Okay, good afternoon. So if you remember so today we will continue our study of piecewise affine maps Our main theorem is that F piecewise affine full-branch map then the big measure is invariant Okay, so did you do the exercise to prove that the big measure is invariant? Yes, so we proved last time I left it as an exercise Okay, to show that the big measure is invariant It's quite simple exercise today. We're going to show that it's a garlic Which would not be difficult, but it's quite subtle and it's an interesting argument So first of all we're going to say something about the combinatorics of piecewise affine maps So remember that piecewise affine maps piecewise affine full-branch maps Have the structure where they have a certain number of branches and the key property is that each branch Maps to the whole interval in a fine way. It can be finite number or an accountable number of branches So the lemma let F a C0 Means continuous or respectively C1 C2 affine piecewise Sorry full-branch map B a piecewise full-branch map then there exists and exists Partitions Pn mod 0 a family of partitions. So for each end Pn is a partition of I such that P1 is equal to the original partition. So P1 is this partition here and for all and Greater than equal to one For all partition element omega n in Pn We have the map f n From the interior restricted to the interior Okay, Fn From the interior of omega n to the interior of I is a homeomorphism respectively C1 or C2 or affine Defil so what is this saying? This is just saying something that actually is very simple. It takes a little while to say it precisely but the P1 partition here is a partition into intervals of Into subintervals of the interval such that after one iterate f1 each interval is mapped on to the whole original And then What does that mean? We've looked at the structure already when we studied the topological structure here if you look at this interval here Each one of these intervals then because after one it that it maps to the whole thing Inside here. You can there is a Subdivision of this interval which maps exactly to the subdivision of the entire interval, right? So there is a small interval here that maps to the first one There's another small one that maps to the second one third one that maps to the third one and so on So there's a small copy because what this map does is it maps this small interval in an affine way to the whole Interval so inside the small interval There's exactly a small copy of the big partition here in such a way that each element of this Partition P2 is mapped after one iteration to an element of the partition P1 And therefore after two iterations to the whole interval like this Okay, and then you can continue you can repeat the process so that at each level Pn Corresponds to pulling back this partition n times So each element of the partition maps to an element of the previous partition Pn minus one and then After any that's exactly on to the whole thing in a bijection as a bijective way Well, in fact as a homeomorphism or C1 diffeomorphism or affine Okay, so let me write just very briefly the proof We can easily write the proof by induction So by induction for n equals one we have that This is true by assumption true by definition of f Okay, by definition of piecewise full branch For n equals one and now suppose it's true for n minus one. So let Omega n minus one Belong to Pn minus one. So there's a by induction. We suppose there's a partition Pn minus one we take one element and we have this property here that f n minus one from the interior of Omega n minus one to the interior of I is Let's say homeomorphism Is well, okay, whatever it is, right is Is Okay, let's say homeomorphism or in our case affine if you want it to be affine, okay And so what does this mean? All right What this says is that you have a map from omega n minus one to all of I this is f n Minus one and inside I we have our regional partition P Okay, so because these maps are finally or at least homomorphically to I that means that we can Subdivide this element in a way that is the images match up exactly with this partition That's what I mean when I say that we pull back this partition Okay, so we have a subdivision here Okay, that maps exactly onto this partition element and so what do you know about each one of these? partitions each one of these elements, so Subdivide omega n minus one Into subintervals into subintervals which we can call omega n Okay, so that for each Omega n subset of n minus one Fn minus one of omega n so fn minus one Maps the interval the interior of omega n to Some omega in P the interior of some omega in P All again I always like the interior just because I've not specified if these intervals are closed or open or half open And so on into the interval of some omega for some in P Yes, this is this we've already looked at that in the previous course. We're just taking small ones Well, we're defining the partition using the dynamics Okay, I'm just doing it again partly for completeness and also to emphasize the member that we Here as opposed to what we did in the previous course we we can allow countable Partitions