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My name is Dominic Vella, and I am professor of applied maths here in the Mathematical Institute,
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and I'm also a tutorial fellow at Lincoln College, which is one of the colleges in the centre of town.
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I've got this photo here in the front, like just on the off chance that it's raining, which fortunately is not today,
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but also to show you that some of the buildings in the centre of Oxford particularly beautiful and well worth visiting.
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This is a photo of Lincoln's library. So my job today is to tell you a little bit about what applied maths is.
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And historically, it's something that's really been viewed as sort of developing mathematical models of real world phenomena.
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And historically, maybe it's been focussed on physical objects.
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So particular some of the things that people in the department here think about how brains fold and trying to understand
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the very convoluted folds that you get in human brains and are actually observed only in relatively large mammals,
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mammals larger than the ferret or the rat. And the question is what gives rise to this very fine pattern?
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And can we understand how it helps with cognitive ability?
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So that's a problem in solid mechanics. I'm interested in some mechanics, but also how fluids flow.
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And one of the things that I've thought about a little bit is what happens when you pour yourself a cup of juice.
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So this is a movie where you see orange juice being poured into a cup.
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And if you look at the bubbles that rise to the surface, they sort of clump together.
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And then over the period, over a period of time,
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they sort of move towards the edge of the glass that we're interested in understanding how that happens, why it happens and how fast it happens.
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But increasingly applied maths is not just about the physical world, but also about the digital world and how to deal with data.
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So an example may be that some of you might be too young to know about photos on paper,
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but actually a big problem with photos on paper is that they can get creased and damaged.
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And the question is, if that's your only copy of a photo,
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what do you do to try and reconstruct the original image and actually mathematical algorithms,
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the people in the study that try and even paint the image so that you go from
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this damage photo the top to something closer to the original down bottom.
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So what I'd like to do, having described a bit about what applied maths is really trying to do, which is more or less everything.
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Let's give you a taste of how applied mathematicians do that and also to describe a little bit about how
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what techniques we teach you in the undergraduate course here at Oxford that will allow you to do that.
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So the basic idea is that there will be some fundamental principle that we then try it out,
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try to write down mathematically and then from that mathematical expression of the fundamental principle,
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we then try and see what the logical consequences are.
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So a simple example of a fundamental principle is Newton's second law is something you hopefully would have seen at school in science,
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GCSE or A-level physics, for example. And Newton's second law tells you that F equals M.
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It says that the force are not on a body. F is equal to the mass times the acceleration.
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And if you know that the acceleration is the rate of change of lost in velocity is the rate of change of position,
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then you know that that acceleration can be written as a second derivative with respect to time of the position.
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So if I know the force in an object and I know its mass, then I can work out what its position is going to be sometime later on.
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And as an example of this that many of you would have seen in your A-level, I would think a little bit about what happens with a simple pendulum.
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So pendulum is a length of string of length L with a mass m hanging on the end of it and we pull it to one side and let go.
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And the question is, well, I want to write down Newton's second. Also, the question is what are the forces that are acting on this body?
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If I think about the forces tangential to the motion, then I can write down that the component of gravity is minus MJG times.
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The sign of the angle. Theatre features that angle up there and that is the force.
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So it's got to be equal to the mass times. The acceleration and the acceleration is L Times the second derivative of the theatre with respect to time.
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So from this equation, we can cancel em and then we can think about something else that you have seen, hopefully in your A-level,
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which is that for very small angles, sign theatre is approximately theatre and say this equation becomes relatively simple.
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It says that the second derivative of a function is minus some constant squared times the function itself.
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And if you've done the calculus of trigonometric functions,
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you might recognise that the solution of that equation is going to be a trigonometric function, sign or cause.
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So we can write down the general solution as being some amplitude times the cause of omega,
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the some frequency times time plus a Faceshift say omega here is given as the square
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root of the acceleration due to gravity g divided by the length of the pendulum.
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OK, so what can we say, given this solution?
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Well, the first thing we can say is that we know that cost goes backwards and forwards, and it oscillates.
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And that's just what we expect the pendulum to do as well. So that's good.
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What else can we say? Well, we've got this expression for what the frequency of the pendulum is.
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It's the square root of over L. And I think the first time that you see this formula is sort of a bit surprised.
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You're used to things how fast things move, maybe depending on the mass of an object.
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So you might think, well, why doesn't the frequency depend on the mass? Well, we saw that it's because of the cancellation that happens in Second Law.
