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Or if there is a my co-writer honour for me to present my inaugural lecture to you in this very special place.
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In this lecture, I would like to discuss the origins, development and loss of a partial differential equations.
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You change what? I spent my childhood in the beautiful area of Ardmore Bay where the Changing River Origin,
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a different last thing to the Promised Land in the Eastern Time.
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I see. This is actually the place to observe an effect of fantastica nature phenomenon.
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Okay, so I remember when I was three years old, my parents took me to watch the meg and this magnificent change I live with.
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Had a ball on the middle autumn days in Chinese calendar.
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It took the lead to a height of about nine metres our speed at 40 kilometres an hour.
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I was truly impressed by this shop flung.
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Shop around for a they had to propagate in very organised fashion from east to west.
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Alonzo Change. Oliver. The father was my first encounter with this giant wave.
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The now immense. Matthew. Tom. This caused shock from shock wave.
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The Potomac. Okay.
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Anyway, so this is a celestial.
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This is a believer, young man. I was a surfer.
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And this is a giant impact. The waves are.
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So I was also there after that. I was really puzzling about the millions of petals of the form, the body water motion.
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Like Leonardo da Vinci, you know, the livers, legs and even the kitchen sink in my house.
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Okay. So I was really fascinated with all little phenomena, also little phenomena formed by the air and water along the me.
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Okay, so here's a tang form normally formed in west the Pacific Ocean.
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Travelling into the East China Sea often slam into the area found to be.
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Here's a tornado. Actually, I'll also observed a little thing there in Chicago earlier, United States.
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Then shock a wave, along with a supersonic object, as it were, struck them incorrectly.
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Here, the elaborating shockwave formed to allow the and a supersonic aircraft also
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by various explosion by TNT Sofia's explosion and the supernova explosions.
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So. Well, in many cases, it is simply very chaotic and troubling locally.
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But globally we are behaviour. So we are not from such the phenomena of the call flocking phenomena by the boats.
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You came for me. The values were beautiful patterns globally and as a same situation.
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The massive Mexican waves. The pack, the stadium.
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So such observation actually has been successfully used by impressionist artists, especially to put in place an artist painter in 19th century.
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This example. Okay, the painting made by George Peel.
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So long. Then I was also I was curious.
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It was about the various sophisticated geometry or structure created by nature for them, created by the plant leaves like here and the flowers.
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Even for protein folding structure.
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Okay. So we is also a fascinating observation of natural phenomena.
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So I was wondering what are underlying national laws behind this phenomena?
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Are there any efficient way to describe politics?
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Those little dynamical phenomena are changing.
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What? So finding an answer to this question was the wing of my child the dream?
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So first Tom went out to touch to calculus.
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I was immediately attacked by this beautiful mathematical civilian concern with tens.
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Went off of the core concepts and calculus is a derivative.
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Okay. So I believe it actually is a measure of how a function changes in the changes, more precisely, the ability to malfunction if x was it is bad.
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A valuable x is equal to ten delete off the function at the z point.
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For example, we consider a moving particle in the plane.
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So trajectory of moving particles we call the position function choose our curve.
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Such curve mathematically we form as the function like those kind of functions.
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Okay, now the telemetry of this function is the velocity of this particle moving particles.
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Okay. Now, the second delivery of this position function, the first delivery velocity function is acceleration.
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So geometrically, if this is a curve, then the first derivative is the slope of the pending the line.
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Okay. The second delay repeats the the later is curvature of this curve.
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Okay. This is for that. Now, as far as I know, there are four notions about derivatives.
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Okay. So I cleared the hill as a new thing.
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Like I use the dart necessarily with you. There is the lab beneath your study.
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And like a larger use applied. So I use of study for so many different tests for the use of different machines for going for that.
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Okay. Now we are living in the world of three dimensional space.
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So minute quantities. They are not only dependent when malleable.
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You depend on time and allocation. So we depend on several variables.
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Okay. Then this will require the partial derivatives.
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Okay. So our project teams are functioning.
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For example, if I put a two dimension case is a delivery team with a let's get a one off is also variables with others hold a constant.
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Okay. In other words, the project delivered to me. The change of the quantity will expand as the other variable with the other constant for the.
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For example, this is like a surface. You can only present the by a function for two variables.
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So you hold a constant when valuable. There will be this function to present a curve on the surface.
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So very positive. Leave it to the slope of this. Curves in the surface.
