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>> Ok so this is about. The food doing genetic demography. Then metric demographics the term was introduced by Richard Gardiner It just means. The study of genetics but it's based on the. Sections and projections and. Tell you about 3 problems from this area and try to prove something that.

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Several main features of the poor approach to you know some of the 1st. Will be. Of the break of it into Google and direct this. So the problem comes from Isaac Newton a to the. So Newton. And his prince. In 60 and 87. Lemme 28. I wrote the following I will quote it this is a dunce lation from Latin.

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There is. There is no of a figure. Who has area. Of. Right lines. Pleasure. Can be universally found. By Nene's AFIC way sions. For a number of terms. And dimensions Ok so what does it mean nobody knows. And in particular it's not clear what is unknowable. Because this statement is definitely not true for the triangle.

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But time to Liz not the oval of this. This was from elated mathematically by our knowledge in 19 eighties. And let me read you the sun. Will be. I don't know claims that Newton series of Newton results to diminish so he didn't know the dimensions and good. Know there was no calculus.

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Or not and. Are not to show that he has been through and dimension to this corrective. Manus convex and seen its most Ok Otherwise it may be wrong so this is k.. Suppose K's. Contains the origin. And this is the direction unit vector so we take distance t. from the origin and take this area this is called the section function.

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So the distances to the vector is c So this is then minus one dimensional Well you move the section of k.. So this plane is. Going up. Perpendicular to see so. Moved by to see. The birds it's then to go over the section of Kate by the plane where the sky but the seas.

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Of the indicator function of Ok so use this for me. And. Here is the indicator function of 01. So this is the section function and the cutoff. See lots of c.. Is then to go from teeth infinity. C. a z. z.. So this is the volume here. And are not introduced the concept of to break.

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Into a good idea. So suppose k. is infinity. And convex. It is called. Could go if there exists. No no. In. Both. Q.. So the fly through the cordon it's of the vector. This numbered. And the v. C c. They get 0 identical. When t. belongs to the center will.

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Reach is. The support function of the direction of negative c. and the support function a in the direction of c.. So it's 0 support this function a k. doesn't have to be it's not defined. So in other words this is the cutoff and there's another break 5 action of coordinates and coming back there is to the statement that this is what he means.

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And he used this prove the position of the planets in the Norbit is not another great function of time. Ok so this is. Now are not stupid uplands. If. The dimension is even. Down there are no all to break of it. Into the budgets. Of the down Main is dimension is.

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Ball. But they clearly don't bow. It is a pole and I know. And there's an minus sign over too is an integer. So it is a full no no so it's sensible there's a full and I know the cutoff areas and everything is fine. And so it is also true it's.

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Because this concept is invariant up to linear transformations Ok if you transform it still pull in the no so if n. is are there on day I'll just break into Google what does so when I say I was a great calling to. Write in means the same at this most and convex Ok because of the definition.

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They are in the it's not dimension so it's not dimensions only I was a very clean to good book. And the 1st on NOTs problem was solved by by Syria in 2015 so if and is even there I know of the broken into Google does. And the 2nd problem is still open still open.

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However I got enough ski introduced their elected. Of only $9000000.00 to Google the mains. So the definition as Bill and I mean to go down maze of degree and because the proper did the parallel section fine action is the nominal fire it's fixed direction c. So the some key changes from 0 to and.

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Fissions The dependent on the direction heat to the k.. So this is a this is an arbitrary function it doesn't have to be counting it's not thing Ok so this is the kind of a little more general but. The kind it's narrow and the gun nuts. Reduce the special case the fine loads question to this one and this special cases where the demand doesn't have real single wire it's I'm not going to explain it basically this is a function it can be extended for every 60 extent that there's a realistic function of t. to the Co.

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So this was the question and the gun nuts proved something about them and I milled into Google. Demands 1st of all every side I may it's convex every said I mean it's convex and 2nd prove that the fan is even and there are no polling on milling to go with the mains so I don't even dimension there are no domains like this is arbitrary Egyptians Ok.

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So there are many problem what happens and not dimensions was sold and no paper. Was that Xander. D.s.. And it is true that the only building on the way into Google demands. It's an old and dimensions. And I will prove. When I touch this something. It's my fault.

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There are going symmetric cases kind of easy. But it will do for in the non-symmetric case also I will say a little. Cable bill Oregon symmetric and here's the farm. Earl ating the derivative So the parallel section function visit to transform so. K. there is the corresponding norm.

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Which I do note by it's. Not my. So now we need several things to use the distributions we use the short space so the 2 are. Space for actively decreasing infinity differentiable functions distributions continuous linear functionals. On this if f. is a locally into Google function in Iran it Sun functions by integration locally into Google and not increasing too fast for the not really increasing.

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Its end to go low bar and perfect. Yet. If 5 belongs to the space of functions so is that for a transfer this is known so the Fourier transform of a distribution. That is defined as a function no. This is the definition given the distribution after its forward transform merits and functions like this so also we need to know.

