes more sophisticated and the computers behind it become more powerful it can do more and more things that really do seem to be intelligence and in some senses really are intelligent but by their nature it's very hard to see them being genuinely creative you've got to be careful because when you've got that degree of complexity they can seem creative because of unexpectedness and surprises nevertheless evolution over millions of years has prepared our brains to be supremely adaptable you know the word about today's plasticity we can adapt very quickly in a space of weeks months and years to a new circumstance I see nothing that's going on now that's different from things in the past where the new circumstance has just launched us on something new can we see it now by definition no if we could then it wouldn't be different but a new generation grows up they look at the world differently and they will do different things because for them AI is part of the background you know I mean the iPhone is part of their background the smart phone I remember getting excited when Steve Jobs got up and showed the first iPhone I thought that was really cool I mean my kids would laugh if I took that out of the door I still have my first iPhone I have my first iPhone it's a piece of art there's been a lot of violent weather in the past few months hurricanes as we both know with family there's been I don't know as we technically count well yes you know wildfires certainly aggravated by weather patterns can mathematics help us to understand about that well first of all it can look at the patterns and make predictions I mean this is what the EPA is all about it's to be to study what's happening in the environment and to make predictions as to things that are happening you know the recent wildfires in Sonoma the actual timing wasn't known some winds came up during the night it was hot but you know I ride my bicycle all around Sonoma all of the time and around the Bay Area these posters saying fire danger today hi everybody knew that after the long hot dry summer there was a fire danger people were not unaware of that it wasn't a surprise it was just one of these unfortunate circumstances that during the night the wind came up there was something that caused it and it just moved very quickly but scientifically we knew that was going to happen sooner or later somewhere you know right now it may be happening in Southern California we've got used to it so it was a tragedy it was a on the moment it was a surprise but nothing happened in California that people hadn't sort of been expecting at the back of their mind it's just part of living in California you know if you live on the east coast hurricanes and floods if you look on the west coast actually we don't get hurricanes we do get floods and we get fire and we get earthquakes but we know about these things you can't always predict them but you can take steps and indeed California does take steps they're very careful about giving the warnings and different kinds of days you know when it's a dangerous day we don't know what caused those fires but what made them dangerous was the wind in the middle of the night when people were asleep and there are mathematical principles that can help us with our calculations of likelihood you can make changes to the road systems that make it less likely that the roads will get clogged and the sorts of research it began with research into buildings when you have buildings the worst thing in the world is to have large doors you want small doors or lots of doors with posts in between them because if you have a large door people will crowd through it and then they'll just block it if you have a small door people will go out in a much more orderly fashion even though fast so you can simulate these things and you can find results that show you how to build buildings or that you minimise the chances of snarl ups when lots of people want to move out at the same time I asked this question and we're inviting questions from you folks too I don't know where Cindy went but I guess people will just stand up and bark them out ah okay I asked this as you know the father of daughters and I've tried to be scrupulous without bothering you too much with their homework but are you concerned that and of course as you know they go to all girls schools are you concerned that women are discouraged from their exhibit in an interest in math and pursuing it vocationally very discouraged and I went through it myself when my two daughters were at school and I saw it happening even with a father who was a mathematician by the way because I'm an inveter and an educator I would never do their homework for them I would never answer their questions except with another question so my wife had to take up the slack because with teenage girls that's actually not a good strategy it's easier with students the worst they can do is give you a bad grade at the end of the semester with your daughters they can make life hell and their wife is always on their side so there's a danger to doing that it really does bother me and it's endemic, it's not unique to the United States but it's not unique to all countries either some Eastern European countries don't have that it's part of our culture and our tradition to make those assumptions about ability we even had a Harvard president a few years ago saying there were some biological reasons and had to quit the presidency as a result so we're living in a world where these assumptions are made and unfortunately they just get passed on not willingly, not