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Okay. With the same with other variable. Handle the delay with the slope of the other variables.
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Okay. So for positivity we only use noting along the de oc which first introduced by legend for answering stamp the 86.
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Okay, now we use partial derivative with respect for that time means the fixed excellent to take it that it was this valuable.
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Then we will on this form some time. Okay. With other x variables we allowed the default.
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So this is the same for high dimension case. We can do the same thing now.
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Sometime we use the reading. Okay. Now we. How many piling up later is a vector as the partial delivered all those variables.
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Okay. So that notion for that. Then when I first learned the first the course of partial differential equation,
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I find was that there was clearly I showed a study to answer my child for the question.
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Okay.
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So this is a positive delivered partial differential equation in the later with a function, as I mention, some animal function we really want to know.
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Okay. The only dependent not only Pam also depend location, other variables.
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Okay. Then for example, like those quantity frequencies temporally choose for propagation of sound and heating velocity, density,
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pressure and momentum for motion of fluid and displacement, surface tensions covetous for moulting of elastic and materials.
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So in order to understand the those changes later, those functions will acquire positively with the project definition of creating that relation.
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Involve this, I don't know, functioning and the positive derivative with respect valuable t and x slow equal sign for example here decreasing.
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I can write of the as first of the equation releasing the present by as a
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function OC between partial delivery with you articulating upstream so forth.
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Now I'm in a physical system. Usually we want to know more than one and no functions which satisfy more than one relations.
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Okay then this kind of positive system about the differential equations.
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So clearly positive involves the sum and no function of depend more than one variables.
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So the more general automotive initial equation the way learn that in differential equation.
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Well this element of me as choose starting actually once I meet a novelist American literature after she got to know me my list which
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is a partial differential equations she look at that of me with a full of puzzlement and ask why you study different interpretation,
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partially not the holy. Now there was a little galaxy that turns out that we had a really very pleasant time together.
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Finally, agree with me,
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the term partial differential equation is really I got a term and you mean literally are now also so we cannot avoid equal sign and she
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I would like product to mention that actually equal sign was invented by Oxford the okay labour the recall and in in his book in 1557.
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So at the time I've got his nose in the he also to the plus.
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On the minus sign. Of course, our time equals are much longer than nowadays.
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You mean plus time?
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Well, I hadn't really had a time to read this the longest paragraph because it wasn't something that early modern English before Shakespeare.
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Now, partial differential equation allows actually mainly for two sources weighing the
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from the first one following fundamental law of and the principles in the sciences,
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including macroscopic law and principles macroscopic massless coupling law principles mach also
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macroscopic scaling and the proceed procedures from macroscopic massless approach process.
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The second source of fundamental mathematics itself as the mathematical rationales are principles include compatibility condition,
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consistent condition, construing critical points as the warehouse, the mathematical tools instrument and the generalisation,
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completeness, and also beauty and the curiosity and imagination.
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There is a study of partial differential equation as you start that 18 centuries in the work about Villa,
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the Lambert like lodging Laplace and the mini mathematicians essential tool in the description of mechanics of continuum.
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More generally as the principal mode of analysis of modes and sizes.
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This principal modes as you're still going off of the fundamental concerns of the development of PAGEAU differentiated pacing until today,
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beginning the middle of 19th centuries, particularly with the work of the iMac as a pre become essential tool and other balance of mathematics.
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This duality of viewpoint has a being an essential tool.
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Analysis of Passive Differential Equation. Until today.
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Right now for namely a posture differential equation, satisfactory ceiling has it being established.
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So no effort of several generations. May not most.
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The part of differential equation, as I mentioned, for alternative sources naturally are non-linear.
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So it's much more challenge, more difficult buying for application, more important,
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but now is really coming to mainstream research in mathematics and other sciences for that.
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Now I present the wings and pose of the colossal nonlinear partial differential equation from this category.
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Is from the macroscopic law in computer physics.
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So this is the law, the car conservation law. Okay.
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So it's a third rate of changes of the total amount of certain quantity.
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Kong pending a fixed domain for them in this domain equal to the flux of this quantity across the boundary of the e.g.
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The other war. The amount of such quantity and leaching can be measured by accounting for how much of it is a current present.
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And how much of it they are, the leave, the leading and if it's the period of time.
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Well, typical examples to suit your fundamental of nature the conservation law,
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mass conservation momentum, conservation of energy which it can formulate that this form.