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That. The norm to be. P. is greater than mine and. Is always local into Google. Ok. And it's not doing as degree p. So it's for a transform. Sun that function by. A like the norm itself but that which is by integration and the transformer for my genius distribution is also come my genius the degree.

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For u. turns from a semi genius of a degree minus an minus b. and the last thing the thing it to know is that if you look at become a genius functions on this year it's after dimes to the extent that the function f. is on this here and the extent of my genius function of degree.

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And it continued to this most in this year than before it transform is the g. 8 times. The minus and minus b.. It's also come my genius function and G.'s also infinitesimal I will let you to prove this. To later but I will not mention it so so actually we can say that G.'s I find this here.

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Extended to the quote I ran as a comic genius function of degree and I understand but so I will write that of c. meaning j a c and the degree of homogeneity will be clear Ok so in principle this is the illegal to write the value of a distribution at the point but this particular distribution is a function the sphere extended to the color and.

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And. The point is that there is a formula. This is. Not very dark. I can see myself. Suppose gays infinities. In Iran and am is an even integer and not Dick will and minus one. And then. The battle section find Sion has a do both during the orders that do are gion which are equal to negative one to the power I'm over to buy and mine the same minus one and this is on the reason why and there's nothing 10 minus one and the 4 you transform of the norm of.

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The bar minus sent by Sam plus one even non-negative integer so starting from 0. You poor transform. The point. So. Case infinity this move which means that this isn't this most function of the sphere so the extent of my genius function of degree minus sent but some plus one it's for a transform is another.

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And seen it this time with a function which is written here and extended to the command genius function of degree minus and mine is this whatever it is Ok which is mine is someone's one. And. This far and away something that I will use the 2nd day for a completely different problem and the 3rd day for just another problem.

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And it was introduced as the gardener and Schoenberg. And was used to solve the Blues in one piece to pull. So let me 1st show cause if this works for. The pub and I'm really into the overall problem. And done I will try to prove it to. I 30 minutes it's very questionable.

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Ok. So if I finish it 630. Ok I'll not. So. If not mailing to Google convex boy day then the derivative of this function of the order in which is greater than than is 0 for every direction. So a cake c. and am is even. 0 is 0 if m. is an even number greater than and so it's starting from some point all even.

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Your even 0 in the close here and this immediately so the problem. Since I'm using this formula and this formulas for symmetric under a good. This solves all of the symmetric case Ok I will say a little later would you need so non-symmetric case so what does this mean that means the transform is 0 in the sphere.

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And this is a genius distribution so it is 0 everywhere except for the. Origin maybe this for a transform I supported in the origin. Everywhere else a to z. because it's a comedy in this function this is an interesting with function because ks infinity it's most so every distribution supported in their origin is a fine outside of the derivative of the delta function so it's transform is appalling.

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So the 40 transform of the Fourier transform of the original function. This is a Bonanno. Genius Bonanno of degree minus sent by Sam plus one now if an is even Amazo So even plus one this is but so this is a Bonanno of or degree so it cannot be positive.

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Normally is positive for a fan is even it's impossible this is the 1st part. So one is and is even minus sent but one is odd and p. changes sign. So that the different so this is kind of feeds you what if and is. So I will keep this them I will raise this wonderful thing so sorry.

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If you want to read this. Many papers about Newton's strain Selema. Different that pinions why it's wrong why it's correct that lived it started the critique of it's. So. Odd. Minus. But Sam plus one is even. And we need only took on 2nd. So say that this is Ok ks an even number so the norm to the k. is a bone lot of male.

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And the norm to the plus to his opponent will. For the same reason that supports it in the original Now let's take any 2 dimensional subspace of fire and age so we can say that this is the norm a few v. and p. is that Bowen on the a few v. coordinates an h. there is stiction a football and I know there's a.

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Because linger equations substituting your equations and. With us do this Q a few v.. And this is a come out genius pulling on no this bald spot and I was a genius this is my genius so we can say that this is the norm a few over v one and you are V's and you variable which I also call you just to confuse.

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The norm of you one of the k b a few. And the norm of. You one of the. 2. You and this is my genius but and I know of degree k. and this is my genius but a non they'll love degree Kate Plus do so suppose. You are not is a 0 of be.

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Ok 1st of all there's 0 cannot be real because if it's 30 Then the norm is something one is 0 Ok so the zeros are to complex complex numbers. And suppose you know that this is 0 with a multiplicity of. That and the same you know it is 0 off q. because p. to the power of k. plus 2 equals Q To the power k. this is 0 this is also 0 Ok so let's suppose it's multiplicity Alpha for q. and a for p. and smoke lifted to but better than the multiplicity is Alpha took a plus to.

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Be in b. to to k. Yeah so. Over to. The multiplicity here you to the power to do is you to the not the. So and k. is even so. Is their kid a or k. over 2 It cannot be because and. It cannot be 3 k. 4 k. and so on but cannot be k. because then it's the only root of p..