here. This is some omega for the original partition some element of the original partition I because this is the original partition P here We have omega n minus one maps to all of I so we split omega n minus one such that each element maps to an element of the original partition This original partition therefore, what do we know About fn Where does it map omega n? To to the whole original interval right because after n minus one iterates It's maps to one of the original partition elements and by assumption the original partition elements get mapped in one iterate to the whole thing Okay, so this maps to the interior of I and of course the regularity is always the same right because we assume that this is a homomorphism Okay, and therefore when you restricted the restriction of fn minus one to a sub interval is also homomorphism on to this and Then from this you apply f once more but f from this interior of this to the whole thing is a homomorphism So the composition is a homomorphism So this is a homomorphism if we're looking at homomorphism or it's affine a composition of affine maps is affine composition of c1 maps is c1 and so on Okay, so we get the right regularity So this very nice very clean combinatorial structure is the key to the full branch to the full branch maps Right, this is a very nice. So all the iterates are themselves full branch maps. This is what we're saying If f is a piecewise affine full branch map Then all the iterates are also piecewise affine full branch maps on some finer and finer partition because Each one has an iterated maps each small interval to the whole to everything. Okay, so that is why One of the main reasons why full branch maps have such an easy structure to study Okay, so we're going to use this now to prove a goodicity. This is going to be a key property So first me let me prove one more small lemma. So f i to i piecewise affine piecewise Affine Full branch. Okay then supremum then Supremum overall omega n in Pn Of the size of the length of omega n converges to zero as n tends to infinity And why is this? Yes, but remember that the intervals decreasing is not enough to show that they decrease to zero right The intervals you can have a sequence of nested intervals decreasing but not to zero Remember we use some additional property. What do we how did we do this calculation before? The derivative is strictly bigger than one and so by the mean value theorem This is less than equal to lambda to the minus n Okay, and how do we know the derivative is bigger than one?  That's right Exactly, okay as long as this partition is not trivial So there's at least two elements in this partition Then both elements are smaller than the whole interval and they get mapped in an affine way to the full interval So you must have the derivative bigger than one. Okay, the more partition elements the bigger the derivative so proof Since f is is Peace wise Affine there exists lambda greater than one Such that derivative f prime of x is that an equal to lambda for all x in omega for all omega in p right and so The size of omega n is less than equal to 1 over lambda n for all n greater than 1 and for all omega in P and Okay, we have uniform bounds on how they shrink the Lebesgue measure. You don't like this Yes, thank you absolute value of the derivative Okay, so now we're ready to prove a good this city So how do we prove a good this team? Remember? What do we need to prove?  Algorithm what do we need to prove? Zero one. So how do we prove that it's zero or one? Well, there's various ways in various situations that one standard ways the following we need to prove is a one So let's assume that it's got positive measure and Prove that it has that the measure one And that's one way to prove that I did zero one is to show that if you assume the measure is non-zero Then it has to be one Okay, so we'll assume that we have that the Lebesgue measure of A is greater than zero and now we will prove that the Lebesgue measure is equal to one Okay, so Here we have our interval Zero one Here we have our partition P and all the refinements of our partition So We assume that a has positive measure. So have you heard of Lebesgue density theorem? So Lebesgue density theorem Theorem is very interesting and quite subtle and we will not prove it here because it's a Theorem from measure theory. We cannot prove it here, but it says the following if you have a a set of measure of positive measure right then For so if measure of a is positive then For m almost every x in a We have the following property that if you take Interval x minus epsilon x plus epsilon and you look at the measure of a Inside this interval So you don't know where this set is. Okay, but you take a set a point x that belongs to a Right, and then you take a small interval here Okay, and you look at the amount of a that's in see inside this interval and you divide By the size of this interval, which is 2 epsilon this converges to 1 as epsilon goes to 0 So one way to say this is that almost every point is a Lebesgue density point of the set a Now what does it mean and why is this a little bit surprising? What it says is that if you