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Mathematically, we can farm into this conservation as a following.
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So pull out the know that you as the benefit of this quantity value integration.
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This quantity as a whole.
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The math is the total amount of this quantity.
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Now the chanting method means you pick a delimited with a square t, right.
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So the statement that the total rate of Chanda to the equal pool, the flag of the quantity across the boundary.
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Okay. So this is a mathematical statement for this conservation law.
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Now, use, uh, uh, simple calculus manipulations and the use this law holds for any domain.
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Okay. Then we can build from here together.
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This first are the partial differential equation f of the function of view in general.
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So this is not only the partial differential equations, not in sorting system.
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Actually, they are more than two more than one quantities.
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And instead of our several laws now in the farm system.
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Okay. So then you can learn the system of conservation laws.
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That is the form like that. Now the notion is a simple to describe but assured massive weather treatment present difficulties for that this company.
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One typical example of that is a like fishing for compressible flow, which consists useless formulations and constantly mass momentum and energy.
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This is a system harvest the flow flowing convective motion dominate the diffusion dissipation.
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So for those emotions specifically, while the shockwaves like gases elastica flow and the sheer lattice, so many scientists as you have studied this,
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has been studied this this equation slowed a long time ago and thus made a great contribution,
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especially many British scientist, that made the important contribution for studying the analysis of this systems.
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I would like special mention George Stokes.
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He was the probably first scientist to detect lies, discontinue jumps, now call shockwaves mathematically slow.
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This equation are now mathematical rigorous mathematical study for the constant the knowledge conservation.
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I'll start with the actual scalar equations in the work by Allah the last century for the United Forties in the work by LAX,
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Hope, Atlantic and many others. So I merely say that the fantasy of them has been established slow the effort of several generations.
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For that now greater progress has been made. The following dimensional case a space start of flow started with GlaxoSmithKline leasing
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the by Bianchini please now for we are posting this for solution of small total variations.
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As the warehouse now existence the need for solution of large oscillations by now the modelling and function
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and that they take a technique called compensate companies must still they are many important problem.
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I open like a uniqueness stability for solution of large oscillations.
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Now formatted the magic is doing much more complicated.
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I can more challenge but scientifically more appealing since we are living the world with the full of shock waves.
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As I'm sure you're just existing those minute, those kind of high speed objects, fly objects.
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I also like various explanations for that, right?
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So when those kind of shockwave hits,
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some of the objects obstacles special are some special optical was geometry or them some flight objects metres of shockwaves then shock.
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And the flashing the flashing problem allows that.
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So one of the fundamental problems shock, shock in reflecting the phrasing by two dimensional which is set up is the following.
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Here are the two dimension weight to the plane shock moving from far field at the constant speed.
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So before. Let's do this. Wait. This is a similar plane shock.
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Just moving.
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Then the question is, what happens when this shock leads to the vortex of which and the what the kind of wave padding to form along this which.
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So this setup is really simple, right? Well, however, the the wave patterns are extremely complicated.
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Right. The first two scientists who observed the complexity of those configuration was owns the Mafia in 1879.
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And he's actually experimentally frank to see if this experiment is out.
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He found a tool. Uh, two different types of configuration knowledge, unification, model fractions.
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Now, since the importance of this is the fundamental importance of this problem, I checked as many scientists along 9040s to visit to this problem.
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But especially the experimental scientists, they do various experiments in the labs around the world.
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What did they find? A situation much more complicated, Ma observed.
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Okay, so one case is like this.
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The shock wang. This angle large enough for if we fixed the stress of shock,
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they after later power what they find that this new shock wave formed this caused a leaf rather shockwaves.
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So Wang, this information will move all the left arm of the backwards for that other direction.
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Right. So this is a picture of Lily. Beautiful.
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And then you can see this Wellesley of similar and similar flowers most here nobody living very nicely.
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Now the same problem if you change which angle and it knows more than situation, much more complicated, you see a form instead two way of conflict.
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You can form the full wave configuration SUNY Sharks and the wing vortex sheets.
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So now because it's simple modification.
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Now, if you could decrease the angle a little more that evening, now you observe the this four wave configuration,
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the vorticity waves form that even very recently and new padding has been discovered by the modelling as and your computation and
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experiments experiment through results for them the behind the mask them the form the supersonic bubbles from the generalised generator,
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the theories of shockwaves about that. So the scientific issue we really want to understand the first the structure of those patterns.