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But just to convince you of this, I take my daughters to the park and one of them is roughly 20 kilos and one of them is 10 kilos,
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and I pull them up on the swing and let go of them. And you see that they they fall with or they oscillate with the same frequency.
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OK, so we've seen that mass doesn't change the frequency of oscillation.
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We could change G by going to Mars or the Moon and change a frequency that way.
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But a much simpler way to change the frequency is to play with the length of the pendulum.
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And this video I'm going to show you in a second here is from the Harvard Natural Sciences demo.
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What they've done is they set up a whole series of pendulum with different lengths, and then they let them go all at the same time.
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OK, so you can see that as we expect from our formula, the length is influencing the frequency.
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The short one is going much faster than the low one or much is oscillating with a lower period.
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OK. And they've chosen the lengths of these pages in a very nice way so that you get these really mesmerising patterns.
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And if you go onto YouTube and watch the rest of the video,
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you'll see that actually comes full circle and goes back to the beginning and then carries on.
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The reason I have included this video here is because it gives you the sense
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that it's not necessarily enough to just think about how things vary in time.
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It might also be interesting to think about how things vary in space as well.
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And if we want to be very crude, then actually applied,
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Massey University is thinking about some of the things that you've thought about at school as functions of one variable,
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but thinking about how we deal with functions of two variables.
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So as in the pendulum example, we might be that something varies in space and time, or it might be that things vary in two spatial directions x y.
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So why do we need to think a little bit about this? Well, one thing that you need to think a little bit about is how we should talk about derivatives.
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How do things change? We have to introduce this curly debate. That's something that comes up a lot in two core first year courses.
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So an introductory calculus, we worry about what difference between this curly divide and the straight divide it is.
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And then in the course called multivariable calculus,
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we sort of talk about doing integration as the reverse of differentiation, but with multiple variables.
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OK, so with that mutation,
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then it turns out that there are essentially three different kinds of differential equations that come up with three different equations that come up.
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One is called the wave equation, and if you look at it, you can sort of say I've got a function f that varies with x and time.
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And if you look on the left hand side, you see the second derivative of F with respect to time.
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That's sort of coming from Newton's second law. It's a bit like the acceleration time in Newton's second law.
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And on the right hand side, I have a second derivative with respect to spatial code in the X that turns out to come from the forces.
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The second equation that comes up is what's called the diffusion equation or the heat equation.
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Again, this looks similar, except on the left hand side you only have one time derivative.
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And I'll give you an example of or try and outline how this equation is derived so that
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you get a sense of a different physical principle that's useful in applied maths.
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The final example equation that comes up a lot is what's called the +s equation,
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and that two looks a lot like the wave equation, except that the two times are now on the same side of the equation.
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So it's as if time was sort of an imaginary space coordinate, if you like.
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OK, so these are the three equations that come up a lot. And again,
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a cool first year course called very serious and partial differential equations is really focussed on trying to
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understand how you would solve the different techniques you would use to solve each of these three equations.
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Obviously, we can't talk about that today, but what I want to give you instead is a sense of how the solutions of these equations behave.
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And some examples of where you might have seen these in your everyday life, where it.
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OK, so what about the wave equation?
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As I said already, the physical principle behind the wave equation is often Newton's second law, but it turns out that in one dimension,
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you can write down the general solution of this equation as some function big f of x minus plus another function big g of X plus c t.
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And then the question is just what are the big f g?
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Now I can't show you why that works, but hopefully you can see that this f of x minus c t is a way of moving to the right at some speed.
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C, You know that when you take a function of X minus A, that's like just translating the whole graph to the right.
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And if you do a function of X plus a constant, that's like moving the graph to the left.
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So what we're doing is we're moving the graph Big F to the right with an amount that depends
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on time with this constant C and we move in the graph Big D to the left again at some speed C.
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OK, well, what can we do with this? Well, one example that you might think about in the third year course is actually basically to try and
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understand what this transmission or this this sorry troubling waves do when you're playing your music.
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I'm sure that it's a familiar experience to parents and to children alike that you often get complaints about your music.
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And the question is, why is that what's going on there?
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So what we have is we have our music in our bedroom, which we think we are playing at a reasonable volume, OK?
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And it's hitting the wall and we hear it all being reflected by our parents hear stuff being transmitted.
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And the question is what bits are being transmitted and what bits are being reflected.
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And again, for a calculation that you can do after the third day course,
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you find that the ratio of what's transmitted to what's reflected is inversely proportional to the frequency omega.
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OK. So what does that mean? Well, it means that if you have very low frequencies, that transmission to reflection is very large.