Look at a very small region around x Then 99% of that region belongs to a Okay, if the set a is an interval, this is obvious Okay, if the set a is not an interval This is not at all obvious if it's for example a Kantor set of positive measure or somehow a more complicated set because Suppose I take a set a very strange set and I say the measure of the set is one half For example Then why should it not be that if I look at a very small scale? I always get that the measure of of that Interval of a compared to the complement is also half and half at every scale What this says is that this is not true The measure of a might be a half at a certain scale But if you focus on to a specific point then either the measure of a the more you focus the measure of a either Goes to zero or goes to one for almost every point It goes to one and for the complement for the other points It goes to zero the density of the set a Okay, so this is one of the basic theorem of measure set theory, which I acknowledge is I appreciate is not Composers really subtle actually it's not at all intuitive or obvious but we will use this in an essential way here because That's what we have here. Okay here. We have a set of positive measure and We have the Refinements of the partition So if we take almost every point with respect to Lebesgue is always in the interior of these refinements So by taking a deep enough refinement, it's like taking a sufficiently small neighborhood of the point x right so using the Lebesgue density theorem so For m almost every x in a There exists and and for all epsilon greater than zero there exists n sufficiently large such and Omega n in pn because n sufficiently large means that omega n is sufficiently small Okay, so that the Lebesgue measure of omega n intersection a over the Lebesgue measure of Omega n is greater than or equal to F So what we're doing is we're using these partitions to zoom in to very small scale And when you zoom into very small scale most of the measure Sorry, what I meant here actually is one minus epsilon what I want to write here is One minus epsilon because I wanted to be totally close to one So I'm saying that at a very small scale I can I Can focus in and I can get an interval where most of the measure of this interval belongs to a or the qu and equivalently in particular this means that the measure of Omega n Intersection the complement of a Over the measure of omega n is less than equal to epsilon Okay, this is just a complement Now, what do I know about this little omega n? So remember that I know that omega n maps to I under fn and This map is a fine Right, so here. I know that I have the complement of a inside here Okay, and so what I have is because fn Fn from omega n to I is an affine Byjection Then the measure is preserved and I have that omega n Intersection a complement over sorry Sometimes I use that and measure of omega n is precisely equal to the image of both So the measure of the image of fn of Omega n Intersection a over Measure of fn This section Compliment Okay, this of course is just the same as the measure of fn Omega n intersection a complement Over the measure of I which is equal to one and this Is by assumption less than equal to epsilon? Okay, so this is equal to one so this is less than equal to epsilon and what is this so So they call that f minus One of a equals a by assumption What's the question here? exactly here Exactly, that's what we're going to do So the reason we can do that is because f minus a equals to a so f minus one of a c equals a c Because when a set is backward invariance, so it's this complement and so f minus n of a c equals a c Okay, and so We have that Fn of Omega n intersection a c Is equal to fn of Omega n Intersection f minus n of a c and now I apply fn to both sides That fn of Omega n is the whole interval Fn of f minus n of a c is just a c Okay, so this is just equal to a c if you think about it. It's quite obvious I so the fact that f minus a c is invariant under the map means that you have a c here Okay, and some sub in this is some sub interval of this But because these maps are finely to this the part of a c that is here must map exactly to all of a c here Because this interval maps exactly to all of i so the part of this interval that belongs to a c must map to a c because a c maps to a c a maps to a Okay, so it's just that the fact that this is invariant Okay, so if you take the image of the part of a c that belongs to Omega n You apply you iterate it n times This is this maps to all of i and a c simply maps to a c and so from this Okay, and so We get that measure of a c is less than epsilon f of what? Yes You're right. It's f of minus n of a c intersected Omega n well so f Minus n of fn of a Is not this is not equal to a but if you take the pre image of a c and then you apply n You get the set itself in one direction. It works in the other direction. It doesn't Yes Yes Yes What's it? What is the difficulty here here? This equality here This equality here this one Because fn is affine fn is affine so and then affine map all the Length is preserved on an affine map the ratios of lengths are preserved Right because