thinking studies have been done of teachers who were regarded as good teachers female teachers, teaching students and without realising it to respond to boys' questions in a way different to girls' questions it's just part of being a human one of the strengths about humans we're talking about AI one of the strengths about humans is we go through life by the way, not to put you on the spot how long does it take to serve those desserts if you calculate the I just thought we could use a little object don't ask me to do arithmetic in public I always suck to the arithmetic and I still do and the electronic calculator helps me become a mathematician these things just get passed on they're part of it and the studies have shown that they just treat the two genders differently but it's partly because our strength as people as humans is we are able to make all sorts of discussions very quickly we use heuristics we are not logical creatures we reason using heuristics but over the generations society was structured differently we built heuristics about boys can do this girls can do that but now the technologies have reached the stage the way we live our lives means that in many countries there's really no distinction between the two genders or there shouldn't be but unfortunately it's part of our cultural heritage that those things are there so even people with the best intentions in the world subconsciously make those errors and they get passed on it's a really difficult thing to overcome the emotion the needle is moving but it's moving very slowly let's take a question if we can I'll try and am I missing someone oops sorry what would that do Keith is marvelously complete and there are no questions what are you working on now I'm putting most of my efforts into building mathematics learning video games I co-founded a company 12 years ago called brainquake that makes mobile games for teaching not just teaching mathematics helping people learn to be mathematics and for assessing mathematical ability I actually learned how to do that working for the defence department for 12 years post 9-11 I worked indirectly for the CIA then I worked for the navy then I worked for the army and it was always about improving the ability to analyse intelligence in the case of the defence department the problems are so big and so intractable that it was really hard to do anything other than improve intelligence analysis by one or two percentage points because of the complexity but in thinking about all of the difficulties of doing that I had various insights into how you could help a kid in a classroom because a child in a classroom is like an intelligence analyst at the Pentagon you're faced with incoming information you have to make sense of that's what the intelligence analysts do that's what the people that's what kids do in the classroom and so all of the 12 years I spent working for the defence department built up a lot of expertise in how you would build video games to engage kids and get them to develop mathematical skills actually not mathematical skills but metacognitive skills and mathematical ability we were a little bit unusual we don't build games to develop skills we build games to develop thinking capacity the ability to deal with a novel problem because in today's world on your iPhone you can access a resource several resources that would take any university math exam in the world and get at the very least a B plus and very often an A we now have systems that do for the whole of mathematics what the calculator did for arithmetic in the 60s the world our kids come out of schools to study in is a world in which those tools are available on the cloud that means doing mathematics is a very very different activity than it was even 20 years ago every now and then there's a popular entertainment that comes along where the mathematician is the hero I've worked on some of those I know you've been a consultant I know and they're invariably for reasons I understand well because they're patterned on you they're lively sexy incredibly charming and accomplished people yeah of course some of these people know me so you're not going to get away with that I understand the groans despite the fact that they've so often consulted with you what do popular entertainments often get wrong about math and mathematicians and people who use it in their lives in some I mean the ones I've been involved with some of them they're actually getting more things right than they used to they almost always now have a mathematical consultant they did with the manager one recently with the more math connection with that photo we saw so they tend to be getting it more and more accurately correct now the days when I mean there's still a lot of bad stuff you know people like Ron Howard for example when they make a movie now they really go to a page to get the mathematics right so there's been far fewer movies where I've sort of ground about it where I thought no that was really a bad portrayal it's still the case I worked on this CBS series numbers for several years to make it exciting they have to compress everything into something that could happen over a few days whereas mathematics is slow it takes place over weeks or months or years so the only unrealistic thing that's sort of significance I think is it gives the impression and CSI does the same thing in chemical forensics biological forensics it gives the impression that this is fast and it's exciting it's not it's slow and exciting so the idea of speed you have to telescope speed to make the thing