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The second transition collectively between the patterns.
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Man. We want to live in a hopefully we want to understand the dependence of those things.
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Are those parameters very important?
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We can model which angle. Now I show you a picture that is a dependent.
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We can also depend how strong string as the the the stress of the shock wave, the tonality,
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the present, the fruit America my number and it's all dependent with a kind of flow.
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Okay, we're talking about. So those issues have been studied slow in the earliest interdisciplinary approach.
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Okay. If I show you the first two approaches, then there are also a lot of work.
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Uh, slow. This is a larger and small scale in scientific computing.
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I just released the oldest contributed now from more than 400 papers in this collection.
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Also asymptotic analysis early now 1942 for the lab.
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[INAUDIBLE] talk to the many people that trying to think, uh, and analyse those patterns.
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So what we are thinking, whether possible, we develop a rigorous mathematical analysis to construct the those patterns globally for existence,
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stability, regularity in the bifurcation, those this legal analysis is very important somehow in specially for them we want to
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understand the transition criteria so we really want really sharp up to certain parameter.
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Jim Wang Other project, you always have alums.
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Okay, so this is very important to really understand those issues, the regular mathematical analysis.
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Another layer of anger behind of that is those kind of way of getting actually the call patterns in
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general solution of two dimensional conservation laws listing 20 years that are largely suitable.
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Image problem, two dimensional image problem for this hybrid a body conservation laws.
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Now this solution came from an analysis of similar solution.
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When you're through the analysis and in your macro simulation, you would find by you choose appropriate email.
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Problem means is a special classification of data.
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It means that you initially not only depend the piece with constant you depend and go okay so with well the by agreement that the 19th century.
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Now this is a recently so it's true that you can just choose those appropriate
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data you will form all those pet terms assure you in the previous classes.
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Okay. So but also the solution actually is the asymptotic state attractors and the building blocks and look for general solutions.
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So we have somehow we have to understand those configurations.
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Now, first we started for legal reflection. So Bomani, he had idea he had some conjecture in parliament.
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Well, he proposed casali conjecture.
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So he first posted this as a necessary condition in order to get the legality of this kind of form.
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Well, this is necessary. This information will touch those of the weight and the form, the two wave configurations.
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So we asked whether Huang's it's a possible you're given insulin shock given then can we find the sleuthing angles such that this is possible.
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It turns out this plot, this question actually can translate as Anglo problem.
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You can find a critical angle. Where is this angle with angle bigger than that?
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A critical angle is always the case. Okay. Less than impossible.
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Okay. So this is the first argument here. Now, when this angle the now you are usually this still to you now unique.
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But in order to get your neck in the liquid, this one to the strike is stable and this angle goes to pi over two with the other one.
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Then we choose to show this shock to the weak shock.
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In this podcast, supersonic means hot mess metal, hyperbolic leaching.
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Okay, now, when this angle becomes smaller, now this region becomes smaller, smaller than after.
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So the angle, this curve will beat that. So this cause the angle cause sonic angle.
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So the nightmare conjecture. Well,
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maybe they are there because it's a reflection configuration when we which angle bigger than that sonic angle leading because after
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that it was this will be after hopefully this would be elliptical then we'll ten he's saying we should change some structure.
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So last eight years I joined with the mission to fill the mag as we think about this problem.
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So we in two papers and the last papers we first the process issue this is we can really come out of
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this kind of configurations and which is stable you mean as Anglo's pi over two for larger angle keys.
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So in the wing months ago we just finished our research monograph and the way to solve that is conjecture for potential floor value.
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Listen to. Are now formally flashing companies to be much more competitive.
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So unlike a incomprehensible case, the competitive option, we still haven't understood the well enough.
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Okay. For the apple specially to ask what are the lightest basis of of vorticity allow this 3G and is it possible.
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Well, this vortex fruit is called dark, for example.
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So this awareness of passion with requires of further understanding with those kinds of things of possibility.
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And we can further understand those more complicated, complicated configurations.
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So I hope to show you some ideas for trying to solve those kind of multidimensional problem, require overcome some core difficulties.
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We we are also facing other mathematics especially are the part of differential equations,
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a mixed type composite equation and the fully bounded Antigone.
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And then we show the new understanding of complex of our texture and the vorticity and are
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bounded up and hammering inequalities are our values and I mean the numerical analysis that.
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The conservation law is also. Those glasses have many connections.