if this affine is stretching by a factor two hundred all the measures are stretched by a factor two hundred So the length of this is stretched is increased by a factor two hundred But the length of this is also increased by like factor two hundred So it is the ratio that is preserved It's not the numerator and the denominator. Of course, they're not preserved It's only the ratio that is preserved because f is affine And in fact, this is a key point that we will come back to because we want to generalize this theorem with removing the affine Assumption at the moment. We're using the affine assumption in a very important way So the affine means that the ratios of measures are preserved So here we have this set here This is a c intersection omega n and we have omega n Okay, now we map all of this in an affine way to this So the image of this set in here the ratio between the size of the image and the of AC of This set and this set are preserved by the affine map and here This is nothing like this is just the fact that fn of omega n is equal to the whole thing which is equal to one In this case, okay, so back to this step here We have We look at the image of this set Okay, which we which is this because AC is equal to f minus n of AC and then What is disturbing you here, you know, maybe maybe it's easier if you if we look this Let's write. Let's write this the set theoretic definition of this. Okay, this is the set of all points z such that z is in Wn and fn of Z belongs to AC This is the definition of this right so Z is in Wn and fn of Z Is an AC so when you apply fn you get AC So you're saying you need to it's not clear that this gives all of AC maybe But then this depends on this Omega n is fixed, but this is true for any omega n of course So f minus n of a of c of course has many components inside each omega n There is a small copy of AC. So this is the way to think about it. Okay is that here You have your interval I like you say here You have all these all your omega n's right and then you have your globally you have your AC Okay, which is in this set and if you take the pre-image of AC this pre-image has some compose some piece inside each AC and So it's clear that this AC will map when you map this Bijectively onto this this will map to AC because this is exactly the pre-image of AC on the left hand Okay, so Maybe it's possible that you know formally They might be necessary. Maybe that's not exactly the right way to write it when you think of the set the retic definition But the meaning is clear and the fact that it's true is clear so Because these are exactly the pre-images of AC inside each element and you map them back to AC So you're using a crucial way this property here and the conclusion is that this less than epsilon and the conclusion is that we chose epsilon arbitrary, of course Dependant we have that the n and the omega n will depend on epsilon Okay, but the bound on AC does not depend on n or omega n so for arbitrarily small epsilon We get this so this implies that the measure of AC Equals zero and this implies that the measure Equals one so we use two ingredients in the proof and it's worth spending a couple of minutes thinking about them because we will generalize this proof in the next lecture So what are the two ingredients in this very last part we used this fact to To allow us just to make this last Statement right to show that you we're estimating AC and AC has a small component inside Omega n and each of these maps to all of AC But besides that that is just part of the way we've chosen The set a and the set ac. What are the two ingredients that we used? so first of all We use this statement here. This is one ingredient Yes Sorry Exactly This is one ingredient and this is the second ingredient everything else is just you know everything else is basically just Understanding the sets and mapping and these are the really the two ingredients and what I mean by the two ingredients Is the two places in which we use the assumptions on the map? What assumption on the map? Are we using here? What property of the map are we using here? All we're using is a very weak property We're just using the fact that these will be arbitrarily small We apply the Lebesgue density theorem and to apply the Lebesgue density theorem. We need to say that for every epsilon To get this we need to say that we're in a very small neighborhood of the point x we're in a very small interval Okay, so the ingredient we use here is the fact that the size of this is going to zero okay, and In the previous lemma we showed that if F is piecewise a fine then these are going to zero But as you know these go to zero in the much more general conditions like what are the conditions? That imply that omega n goes to zero just if the derivative is bigger than one But we don't need piecewise a fine so piecewise a fine is a very strong Tool just to prove that these go to zero and so we get this Okay, so I'm we're not going to do it now But I'm just mentioning it for the future for the next lecture that it is very easy to get this part of the argument Without requiring F to be piecewise a fine all we need is to be