fun to watch and engaging that is totally unrealistic but apart from that I think they get it by and large pretty well accurate these days I mean you could nitpick but I'm just talking about the essence are there any more questions yes sir the implications to you the ability of these techniques actually I wrote about this in the Huffington Post at the beginning of this year I had two articles one was about was titled everything I learnt as a math major became obsolete in my lifetime which is absolutely true in the sense of I no longer need to do any of those things but the projects I worked on the follow up article was what does that mean for mathematics education and it really means since you have at your fingertips the ability to run all of these procedures you have to be able to use those procedures and those tools that means you need a very high level but penetrating understanding so the simple take home message which is already on board with many teachers is don't teach for procedural execution teach for understanding a machine will do the execution you have to teach for understanding because a machine is a machine is a machine and if you think it understands you're in a dangerous place because it doesn't and it can't so you have to teach people to be able to make intelligent use of machines in order to solve problems real world problems or mathematical problems so it's teaching for understanding you still have to know what it's about for example you need to know how to solve linear equations in two unknowns in the classroom you never need to go to three there's nothing special about three most equations in the world have hundreds of unknowns teach it for one to two that gets the understanding then just let the machines take over from then so it's really teach for understanding that one thing alone should make a huge difference as usual Keith I have to drag us out of there if I get to say something in your honor tonight I was startled to hear and I guess I've heard this before that American children are so far down in the world when it comes to mastery of mathematics skills I like to think because we inevitably hear people come to work to us as interns and young producers who say you know I grew up listening to NPR in the back of the car I was kind of a prisoner my parents made me listen I like to think if just 10% of those youngsters could hear you and decide I'm really interested in math I think I'm going to make a life in math it could make a difference I do know that there were some kids who've done that because they occasionally tell me that they've heard me on the radio or they've read the books and so that's kind of nice when you hear that 10% seems a bit high but I'll settle for two or three percent I'll settle for half a percent Keith Devlin, thank you very much Thank you Scott, much appreciated And my final task of the evening is to call Tim Nissen back up who has some awards that we would like to present to both Keith Devlin and Scott Simon so if you would both also join us in our gratitude for your both of you doing some tremendous flying today to get here Thank you Alright, so you will notice everybody will notice of course you may, I'm sure you have it's been impossible to ignore the pineapples in the middle of your tables and there are some supporting pine cones and Keith Keith can describe the importance of these for this evening's event Okay, interesting tactic I'll get an award but first I have to work to earn my reward Yeah, you're looking at the pine cones and the pineapple because those are two examples of things you find in the natural world that embody the Fibonacci sequence If you were to take the pine cone out I'm not sure whether you could or you should take the pineapple out and you look at it you'll find there are spirals going round one way you have to maybe squint and spirals going round the other way if you look at the pine cone, you look at the base you see the same thing happening and sometimes you can see them with some pine cones from the side it's a spiral round and if you count the number of spirals you'll typically get a pair of numbers like 8 and 13 successive Fibonacci numbers it was actually only relatively recently maybe a few decades ago that scientists discovered the mechanics that led to the appearance of the Fibonacci numbers it's because the Fibonacci numbers are approximations to the golden ratio and nature loves the golden ratio because when it grows to optimise nutrition the ability to collect from the sun it tends to move things around in what you can prove mathematically is the angle that's most suited to optimised life and that angle is the golden ratio and since you can only have an exact number of things like spirals you'll get a whole number of approximations so you get Fibonacci numbers so we do now have an explanation of why some things in the garden like pineapples and pine cones have Fibonacci numbers in their growth structure and so as Keith has brought so many touchstones of mathematics to us like a pine cone you can find on the forest floor or a pine apple you can find in the grocery store we present an honorary pine cone and Scott Simon as a broadcaster Scott Simon as a broadcaster of so many ideas to the nation and the world as a pine cone broadcasts its seeds and there is a pine cone for you sir you've never got one of those before have you Scott? oops and mine fell off let's give a thank you to both of our presenters Sharon Honorees and thank you to everyone for coming tonight and for supporting the nation's only Museum of Math we hope we'll see you the last Tuesday in February at the MoMath Masters mark your calendars and we'll see you then