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There are lots of complex numbers so there are at least 2 of them the root itself and the conjugate So the multiplicity scare were 2 which means that you have. The Bonanno is you minus you not to the k. were 2. Dimes you minus you're not going to get the care we're due and this is a few the k. so b. is.

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Another good it's a quadratic norm to you and you can't you get Ok Which means that every 2 dimensional section of idea is another. And then there is a wonderful theorem of Jordan unfun Newman saying that if something satisfies the pilot ground war then this is another never to mention Ok so since it's another good in every 2 dimensional section because of the parallel Graham Law So it's another certain every dimension and this finishes.

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Now if. The body is not Arjun Singh I think there is a different formula so the different formula was. To vary Borgen. Yes can. But unfortunately it doesn't give such a clear. Case so it's becomes a serious soldier brag of them and. He is my squad school friend.

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He's an alter braced a very good deal to raise so and I understood it up to 5 of those in the sky. And the sun. So this solves a non-symmetric case and now and there are many 2 hours. I will try to prove this formula. You know when I started there's no stopping point.

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Ok. So I have to live going to. The same formula but for fractional do it it's. So that the fractional derivative of a function. F. is in think this moved the origin so here the real part of q. is going to the negative one and Q.'s not pick will.

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And so on again. So the fractional David defaces line over the gun the function of minus q. and that action of. The plus the minus one minus q.. And this is the fraction of the q. So when a negative one and 0 keep loss is equal to t. to minus 5 mind one piece by the way Ok so it's just then to go.

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Because this function becomes locally into Google but between negative 0 and one. And then you do the regular ization So Def of 0 take them to go from 0 to $1.00 add whatever you just subtract the 10th add then to go from one to infinity computed this and this new pharma level works for.

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The real part the feud between 0 and one if you continue with the Taylor series of the function f. care it extends further Ok so I'm not going to do it I just say that this is an entire function. Entire function of Q So and a c. Q a 0 considered as a function of c. apply to function field c. is.

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An entire function. Because this is the fractional derivative of something that is infinite just moves around 0 and 0 so if you have. The norm don't mind a cent but you plus one cat applied to the this is the integral This is looking to go because it is great for the negative one.

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So this is then to go above the norm apply. And this isn't only to find of q. also because they're just different shapes you get a log of the norm right and it's got convergent still so to prove that this is equal to that which is a formula with fish and it's enough to prove it for q. between negative one and 0 and then you cast on the way to functions there you quote on the whole life and continuation.

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And how it really does the will to do it in 6 minutes they did 1010. So all they have to do is. This formula q. between negative one and 0. So the fractional derivative is then to go. Over the call line because the function is even if it's not on the tip blast it one cough it's one cough.

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So it's why nowhere to gone off minus Q To make things easier nor is the Constance Ok because I will lose something in your way. The noise the constant the constant the. So. That. The minus one minus q. times the function itself and this form of I write to trim negative infinity infinity.

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Now this stuff to devalue. The function a in terms of then to go like c. was easy. Indicator function of the norm of k. d.c. So this is then to go over k. intersected with this plane of the function one so that. This is then true goal of r n x c The power of minus one minus q. the indicator function normal x. Ok Because if we call this.

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The z. z. we get then to go over there. Of z. to minus one minus Q And then to go of this over points for the cover for the disease and I will write it down polar coordinates axes. So that see goes outside then to go from 0.

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I think you need to arc to. The Sun 1st of all because we do polar coordinates and then to minus y. my new skew comes out of here so our turn minus q. minus one indicator function of. And this is just to go from 0 to $1.00 over the norm of k. So this is then to go from 0 to the normal that the cape the negative one because this is this must do as the new one so r. is less than what I know are the norm.

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Of are to the power and minus q. minus one do you are which is. That. The minus one minute skew and and minus q. minus and minus one and minus 2 soaring. Minus survive and minus one minus Q So it's down minus 2 minus one if like to this so it's normal that k. the minus cent plus q. plus one.

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That divided by and mind is still minus one so I but I don't know constants that this is where and and minus minus one comes from Ok so I can stop here. Because it's not the 15. Core physics where we. Whine. No I meant the middle of the field.

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I just got the formula for this now I have to reply to function fi get some expression and then prove that the norm that applied to the function figure of the same expression. I will started this formula. And I will write this for Milligan and I will write this from that way again so you will not lose anything.

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But. It's complex and those. Know. This is the. And by the way. For probably not going to Google what is the condition of infinity and so on can be removed but in the in the general scheme of this are not problems infinity smooth What is really important so so it was removed by a lot of us can using spirit to her mind.

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And also the body doesn't have to be convex it can be a star board to. Me. Yes you can square this function. A function Square this is a pull and I know that this is just impossible to solve. The square. Ok good luck Ok. Right.