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And if you have very high frequency, the transmission to reflection is very small.
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OK, so as we maybe what you see in everyday life that high frequencies tend to bounce off the wall, low frequencies are transmitted.
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And that's why when you hear music or the traffic light next door's car, it's really the bass that you hear rather than the tune.
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OK, so there are lots of other solution types of solution of the wave equation.
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For example, if you think about a drum, the solutions are irrelevant when you hit a drummer.
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What are called normal modes and normal modes depend on the shape of the boundary,
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the shape of your drum, but they also depend on the way that you hit them.
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So if you hit the drum perfect in 70, you might expect to get this dome shape and that has a particular frequency associated with it.
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If you hit it in slightly other ways, you might get combinations of different moods.
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And these are the modes of a circular drum, so you get this sort of this hat shape.
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And again, each shape has a different frequency associated with it.
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And again, the techniques are used to calculate these are covered in this optional third vehicles called waves and compressible flight.
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What's really important is that the shape of the object is important in determining the frequencies of which it likes to oscillate,
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and that the and that that frequency, this kind of frequency depends on the shape and then the shape of the object gives you a particular frequency.
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So you can see that in the violin,
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where you can put powder on the surface of violin as it vibrates and you get certain nodes of the oscillation patterns.
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OK. And just as a simple demonstration of this, you could imagine taking a simple mug,
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OK, and what I'm going to do is I'm just going to tap it with a spoon. OK, so this is an ordinary mug.
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But the key thing is, if I tap it here, you can hear one note, OK, and if I tap it slightly off off that angle, can you give us a different name?
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OK. So what's happening there is that by tapping the mug in different places, you're exciting,
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different modes of oscillation, and each of those modes of oscillation has a different frequency.
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What's special about the mug? Well, it's certainly symmetric.
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So if I just tap in a different place, it shouldn't make any difference. But of course, there's a handle.
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OK? And say the handle, the mass of the handle effect, which modes vibrate depending on where I hit it and what those frequencies are.
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So I've talked about these different modes,
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what are called eigen modes as being the shape that a drum will have if you hit it and actually you've already walked past an example of that today.
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So if you go outside and look up at the ceiling, you'll see there's this big glass roof, which is precisely the eigen mode.
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Actually, the second I have made of a drum with this strange boundary shake.
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OK. So I've talked about why the frequency of the oscillation depends on the shape,
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and then you might if you're a mathematician, you might start to think, Well,
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if I listened very carefully to the sound the drum makes and I pick out all of the different frequencies that are there in that sound,
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maybe I can also tell what shape the drum is going to have.
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And so that's a question that was posed in the 60s and it wasn't solved until the early 90s when these three mathematicians down here say
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that that actually you can have drums with very different shapes or apparently different shapes that have exactly the same frequencies.
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Okay, so there's a slightly odd shaped drum, but to actually prove that the spectra,
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the frequencies of these two drums is identical is a little bit involved.
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OK, so what about other examples of the wave equation in that you may have come across?
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Well, you may remember a few years ago there was a lot of excitement about. Sorry about the discovery of gravitational waves.
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So this is a simulation of two black holes colliding. OK, and what you're seeing is the sort of background of space time.
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As they get closer and closer, the time left until the collision is shown in the top left.
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And when we get to zero, it will rapidly slow down.
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But what you might be able to see or just start to see is that actually the colour is changing in space,
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and that's really because gravitational waves are being emitted and that's what was detected at the end of 2015.
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So in the second movie, we're just as email. And then you'll be able to see these waves propagating across space time in a very similar way
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to the way in which ripples move on the surface of the pond after you drop a stain into it.
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OK, so these these are the gravitational waves,
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another example that you may have actually seen already today is what's called the phantom traffic jam,
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and that's the situation in which you're travelling on the motorway or on any road and everything's going fine.
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And then all of a sudden the traffic stops and you think, OK, maybe there's been an accident or someone's broken down.
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And then after a few minutes, equally, suddenly everyone starts moving again and you never see any sign of an accident or a breakdown.
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So the question is what's going on there? And the answer is that people are imperfect drivers and as a demonstration of this.
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The BBC One show a few years ago did sort of experiment where they asked a series of drivers
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to just drive around this nice circle and told them to all stick to 10 miles an hour,
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OK? They, even though they've all been given instructions to stick to a particular speed, there are little variations in the speed at which they go.
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And, of course, because they're trying to avoid banging into each other as well,
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what you see is that actually there's a wave sort of ends up with a stop wave that kind of propagates backwards all the way around the circle.