Hello and welcome to the session. In this session we will discuss the graphs of the inverse trigonometric functions, sine inverse x and cos inverse x. Now first of all let us discuss the graph of sine inverse x. Now we know that trigonometric functions are in general not 1 1 on 2, therefore their inverses do not exist. But if we restrict their domains then they become 1 1 on 2 and hence we can have their inverses. Now let us define a function from the closed interval minus 1 to run 2 inverse interval minus pi by 2 to pi by 2 by f of x is equal to sine inverse x. That is we have restricted the domain of sine x to the closed interval minus pi by 2 to pi by 2. Therefore the domain of sine inverse x is the closed interval minus 1 to 1 that is equal to minus 1 and less than equal to 1. We can also say that sine inverse x is defined in the closed interval minus 1 to 1 only and the range of sine inverse x is the closed interval minus pi by 2 to pi by 2. That is the possible values of f of x for the graph of sine inverse x will be greater than equal to minus pi by 2 and less than equal to pi by 2. Now let y is equal to f of x is equal to sine inverse x that is y is equal to sine inverse x. Now let us draw the graph of y is equal to sine inverse x. Now y is equal to sine inverse x implies x is equal to sine y. Now for different values of x and y let us draw a table. Now for y is equal to minus pi by 2 x will be equal to sine of minus pi by 2 which is equal to minus 1. Then for y is equal to minus pi by 3 x is equal to sine of minus pi by 3 which is minus root 3 by 2 is equal to minus 0.87. For y is equal to minus pi by 4 x is equal to sine of minus pi by 4 which is equal to minus 1 by root 2 which is minus 0.71. Then for y is equal to minus pi by 6 pi by 6 which is minus 1 by 2 which is equal to minus 0.5. For y is equal to 0 x is equal to 0 then for y is equal to pi by 6 x is equal to sine pi by 6 which is 1 by 2 which is 0.5. Then for y is equal to sine pi by 4 which is 1 by root 2 which is 0.71. y is equal to pi by 3 x is equal to pi by 3 which is root 3 by 2 which is 0.87 is equal to pi by 2 x is equal to sine pi by 2 which is 1. Now for drawing the graph of the inverse trigonometric function y is equal to sine inverse x we will plot these points. Now by taking 10 small divisions is equal to 1 along the x axis and 5 small divisions is equal to pi by 6 along the y axis. We have drawn a graph of the point minus 1 minus pi by 2 on the graph minus 0.87 and minus pi by 3 on the graph. Now this is the required point on the graph. Similarly we will plot all the other points on the graph. So we have plotted all the points on the graph and by drawing all these points we are getting the graph of the inverse trigonometric function y is equal to sine inverse x. Now here we can observe that y increases monotonically from minus pi by 2 to pi by 2 as x increases from minus 1 to 1. And also if 0 then sine inverse of 1 is pi by 2 so minus 1 is minus pi by 2. First trigonometric function sine inverse x minus 1 is the graph of cosine inverse x. Now let us define a function f from the closed interval minus 1 to 1 to the closed interval 0 to pi by f of x is equal to plus. We have restricted the domain of cos x to the closed interval 0 to pi that domain the closed interval minus 1 to 1 is equal to minus 1 and x is less than equal to 1. So we can say that cosine inverse x is defined to minus 1 to 1 only. And the range is the closed interval 0 to pi that is the possible f of x values for the graph of cosine inverse x will be less than equal to pi is equal to cosine inverse x that is let y is equal to cosine inverse x. Now let us draw the graph of y is equal to cosine inverse x. Now y is equal to cosine inverse x implies x is equal to now for the different values of x and y we will draw now for y is equal to cos pi which is minus 1. For y is equal to x is equal to cos of pi pi by 6 is equal to minus root 3 by 2 which is equal to minus 0.87. For y is equal to pi by 4 which is 1 by root 2 which is minus 0.71. Now for y is equal to 2 pi by 3 2 pi by 3 which is equal to minus 1 by 2 which is minus 0.5. Now for y is equal to pi by 2 x is equal to cos pi by 2 which is 0. And for y is equal to 0.3 x is equal to cos of pi by 3 which is 1 by 2 which is 0.5. Now for y is equal to x is equal to cos pi by 4 which is 1 by root 2 which is 0.71. And for y is equal to pi by 6 x is equal to cos of pi by 6 which is root 3 by 2 which is 0.87. And for y is equal to 0 x is equal to cos 0 which is 1. Now for drawing the graph of the inverse symmetric function y is equal to cos of x. We will plot these points. On the graph we have taken 10 small divisions is equal to 1 along the x-axis. And 5 small divisions is equal to pi by 6 along the y-axis. And then we have drawn a graph that we will plot on the graph minus 1. So this is the required point on the graph. We will plot all the other points on the graph. So we have plotted all the points on the graph. Now on joining all these points we are getting the graph of the inverse symmetric function y is equal to cos inverse n. Now here we can observe that y decreases momentarily from pi to 0 increases from minus 1 also as inverse of minus 1 of 1 is 0. In the closed interval minus 1 to 1 only. So in this session we have learnt about the graphs of the inverse symmetric functions y is equal to sin inverse x and y is equal to cos inverse x. So this will be the session where we will have enjoyed the session.