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For example, I was instead preaching, mostly preaching youth.
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Write it down. This is the actual students energy pencil. This is the Einstein pencil here.
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That equals four pi because it's a four dimensional case we're talking about.
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So those are the equations with the geodesic equation form the core of a mathematical formulation of general relativity.
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So if you like nicely achieved this equation, you can really form a system of pairing coupled, nonlinear, hyperbolic, elliptic, positive equation.
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So the analysis of X exact solutions of Einstein equations, a wing of the activity of the cosmological cosmology.
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So actually last maybe half century that this leads to the predicting our black holes are different.
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More of the evolution of the universe.
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For example, parallel hawking singularity clearly. Now, the beauty of this equation is that this is a type of conservation of energy and momentum.
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By design, just geometrical studies. You knew how to automatically inherently do this conservation of energy and the momentum.
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Okay. So this is really the beauty of this equation.
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I want to mention that another connection of the a little bit was the Texas variation.
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Okay. This is a field of mathematics that that deal with.
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It's to me, the function knows it's opposite to the ordinal calculus.
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We did weighted like for functions deal with a as you do a functional.
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Okay. Those are the function of the a lot applications of problem.
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Right that energy of actually functional things physics engineering industry.
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Although this turns metric function both the optics and the geometry uh, like in the geometry and also cost functioning optimum optimisation.
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So ideas we want to seek to minimise the are critical part of this.
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The functional right for greater progress has been made in the last listen to
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read the case as I should point Adam as you also fact the man who made a great
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contribution in this field especially John Paul he especially the judgement called
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not partial him of party convexity and his insights on the weak convergence method.
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And as you are responsible for listen at the major advances in this earlier.
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Okay. Now what are the connection with conservation law.
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Well actually you check the minimise are the critical point that includes after this the critical part as you said,
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the fastest system of all on the ecological creation opposite by all of the closing, which is the form as a conservation of form.
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Okay. But I like is a significant also seen in see as you find observe the following.
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If you give out a like ology, you give a domain.
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If like a large you don't have soothing symmetrical properties, then such a symmetric symmetry actually has a corresponding conservation loss.
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In other words, an imbalance of variational integrals into ago leads to outcomes a corresponding conservation of a critical point.
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Right. The part of which you find a systematic approach to generate all those conservation levels, of which very useful in many areas that.
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So also the PD.
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Yeah, I would say partial differential equations, uh, came from the, my macroscopic massive scorpion law and the principles right for them.
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A Connecticut Theorem The Bosma equation with the max, the wavelength and the last of a Poisson Landau equations,
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and then the quantum mechanics on the fuel cell in the tilak equations.
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SchrÃ¶dinger equations are the many thanks to function theory and the local conservation law for probabilities, so forth.
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Now, as I point out earlier, that in many cases it's simply chaotic and turbulent locally, but globally well-behaved.
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So should you think about something big, a system with buildings, particles?
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So it's very difficult to use those kind of macros, copula, to describe it individually.
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And put together using some system if you want to understand some behaviour.
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You mean the the well the faster the computation the the figure 100 years for that.
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Okay. So it's now. So how to how to deal with this one?
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This kind of situations. Why this? Well, we use we have a mathematical approach, okay.
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To do that. So ideally, you use the macroscopic scaling and the procedures flaws as a microscopic macroscopic process for larger particles.
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So our plots include the like averaging expectation closure process.
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Now you throw macroscopic scaling, you get a high dynamic limit.
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Now, how much innovation, the mass of the flow fields clean and so forth.
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So that's the idea. You help us understand the midfield upon your new laws and the principles.
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So no macroscopic level. And it turns out, again, a new approach of differential equations and models.
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They allow the application, use this idea to model this kind of idea, putting in essential mathematical modelling of various area.
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Now I come to the second source for part of differential equations which the as you fly mathematical lesson s and the principles.
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Included a compatibility condition, consistent condition,
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constraining critical point s whereas mathematical tools instrument and the generalisation completeness,
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beauty, chaos, imaginations so with just the example belong to the as if from compatibility condition and the constraints.
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So this is a classical problem. Isometric embedding problems which form a following.
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If we are given a curve, then we can describe a length on the curve.
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Okay, so this is called metrics.
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So in that term of definition,
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geometry is a fundamental forms that if we want to know curvature of those curves bearing the various curves only later with the,
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uh, we call each idea this in mathematical thinking, the fundamental forms.