expanding Right there to be the one and then we immediately get this part So this is very easy we can generalize we can say it's full branch C1 piecewise expanding this is more delicate This equality we certainly needs piecewise a fine if it's not piecewise a fine the ratio will not be Constant right because if it's not piecewise a fine it means even if it's expanding It might be expanding a little bit more here a little bit less here So this ratio between the image of AC and the image of the whole thing does not necessarily stay constant Okay So how do we solve this? How could we resolve this is that does the whole proof break down? Or is there something we can do if it's not piecewise a fine? It's not LeBeg invariant Exactly, that's what I'm saying. There's no way we can get this equality if it's piecewise a fine if it's not piecewise a fine We will not get the equality So the question is do we need the equality? Exactly all we need is an inequality there. We don't need an equality Piecewise a fine gives us a quality, but piecewise a fine is very strong. What do we need? We really need here just something like this and Everything else will work right because what we really want to show is that this is less than equal to epsilon Which is what gives us this? So we don't need an equality here Okay, in fact, we don't even need this we need some constant we can add some constant here Okay, as long as this constant is kind of uniform and doesn't depend on n and this we will This will be one of our main theorems in the course will be to Precisely to prove this theorem for much more general C1 Maps that are uniformly expanding and we will show this is automatic and we will study Exactly this point here will be a key point. This is called bounded distortion And it says that the map is not our fine, but is close to our fine sufficiently The distortion distortion means how different the derivative is in one place and the other is not so bad And so we still get this inequality. Okay, so this is We will come back to this I just wanted to mention this So, okay, this proves that good this city so before we go on to this generalization Again, I want to give a little application to number theory But maybe this is a very good place just to take a couple of minutes break. So let's do that Okay, so there's a very nice application of this ergodicity to number theory So do you know what the base? K expansion of a real number is remember, yes, so Let X St. Zero one K greater than equal to two integer Okay, then the base K expansion of X X is Given by X Equals X1 Over K That's X2 Okay, squared Cube okay, well X I belongs to zero K minus one. So certainly, you know when K equals 10 This is just the decimal expansion of the number Decimal expansion. So sometimes we can just write X equals Say zero point X1 X2 X3 X4 Okay base K This is just the sequence That gives the expansion of the number. Okay, so it's the usual decimal expansion, for example in the case of Base 10 and another expansion as you know, this expansion is essentially unique except for some Very special points, right? So in the decimal expansion, you know these points where you finish with 0 0 9 9 And you finish or in the other case, it's very similar the same special points So where have we encountered in the previous course? We already encountered something similar remember In the context of piecewise are fine full branch maps exactly when we did the symbolic coding right, so there's a dynamical way in which we can Study these base K expansions and I will come back to that in a second Fine So but first of all without any dynamics just from this point of view we say that so definition X is Called normal in base K if X equals Zero point X1 X2 X3 base K and and One over N of the number I from 1 to n such that X I equals J limit and tends to infinity equals 1 over K for all J So what are the normal numbers in base 10? Let's think about base 10 a second decimal Expansion, what does this definition mean? What is a? What is the definition say in the case of base 10? What are you measuring here the frequency of the digits exactly a Sympathetically in the limit the frequency of each digit should be 1 over 10 in base 10 Then the number is normal. Do they exist normal numbers? That's right. You can easily construct a normal number like that. Do they exist not normal numbers? Like what? Exactly You just have one digit if we for example you have any decimal expansion that never contains the number five or that only Contains a number zero. There's lots of non normal numbers. Okay, so what do you think is most? likely For the number to be normal or to be not normal to be abnormal you think so Anyone else thinks so? You know mathematicians will call normal. They use the word normal to you don't know if it means Normal should be normal or abnormal What the opposite should be true the opposite what there's more abnormal numbers You think it's very it's a very strong condition things a very strong condition only very few numbers Will satisfy this you think the normal numbers are very rare Why is that? Because you mean you're you're saying well if you just take a sequence