Applied Finite Mathematics offers students an overview of a number of applications of mathematics, especially in the social and management sciences. Applications and modeling are central to the course of study. Topics covered include fair division of resources and costs, voting methods, apportionment of legislative bodies, power of voting coalitions, graph theory and networks, and recursive systems. The course will also extend your knowledge of matrices and their use in applications related to the social sciences, as well as probability and univariate data analysis. You'll be involved in formulating and modeling problems, applying the appropriate mathematics defined solutions, and evaluating those solutions. Computers and calculators are incorporated as computational modeling aids, and activities include lectures, weekly webinar class meetings, discussions, projects, and group activities. If this sounds like a course for you, I hope you'll consider adding it to your schedule.


Well, I work in immunology, clinical immunology, but with an interest in understanding the pathogenesis of disease as well as treating disease. So we have a department of clinical immunology here at the hospital where we investigate, manage, treat patients with spectrum of immune mediated diseases, things like autoimmune diseases, lupus, as well as immune deficiency diseases and allergies, but I also have a laboratory which investigates the pathogenesis of some of those disorders as well. I trained as a medico and having worked as a resident for a couple of years then decided what specialty of internal medicine I was going to concentrate on and made a decision to work in immunology at that time because I suppose a couple of reasons, one was intrinsically interested in the subject, the second it was clearly an area that was dynamic and in fact it still is, there's still a lot going on, a lot to be learned, a lot of things to be understood whereas you can imagine some other areas of internal medicine, most of a lot of the key problems have already been solved. So that was a second reason, third reason was I was fortunate to have some one or two mentors in the discipline who were inspiring and who also lead me into the field. It's exciting for me being here in Canberra working with colleagues here at the hospital and at the John Curtin School because I think we have a very nice setup now, a critical mass of researchers, the infrastructure, access to both models and material from patients who suffer with these diseases to really crack on and make some significant progress in understanding some of these diseases. I see medical science as being integral to the practice of medicine and I think that it's that way because we still have so much to learn about disease, there's no doubt that people are much better off now than they were 100 years ago or longer ago in terms of prognosis from a whole range of diseases but on the other hand we've still got a long way to go and we're really only going to make more progress, reduce overall suffering and mortality by a better understanding of disease.