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Okay, so the idea, each idea.
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So giving surface, of course we can come to judge idea.
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Right now the question for the other way along, we give a metric,
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maybe certain curvatures can we find a surfacing earlier water with this matrix and the corresponding curvature?
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Okay. So this is a really legalisation problem, right?
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You're giving the idea whether I can find a surface for that.
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Okay. Now, as we observe nature, as you give it a lot of very sophisticated geometry,
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as I show you those of the plant leaves, flowers, even things as the plotting for these.
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Right. So the question now, can we produce even more sophisticated therapies?
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Are things you can't even reproduce humanely? Right, by having studios as the nature companies structure.
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So this question of fundamentally differential in mathematics, very different for geometry and topology,
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and now it's essential to why off a sense path to understand evolution of a sophisticated ship of surface and things ship in it you include
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a large elasticity the mature size also in biology especially I want to mention as a U.S. Defence Advanced Research Projects Agency,
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the form the 20 silly challenged problem in science.
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So this problem actually formed in the 10th in the question.
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The statement following beauty are strong mathematical clearly as in chop peak and the digital embedding
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can give insight into plotting [INAUDIBLE] here where the possible okay so this off so that many
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mathematicians I study this is a problem started actually in 1873 I just released to all those contributors
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and in positive list I have people and I want the lack of single out of one column is the by Nash.
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Okay so Nash put all the following thing in.
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Early in the morning in early May. Manafort and Latakia are the key tempo camps seeking asymmetric and Badi into Ukraine's
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basing so his claims this being said to be bigger in capital each the larger enough.
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Okay so this leaves Syria. Some of you, if you have watched the Hollywood movie A Beautiful Man.
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He actually was the subject of that movie of that.
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Okay. So for his surname, as you relate how the cumulative column of original history makers is diminishing
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numbers smaller as you might be 89 by also as you further improve the dimension.
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Now for application, we really want to question whether we can legitimately any dimension,
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the lowest dimension but arestill in the space dimension much higher.
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So you can visualise this the surface in this high dimension space.
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Okay. So open problem now really whether we can reach the optimal dimension.
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The first question. The second question. Can we get lazy when we ask a matching painting?
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What do you mean, a little weak? Okay. See, when Bebe, for example, thinks that if if we move, give a comp example, Najibullah was a C,
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so this possible right now, if we will give example, actually the optimal dimension is to see total in impossible.
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You do a comp example if without stretching out by curvature.
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So this is actually the current very active research activity to try to understand this problem so difficult.
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Why is this problem so difficult?
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I want to point out somehow this involves not only a partial differential equation, often mix the hyperbolic types in mathematical terms.
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Okay. So here's a fundamental C, I mean, differential geometry that's exist as surface actually with the idea, which I mentioned earlier.
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Okay, J.J. Positive, definite answer matches. Okay.
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This is possible. You can construct the surface provided that the coefficients each year should satisfy costs for the AC system.
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Okay. So a gospel that is just a following.
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You give it idea. The second fundamental form each idea of by you through this aiming self by this two equation this
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actually compatibly condition from okay you want to find the existence of the surface of this system,
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then you need a compatible condition. This came from all quadratic equations.
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Then this is a constraint. By the cost equation you give cost curve it.
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You may want to find that your surface keep the cost curvature for that.
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Okay, it provides. This system is not only a passive differential equation of mixed type in mathematic.
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Okay, you have a body equation for that.
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So recently we asked we we thought, well, this is the feature of this system.
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So very like the equation, we observed the flow dynamics.
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So the question whether we can live form of this kind of equation into the flow dynamic formulation.
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So we found the following observation we can make have introduced artificially velocity u v and the density low IMU on this form.
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Now if we choose constitutively constitutively the Chaplin guess means pleasure and thanks to the from minus one over lo.
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Then that decreasing can be formed at the moment.
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Conservation of momentum. And the cost equation is really binary law binary releasing it like other basically like equipment,
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the conservation in energy for this case, those of it. So then we can park this later from this relation we can also define song speed.
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The action is really exactly like a flow dynamics in the transonic flow.
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If it's supersonic then relative to car hyperbolic equations subsonic elliptic equation which mix type.
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The the mixing slows is a gospel curvature where the gospel is positive.
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I'll go to the negative for that. Okay. So with this observation, actually,
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we found a solution that we found out one way to establish existing stability of isometric embedding through the weak convergence method.