of digits This frequency could be anything so what are the chances that it would be exactly one of a K? Very few chances is this your argument Anybody else have the same argument? Yes. Yes, what one yes? Yes, yes So right, so if we we use the dynamics the map Kx mod 1 and You're saying that this should allow us to show that somehow something from this converges For many numbers or for few numbers or for all numbers for many numbers Okay, we're going to see this in a second. Okay, you're along very good lines Okay, let's see in a second. Let me first There will be a dynamic we will discuss this from a dynamical system point of view. This is the interesting thing So I'm formulating the question just from the point of view of numbers and number theory But indeed the answer will come from dynamics Theory dynamics first of all, let me just ask one more question just to to to ask you the questions if a number is Normal in base 10. Do you think that it would be normal in base 9 or in some other base? You think it would be normal in base 2 What if it's normal in base 10 then it's normal in base 2 So what we're going to prove if it's the following theorem there exists a set N N in 0 1 with the Lebesgue measure of N equals 1 such that X in N implies X is normal in every base K Although I as you shall see even though the it's very surprising that N does not depend on K That's actually the easy part of the proof We'll see okay in a second. So let me remark. However, that as far as I know Nobody knows any such number So I don't think that anybody knows one number That I can tell you this number is normal in every base So this is an abstract theorem and Probably any number you choose will not belong to this set any explicit number you choose choose the number one half or Any rational number or choose the number pi or whatever number probably it will not belong to this And I don't think anyone has tried recently I think I saw some Paper in which someone was discussing this time to find some construction to find some number, but it's not It's not known. Okay, however, I think it's good that we had a little discussion about it because indeed This is quite a strong property this convergence here Okay, so let's see how we will prove it using the dynamical tools that we have so far. So first of all as you remarked It's remarkable that this set is independent of K, but Remember that if so it is enough if we can show So it is enough It is sufficient Show that for all K greater than equal to 2 there exists and K with measure of Nk equals 1 Okay, such that X in Nk Implies X Normal in base K. So why is it enough to show this? It's enough to show that for every K almost every point is Normally in base K We don't need Nk to be close because if you take the intersection of all these Nk's All we need to show that this intersection still has measure 1 and how do we know that the intersection has measure 1? No, no, no, no, you don't need no This is you don't need any of that with measure is much more simple the complement has measure 0 and The complement of the intersection will be the union of all the complements and the union of a complements of sets of measure 0 has measure 0 Okay, so we write that down indeed the measure Lebesgue measure of the interval minus N the of the interval Minus the union sorry interval minus N so let Let N N equals the union Sorry the intersection Of all the Nk K equals 2 to infinity so this is the set that we're looking for we want to show that N has measure 1 So I minus N is just equal to the Lebesgue measure of the union So the measure of the complement is just the measure of the union From K equals 2 to infinity of the complement of Nk I'd take the unions of all the complements and then you have the complements of the intersection And this is just less than equal to the sum K equals 2 to infinity of the measure of I minus Nk and this is just equal to 0 Okay, so this is the easy part so we just need to choose 1 Nk Okay, so let K greater than equal to 2 let F Zero one to zero one the F of X equals K X Not one Okay, like you said and then If you remember What is the structure here? So this is the case K equals 3 For example, if you remember we have three intervals I 0 I 1 I 2 So let I j equals the interval J over K J plus 1 Over K are these intervals here? Then you take some point here X and let X equals 0 X 1 X 2 X 3 and so on is the base K expansion of X Then where is this point? Where does this point? Exactly exactly so X belongs to I j if and only if X 1 equals J and F I of X Belongs to I j if and only if So f of X Belongs to I j or a different I j we can call it. Okay, it's still I j for any J So what is the property f of X belongs to I j if and only if X? I plus 1 equals J, which means That the frequency of digits of J in this expansion is Exactly the frequency of how many times the orbit of X belongs to I j Every time the orbit of X belongs to I j so you choose for example to I To every time the orbit of X belongs here It corresponds to the fact that there's a 2 in the base 3 expansion So the limit that we're looking for the asymptotic frequency of J inside here is Exactly in the