I want to play you what Joe Rogan said, and the left is very, very good at defining an issue and repeat, repeat, repeat, repeat, repeat. Don't worry about facts, don't worry about reality, don't worry about contradictions. Just keep saying it, and it enters into the culture, and then it's just there, and nobody knows where it came from, and nobody challenges it. It's just now the truth. And they are definitely doing it with regard to this concept of genocide. To the point where, to the point where Joe Rogan now is joining EOC in blaming Israel for genocide. Now Joe Rogan is not the smartest guy on the block, this is part of it. So let's watch this, and then let's analyze his quote reasoning, and I'm using quotes around reasoning because there's no reasoning here. So let's watch this and see what Joe Rogan said. I'm going to try, I really am, I'm going to try to play the whole thing without interrupting, and then we'll go back and analyze it and talk about it. So here is Joe Rogan. Like look what's going on with Candace Owens and Ben Shapiro. What did she say? I want to know what she was fired for, because was it criticism of Israel, did she show that Edward Snowden video that he put up on Twitter that shows them drone bombing those kids that are those men, I should say unarmed people that were walking towards the rubble that clearly weren't causing any danger to anybody, they just bombed them. I have to stop here because otherwise you'll forget. You saw the video, this is the video, it's been circulating the last few days. I mean it's a weird video because here's the video, I'll show it again, they're the four men. It shows them drone bombing those kids that are those men, I should say, unarmed people that were walking towards the rubble that clearly weren't causing any danger to anybody. Clearly not causing any danger now. Think about this, four men, Israel assigned a drone to follow them. Wouldn't you ask yourself why? Is Israel just randomly looking for civilians in Gaza and shooting at them?  Why is Israel expending the resources to assign a drone to follow these four men? How do we know they're not a danger to anybody? How do we know that? Now Rogan says they're not armed because Al Jazeba said they're not armed, how do we know they're not armed? How do we know what they're holding underneath their jackets? But okay, if they're not armed, maybe they were armed five minutes ago before this video was shown and they've stashed their weapons. Maybe they've just come out of a tunnel and they're heading towards whatever. How do we know? Maybe Israel has used its facial recognition software or other intelligence to identify one of these men as a Hamas leader, as a commander. Maybe these men are heading towards a weapons cache and are about to kill some Israeli soldiers. I mean, I don't know, but how does Joe Rogan know? I mean, naïve, no, he's not naïve, come on, Andrew, or am I naïve, maybe I'm naïve.   Where is thinking, thinking, thinking, you know, that? This is not difficult. A war zone. Four men are walking. Oh, they don't wear uniforms. They're not wearing uniforms. They must be civilians. I mean, that would be true in a real, you know, war between, I don't know, European countries or something. But we know that Hamas never wears uniforms, at least since October 7th. No Hamas fighter has worn a uniform. They all are in civilians. I know you meant Joe, but I don't think he's naïve.    They drop their weapons and they walk into a crowd. They pick up weapons somewhere else and they go fight. Yes, Joe Rogan has 14.9 million subscribers. He sees this, right? And it's Joe Rogan. He's like super smart. He interviews the biggest names, the biggest names in the world. He is the biggest talk show host on the planet right now. Him and Tucker, I guess. And what's he implying that Israel just killed them? Why? They're not hurting anybody? It's not a danger to anybody? Really? How do we know that? You know, it's just God. Why isn't he interviewing some Navy SEAL who can explain this to him? It's insane. But this is the attitude. And this is a constant drone. And this is he's concerned about Candace Owen. This is Candace Owen. Why was she fired? Because she's an anti-Semite. And Ben Shapiro and the people at the Daily Wire had enough because she's clearly an anti-Semite. That's why she was fired. Anybody? They just bombed them? They just bombed them. It's just like for Biden or whoever you like, you're supposed to cover up for them because you have... But the whole thing is like they're always saying they're only targeting Hamas and everybody else is a casualty. Well, if those guys are just unarmed civilians and they're walking alone, that's what they appear to be. Dresden. And how do you know this? They appear to be just people just walking alone. Civilians. How does he know this? I mean, really, how does Joe Rogan know this? I mean, does he have any frigging responsibility with 14.9 million subscribers to think before he speaks, to do a little bit of research before he spouts this nonsense? Then, and you just blast them from the sky with robots. Yeah, you blast them with the sky with robots. Costa Fortune, these robots. Every one of those missiles, that was one of the guys who escaped from the first bombing. They went after him and killed him. You think they just do that to an arbitrary civilian, drop $150,000 bomb on him? Just because, for fun, what do you think they're doing here? I mean, but he says it with no hesitation. No, well, maybe I don't understand. Maybe we can get somebody to explain this. Maybe we can call somebody who knows something about this. This is the tragedy of war. Yeah, this is insane. And no one knows what to think now, because if you can't talk about that, if you can't say that's real, then you're saying that genocide is OK as long as we're doing it. Yeah, there it is. Genocide is OK as long as we're doing it. He called the genocide. That is what we're saying.  