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OK, now of course, the mathematics behind this is slightly different to the mathematics of the wave equation.
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I presented a few slides ago, but the features of this wave propagation and various other things are really generic features of the wave equation.
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What I want to do now is to move on to the diffusion equation.
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It's often also called the heat equation because it's used to describe the mathematics of how heat flows.
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And I want to go through the derivation of the heat equation a little bit just because
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it gives you an idea of a different physical principles and Newton's second law.
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So what we do is we think about a metal bar, OK, that's got there's no heat loss through the sites.
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I'm just going to imagine putting some heat in at one end, and I want to think about how it flows along the length of the bar.
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OK.
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So if I think about the temperature, which is going to vary with space and time and I think about how does the temperature in this little cylinder,
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this little cylinder with the dust curves on either side vary between time T and time T plus Delta G.
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Well, if the temperature has changed and that means there must be a change in the internal energy.
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The change in internal energy is a change in temperature times the area times the length times,
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the density times what's called the specific heat capacity. How can that energy have appeared a contest have appeared from nowhere.
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It must have either flown in from the left or not flown out from the right.
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So really it's a difference between how much heat flows in from the left and how much heat flows
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in from the right that will determine how much the temperature changes in an instant of time.
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So I've just introduced this Q of X, and I haven't told you anything about what it is,
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but Fourier told us that actually heat flows from hot to cold and in mathematics,
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we write that as Q the heat flux, the flow of heat is minus d t by the X case.
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Heat flows from high T again down to low T.
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So if I substitute this into the right hand side here and then 10 X do that, Delta X tends to zero.
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And Delta T 10 to zero, then what I find is exactly this diffusion equation.
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Well, what can I say about how that or the way in which the solution to this behave?
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Well, the way in which you will have seen the consequences of the heat equation in everyday life is when cooking,
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and it's typically important in our house because I really like this brownie recipe.
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OK. And so what you see when you look at the recipe for brownies is, as you expect, it might tell you the ingredients.
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It tells you what you need to make this brownie. But then somewhat surprisingly,
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I think the first instruction is that you should lightly grease a 20 centimetre shallow square cake tin and line the base.
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So that wasn't what I was expecting. I was expecting to do some mixing.
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So the question then is, well, why does it tell us what size tin we should use?
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And the real particular problem in our house is that we don't have a 20 centimetre cake tin.
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We only have a 15 centimetre cake tin. And so the question is what should we do if we only have a 15 cents make sense to me, the cake tim.
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Obviously, one solution would just be to make two sort of scale the ingredients, but that would mean having less brownie.
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So that's not really a solution. We want to think about how we should change the cooking of the brownie to accommodate this.
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OK, so to a mathematician, cooking is really just heating up the browning, you take a thin slab of brownie mix, you put it in the oven, the oven hot,
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the top and bottom are getting heated that he has to diffuse through to the centre
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and say that the centre gets sufficiently hot to cook the eggs and whatever else.
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OK. So basically, what you're waiting for is you're waiting for the temperature to diffuse through the thickness.
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That means we're going to use a diffusion equation,
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and the diffusion equation tells us that the temperature changes with time according to the derivative with respect to the thickness of a y.
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Well, second derivative with respect to the thickness.
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Now, as I said, we've not don't have time to talk about how to solve this, but I can say that basically in terms of the solution,
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what matters is the differentiating is a little bit like dividing by time and
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differentiating in respect to why it's a little bit like dividing by the thickness h.
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So I differentiate respect to time once I divide by time, once I differentiate with respect to y twice.
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I divide by the thickness twice and the important things.
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So that tells me that the time it takes to cook is going to be proportional to the square of the thickness of the brownie mix.
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OK, so that's the sort of conclusion from this very simple mathematical model.
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And of course, the recipe is already hinted fixes a particular volume of mix.
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So if I change my area of my tin, then I'm going to change the thickness of my brownie mix.
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And that's going to mean that I'm going to change the cooking time in proportion to the thickness squared.
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So in particular, if I use my 15 centimetre cake tin, it's going to mean that I need to cook three times longer.
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OK, just because of the way the area scales and so on. Now, of course,
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there are lots more serious applications of the diffusion equation and one that's particularly important
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to the people doing research here is what's called by interaction between multiple chemical species.
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So what they do is they write down a different diffusion equation for the diffusion of a chemical one.
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And then for a chemical to and there's an interaction between these two chemicals.
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And then you can do various analyses on this. I'm going to show you a video of a numerical solution of this equation down here.