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Now those conflicts tend to sum up curvature. It's very normal for that pole.
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You hear the very simple this is the geometry ended up with almost every day.
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Not so much. Okay. So one side of the court coverage of politics, the other side, the gospel, the negative.
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So this law changes side on the surface. So we cannot avoid padgett differential equation mixed by elliptic hyperbolic equations for that.
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Now for high demand in situation much more complicated. Okay.
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So besides the gossip equation, quadratic equations very with the compatibility condition for normal butter.
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So this is the cognitive equation which forms the bigger systems.
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Looks really awful by. I want to point out they have very beautiful geometric structure for their systems.
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So this thing that we observe when structure for that, we observe that it is this geometric start to enable us that give us actually get us
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to look that up with continuity and the stability of asymmetric bed for the problem.
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Okay. So now I come to the PDF with the mathematical tools and the instruments.
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Okay. So literally closely following many mathematical problems.
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Listen to example the Bengali conjecture by poor man.
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He began his work in 2002 and 2003.
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He's a pullover really based. I'll let you fly equation to study dynamical behaviour of solution of equation to solve this conjecture.
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The other example I don I sincerely enforce most for the emerging manifolds and the like is it's a very important and complex analysis surfaces.
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What you see in which the lead leader the lack of value in policy or implement largest union at the single index ceilings.
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There's a lot of those kind of mathematical look towards a ladder.
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If the really essential are those problems, you mean an adequate relation between algebra,
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geometry and the solid, solid concealing slow collective equations.
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And now for stochastic project, stochastic PD and also the scientific computing the as actual one of the main
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sources for project the for scientific motivation for time through computing.
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Now I've got P.D. also, as a mathematician, you really usually will say All I want to do generalisation.
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I'll make theatre more beautiful than completeness, right?
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I'll curiosity and imaginations. So those kinda ideas actually do actually mean, uh, lead me to generalise general or mathematical.
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C earlier and further development and the more wide applications are tend toward the for that.
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So I hope I have convinced you.
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The calculus there is clearly a mathematical element conveying concern with changes.
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Okay. The power differential equations of a mathematician foremost aim to describe a change.
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So behave yourself heavily.
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Material objects in nature with timescales ranging from picoseconds to millennia and with a lengthy scale ranging from subatomic to astronomical,
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can be maddening by nonlinear parts of differential equations equations with similar features, right?
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The law of partial differential equation have become increasingly significant within mathematics as a science since.
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So the mathematically you only have a partial differential and then it has a little history.
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And in listing the case this is related to somebody has to experience a vigilance clause.
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The research is a much more at a brisk pace.
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So if you want maybe a more kids as of tomorrow, reschedule another event.
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Five lectures by five distinguished mathematicians.
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The 915 in the morning for talk from them to our morning and the mathematical oracle by professor lawyer carefully from Texas Austin.
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So this event was organised by the Oxford Centre for Nonlinear Parts of Differential Equations, which open the October 1st, 2007.
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Uh, followed by the actual 2003 International Review of Mathematics in the UK,
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which indicated the critical attention is needed to both analysis a part of the future equation in the UK.
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So finding intellect is a professor. Sir John Paul.
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Thank. Now there are as you as the 11 faculty members in the centre, they are experts in partial differential equation as early as now.
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Also, PD is a part of the Mathematical Institute.
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There are four centres massive at the centre and the event 11 research groups.
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Okay, so this is the picture four new mathematical things that the new beautiful mathematics institute.
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And as all of us actually are looking forward to moving in to solving 30 sona of that with support of university staff of a university.
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Well, as a cable fellow, I would like to acknowledge the cables plan to establish at the Vantage Study Centre.
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Okay. Excellent side, which will hopefully be this building the other side on UMass, the Matthew Institute building, colossal Woodstock.
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Right. So for this that yeah, we're very excited about that.
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And of course, the more support is required to make this plan to ensure that they are additive for that from community that.
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So finally, I would like to thank the University of Oxford, set this post of a professorship for analysis of differential equations.
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There I find the Oxford is really unique, a place with a stimulating academic atmosphere.
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Firstly, the faculty and the excellent body of students.
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I feel very honoured, a great honour to be a part of the Oxford community and looking forward in the coming year to work with all of you
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and the Oxford community mobilised to make the UK community and the European Community for challenging scientific,
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socio educational issues, especially those issues where partial differential equations can play alone.
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Thank you very much. Please don't.