I hope the irony is not lost on you. It's so nuts. It's so hard to imagine that someone where a culture, a country was officially founded in what, 47? 48. 48? OK, officially founded. So that's so recent.  You're willing to do that at least on a small scale in Gaza. There's nothing left. I mean, what do you do about that? He basically said Israel's committing genocide. He basically said Israel's doing to the Palestinians what the Nazis did to the Jews. He equivocated between the two. Israel's doing it on a smaller scale. Only 2 million people, I guess, versus 6. But it's the same thing. There's no difference. 14.1 million people. I don't know how many people have watched that. Many, many, many millions, probably many more than 14.9 because it's making the rounds. This is what Joe Rogan thinks now. Joe Rogan's not exactly a rabid leftist.  He is now, I don't know, mainstream center, right? I don't know what you make of Joe Rogan. But now, whatever Joe Rogan is, yeah, Israel's committing genocide. No question because they killed four guys. And I didn't see any weapons. And I don't see any, like they didn't wear those jackets. You know how the FBI wears jackets and it says FBI? They weren't wearing the Hamas jackets. There was no Hamas jacket on them. They weren't wearing the Hamas hats with the little thing to let the Israelis know where to shoot them. So they're obviously not Hamas members. So Israel's obviously now going around Gaza with $150,000 bombs and killing the Palestinians one by one in an attempt to commit genocide. And I look to see, has anybody criticized this? Not much, not much. Everybody's just silent. I mean, generally I think what's happening right now is that the genocide rhetoric is winning. It's demonstration after demonstration in New York and everywhere else, sit-ins at Stanford University and everywhere else. And I think everybody else is fatigued and they're just like, okay, give them the genocide thing and let's go on with our lives. And the victim here is going to be Israel on a grand scale. On a grand scale. The photos, by the way, if you watch the video, I mean, there are not a lot of weird things about this video. This video was released by El Jazeera when asked how they got the video, they claim they got it from a down drone, a drone that was knocked down. But if you know anything about drones, they don't actually store the video on the drone. The video is sent back. And if they do store the video on the drone, it's encrypted. So there's no way they got this video from a drone. My guess is somebody in Israel leaked it. But the drone follows these four guys, shoots two of them, get away, and then it finds the other two and shoots them. Why would it do that? Unless these were, for whatever reason, important targets that Israel thought it was necessary to kill. So yeah, I mean, but it didn't take me long to do a little bit of research about drones and about the storage and about where El Jazeera got the video or any of that. But why would Joe even do it? I mean, he's only got 14.9 million viewers and they are not particularly care about the facts. So why make the effort to provide them with facts? That's just a waste of time. He's gonna make his $250 million no matter what.


Hi my name is Chris Teitzel and I'm here to keynote tomorrow as well as talk on security today and yeah it's been been a very fun trip to come over and get to be with the European community again I love love love love coming over from the US and getting a different perspective as well as catching up with old friends. So this is actually my first triple camp London and I'm very I mean already impressed with how well it's been run and how smooth everything's gone but really in the depth of conversation I've had deeper conversation today and this morning already than I have in many other conferences and in ranging from a wide range of topics technical and not I think that it is attracting a certain group of people that are thinking deep about a lot of the topics and issues that we deal with in the community so within Drupal specifically we're seeing more and more that as Drupal is getting used in enterprise scenarios and enterprise situations that there is a higher level of data security that's necessary obviously we're coming up on GDPR which is going to be very important for compliance and regulatory reasons but in addition we're seeing the necessary need to protect business secrets because there are more and more larger enterprises that are bringing their environments and their data into the Drupal environment and so because of that encryption strong encryption is necessary where that's what we do at Locker is provide the ability for strong encryption to be accessible and easy to use inside of Drupal so obviously we're seeing that as a trend in the larger scheme of things we're also looking at how does the connected world interact with Drupal and so now I mean it's pretty crazy but you can talk to your Amazon Echo and have it interact at the Drupal site you've all of a sudden opened up a completely new vector for attacks that could occur to your Drupal site so we're seeing you know Drupal being used as a data hub and the necessary precautions around protecting that data but then also being that data hub how does it interact with all the different environments and open up new and kind of novel ways of attacking Drupal meaningful