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And the key thing is that on the left is going to be a video of what solution looks like if you have a rectangle and on the right,
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if you have a square, OK? And what you see is that very quickly, because you've got these two chemicals interacting,
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you get stripes of one chemical, followed by a stripe of another. If you solve it in a rectangular domain, what is it?
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Whereas if you solve it on a square domain, you tend to get spots.
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So as I say, this is something that's an area of active research.
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But it's believed that at least this this relationship between the domain size and what pattern you get is seen in the natural world.
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So if you look at big cats in particular, a lot of them tend to have spotty coats and stripy tails.
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And it's thought that this is really a consequence of the change in aspect ratio as you go along the tail.
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OK, so this is covered in a various third and fourth year courses in mathematical biology.
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The final equation is what's called the +s equation, which is used to govern the potential around electric fields,
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how fluids flow past aircraft wing and then also wings, and then also things like the evacuation of paint.
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So this is a time lapse movie of paint drying, and what you can see is it's more interesting than it's meant to be, right?
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So basically you you can see the sort of wet front propagating and so on.
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So the important characteristic of the plastics equation is that it really doesn't like curved regions.
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So you know that you might have heard in science or physics that when you have a very sharp conductor,
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charges accumulate in that region of very high curvature, and that's really a consequence of the plastics equation.
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The consequence of that is that tool. Shop buildings tend to get struck by lightning.
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But in your everyday life, there are some other examples as well.
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So if you spill a drop of coffee and then it turns out that the contact line the region around the edge has an evaporative singularity,
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an evaporation rate and that pulls liquid to the edge and leaves behind a very dark ring of coffee right at the edge.
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Another example of thinking about cooking again is that when you cook potato wedges,
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you see that they don't tend to get cooked evenly all across the surface. And instead, you get very but you get sort of browner or burnt edges.
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And that's because these sharp edges are where the evaporation happens most.
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That means as the smallest water content and hence the highest temperature.
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And so those regions get burnt quickest. Again, that's just a consequence of the +s equation not liking sharp corners.
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OK, so of course, I've told you about some applications, but there are other aspects of applied maths,
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including how we deal with problems that we can't solve analytically. And as you might expect, that involves using a computer.
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But the important thing is to use the computer intelligently and that's again crudely this the field of numerical analysis.
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And there are a series of courses on numerical analysis in the undergraduate course.
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At Oxford, for example, the second year numerical analysis course really asks questions about how should we deal with matrices efficiently?
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How should we think about a function on a computer? We can't tell it the whole function.
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We have to tell it the functions, behaviour at certain points. So how do we do that in a clever way?
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And then in the third year, there are various options again,
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which are really about how to control the errors that you make by solving a problem on a computer.
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And again, there's two different courses one that, again, loosely speaking,
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is focussed on solving the diffusion equation and then the second one that's loosely speaking focussed on solving the +s equation.
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OK, so I've told you some examples of problems of plague mass in the real world,
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and I hope I've given you a sense of at least why I study applied maths,
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which is that I think it really gives you a new perspective on the world around you.
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So whoever that's thinking about why potato wedges are bent on the edges or
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why a boat travelling alone to sea has a particular wave pattern behind it?
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And I think when I was your age, I think I was very worried about whether I wanted to do science or whether I wanted
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to do maths and actually doing applied maths is kind of got the best of both worlds.
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You sort of get to learn about interesting scientific problems and to develop mathematical models of those problems and then to solve them.
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And of course, because it's mathematics, the key thing is that you're allowed to make abstractions.
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Just think about that model of the brownie. Of course, the real brownie is not an infinitely thin, infinitely long, thin slab,
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but it tells me the essential mathematics that's going on in the cooking of a brownie.
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I realise that there are a lot of parents in the audience as well as I just wouldn't to say
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a few words about why the skills learnt and applied maths are useful in the job market,
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essentially. And there are various reasons. For example, the way in which fluid flows are described mathematically is used a lot in finance.
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A key thing that we get students to do is really to learn how to code, how to deal with data.
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And again, that's a key skill. And crucially, a lot of our students go into a whole range of different careers,
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whether it's in finance, software engineering, cryptography, teaching and so on.
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So I'm almost done, I just wanted to say that if you're interested in some of the applications I told you about today,
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there's a very nice short book called Applied Maths, a very short introduction by Alan Greeley, which is a very short book.
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And it's really in the same spirit as a talk that I gave you here. And that's the main reason that I mention it now.
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But I also mention it because I am sort of my boss, and it's good to keep him happy.
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Thank you very much.