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Hello, I'm Fred Wan from the Department of Mathematics at UCI, UC Irvine.

Â Our discussion today is about complex patterns and

Â how everyday in our everyday life there are plenty of complex patterns around us.

Â As you can see some of them in back of the screen behind me.

Â These complex patterns, some are interesting, others are beautiful and

Â they are as beautiful and as interesting as my camera skill allowed them to be.

Â So we do know quite a bit about these patterns,

Â especially the physical ones that we see like water waves.

Â Water waves in the ocean, the rainbow in the sky and so on.

Â But we know a lot less about

Â patterns associated with complex biological systems.

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We know a lot more about the physical patterns,

Â like water waves in the ocean and rainbow in the skies.

Â We know where they come from, how they are generated.

Â We also know maybe how they will evolve as time goes on.

Â But we know a lot less about biological systems there's the pattern that's

Â associated with them and for them we see some of them have very little patterns and

Â others very similar like the lions would have very little patterns and

Â the leopards would have lots of spots and so why are they different?

Â And this is the elephant and this is the leopard.

Â And how about the pattern changing with time?

Â That's another question, like the Penguins.

Â When they were small they are kind of uninteresting.

Â And when they grow, they become much more complex.

Â So while we do not have answers to all these questions,

Â we do know something about them.

Â And we wish today, I'd like to share with you what little do we know about them,

Â especially those come from theory formulated my mathematicians.

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To appreciate Turing's amazing accomplishments,

Â we need to begin a little bit with some background information that

Â led up to Turing's remarkable theory on pattern formation, and that is this.

Â It is known mathematically, mathematicians have proved this over the years, that

Â any complex pattern like the one that we see in the back of me, how simple or

Â complex a combination of set of basic units, components you might say.

Â And these component may be cosines like this one.

Â 1, cosine(x), cosine(2x), cosine(3x), we're familiar with them.

Â From trigonometry in high school, and yet

Â the triangle in the back of the very top had nothing to do with this curvy things.

Â A lot slated than curvy things and yet mathematics tells us that the triangle

Â is a combination of this other entities, these other components, the cosines.

Â Now, cosine is not the only set of basic components we could use

Â for representing a given pattern.

Â Here's another one, a bunch of sines.

Â We know sines.

Â sin(x), sin(2x), and so on.

Â And they two are different from the cosine that we saw before and yet

Â under certain circumstances they would be most appropriate

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So just which one do you want to use depends on the problem.

Â For example, we can have this triangle and

Â we can use a whole bunch of cosines, combination cosines.

Â If you just use a few like the one, if you use one for example,

Â you got the blue curve.

Â Which is just a simple cosine and

Â they are kinda different from the triangle but if you add more to it then you see

Â that they're getting the red curve which is a little bit closer to the triangle and

Â as you add more and more components you get closer and closer to the triangle.

Â In fact after while you couldn't tell the difference between with

Â your naked eyes between what the combination is and

Â the triangle, you write on the same piece of paper.

Â So another one is a more drastic one this is the so

Â called step function a stair case.

Â And that's quite different from the sins, the sine function or

Â cosine function and yet if you put a bunch of cosines together,

Â sine function together, you get closer and closer to the step function.

Â As you see, you just use one sine, it's like the blue one.

Â You add more to them, you get the red one and then so on.

Â And as you get more and more, you get closer and closer to the step case.

Â Now, on the other hand this one, it doesn't go there as

Â quickly as the cosine for the triangle and there are reasons for that.

Â But nevertheless, the mathematicians assure you that you will have

Â the real thing as you take more and more terms in the combination.

Â There are other, but we don't need to go into it.

Â The basic idea is that any complex pattern is a combination set of basic components.

Â There's a remarkable things in this statement.

Â It's not as obvious as it may seem, but we don't have time to go into it.

Â T here in high dimension, you have x along the line but

Â at points along the line it can be capitalized by x.

Â But if you have two dimension, like a plate or a table top or something.

Â You might need two dimensional characterization and

Â in this case we can have cosine x times cosine xy that will be one component.

Â Cosine 2x times cosine y is another and so on.

Â Now this can get more complicated if you want to visualize here's one possibility.

Â Here's a tabletop that you might want to characterize by a bunch of basic units.

Â And the one unit of cosine x cosine y is right beneath it.

Â So you can think a little bit different perspective from below.

Â So you can see what they look like.

Â There's only one component now.

Â You wanna see more components?

Â We're gonna have the next one, two of the components, the orange one and

Â the green one, the two different components.

Â And you have a bunch of these and add them together,

Â you get the table top so that's the idea.

Â And there's a name for that,

Â it's called the Fourier Series representation of any given pattern.

Â And mathematician assure you that you can do that.

Â In fact, Fourier is one of the first person who tried this and

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what he did becomes what benefit us a great deal today.

Â And all the engineers and scientists use it forever and ever and

Â get to enable us to do many scientific things making many products and so on.

Â So one of the thing that for you use it for

Â is to use it to analyze a temperature change in an object or

Â along a wire that you pass current through it.

Â You can heat it up or

Â in a room where you try to heat up warm up people from when it was cold.

Â So the idea again is that the temperature in a room or along a wire, look at that.

Â Is that the bottom black line is the wire and

Â the red above it is the temperature at different points along the line, and

Â very high in the middle and not, and zero temperature at the end say zero Celsius.

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So what we like to do is, as time goes on, we want to see how this

Â temperature distribution's in the triangular red pattern evolved.

Â We do this with the understanding that the two ends, the y,

Â we maintain at zero temperature.

Â So allow heat to get in and out, we keep it that way and

Â then see what happened to the temperature distribution along the y as time goes on.

Â And what Fourier found in using his four year decomposition, you might say.

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The triangular temperature distribution evolved in an interesting way.

Â They of course, decrease because the heat leaks out of the two ends.

Â And so gradually, you get nothing but a cosine curve, the blue curve.

Â And more and more like a blue curve and the magnitude gets smaller and smaller.

Â All the other components seem to have disappeared.

Â In fact, what Gloria found in bioanalysis is that

Â the component that has the highest oscillation would disappear first.

Â They dissipate away and so while the triangular temperature distribution.

Â It can be broken up into a different component the high with

Â the highest oscillatory behavior dissipates first they go away first.

Â So what your left with is at the end a sort of a consign

Â one hump getting smaller and smaller as time goes on.

Â Eventually all heat disappear from

Â the wire when it gets cooled down and the whole thing flattens out.

Â So in this sense, the situation's not very interesting.

Â Because what had start off as being a interesting triangular

Â profile you now gradually get down to a uniform distribution of zero temperature.

Â Uniformity is never interesting as far as being a complex pattern is concerned.

Â So that's the interesting thing about one single entity like heat.

Â If you allow it to evolve through diffusion or otherwise,

Â it had this behavior of the fastest oscillation.

Â A component goes out first and this is something that Fourier discovered and

Â we still use it together and we'll have an interesting application maybe hopefully

Â we'll get to it in the end.

Â So similar if you have a temperature profile that's triangular like the step

Â function you see that by Fourier decomposed

Â the step function into its different components.

Â Again, the one that oscillate the most disappear, dissipates first.

Â And again,

Â the sine function different component has different level of oscillation.

Â And the one that survives is a blue curve

Â with hardly any oscillation at all as time goes on.

Â So this is the conclusion.

Â Pattern for scalar force phenomena like temperature, heat, in a wire or

Â in a flat plate would as time goes on,

Â the most oscillatory component dissipates sooner, goes away first.

Â So that's the kind of mathematical result.

Â That would serve as a background for

Â our introduction of Turing in contribution.

Â