limit the asymptotic frequency of the visits to I j and what is this limit? What is the frequency of the visits to this interval I j one of a K? Why is it one of a K? because Lebesgue measures invariant and Ergodic and by Birkhoff's ergodic theorem the frequency of the visit by the corollary remember to be closer got it there The frequency of visit converges exactly to the measure of that set for almost every point exactly So there is this is true so the frequency of J's will converge to one of a K for almost every point Well, it beg almost every point which is exactly our statement, which is what we want to prove. Okay, so Frequency of the digit J equals Frequency of visits Frequency of visits of the orbit of X to I j and this converges to one of a K for almost every X and all J and this completes the proof So there's a lot of Connections this is a result in number theory, but the proof is ergodic theory And in fact, there's quite a lot of even recently. There's been a lot of research in Ergodic theory methods in number theory to prove certain things in number and this is just one example Okay, so next time we are going to Remove the piecewise affine assumption and we're going to study full branch maps Without the piecewise affine assumptions and we're going to study the ergodicity of Lebesgue measure in those cases Okay. Thank you


I'm an explorer. I'm exploring. Okay. Yeah. I think it's too much responsibility to be called an innovator. We are here at the LinkedIn studios in the Empire State Building, and I'm here with Marie Goulammeurle, and I'm going to say it my way, but you're going to have to tell me how it really is saying. It's fine, whatever you like, Nadia. I've been here five years and with so many pronunciations. As long as people can understand my French accent, I'm fine. I can understand it perfectly, and I love it. So Marie is not only the CMO of Calvin Klein, but now has taken on another role at the parent company. You're also the chief digital officer of both Calvin Klein and Tommy Hilfiger. Yes, I am. So Marie, what's top of mind for Calvin Klein and Tommy Hilfiger right now? There is one which is hyper personalization and what it means to unify personalization from product to loyalty to memberships to your online experience, your story experience, and it sounds quite easy when I'm saying it like this. It's a lot of stitching and it's a lot of behind the scenes. That's why having two jobs is useful. And I'm super excited by voice. I think it's going to disrupt how consumers choose brands and products, how they prefer brands and products. The discovery piece is going to be trickier for brands to emerge, especially when you're in visual industries like fashion or beauty. We know images. We know videos. And we're still learning videos. By the way, we can talk about that. We're going to have to learn voice. What's the voice of Calvin Klein? I'm not going to make a sexy voice right now. Not now. It's a serious video. So the expression of a brand through voice. More than just find me the latest pair of jeans. How do I discover and love a brand through voice before other players use voice to work on brand preference? So that's what I'm working on. Because you are so innovative. I don't think I am. You don't think so? No. Everybody thinks you are. No, I think I'm an explorer. I'm exploring. OK. Yeah. There is too much responsibility to be an innovator. All right, so you're exploring voice. You're stitching everything together at every single touch point of the customer journey behind the scenes. And you're responsible for the outward branding of Calvin Klein. What is it that you love the most about that? I love that it's a 50-year-old brand that everybody knows the brand, like 90% awareness. Many brands would kill to get 90% awareness. But it's also a brand that has to adapt to new ages. It's a brand that was mastering print and outdoor. And a certain type of body sizes, for instance. So I'm very excited by the exploration of diversity when it comes to body expression, self-expression, outside of traditional media. And that's what I'm spending a lot of time on. So we just had our D&I Council meeting here two nights ago. And so we thank you. And I missed it. You missed it. But thank you for doing what you're doing, because it's such important work. Yes. What do you think our big challenges are in the industry right now? So I keep getting that question around brand and performance. How do you make choices between your brand investments, your performance investments? It's a tricky question, because it's creating silos within organizations. It's opposing teams. It's opposing budgets. I'm really not comfortable having two buckets, two teams. And what I've been doing has been to merge the teams. So they work on one consumer journey. Going back to unification of the consumer experience. OK. So any tips or tricks you're learning while you're doing that? Don't have two teams. They're going to fight over budgets. The conversion team, the performance team, will use the fact that they have higher ROI. And the brand teams will talk about elevation of the brand. So I would say tip number one, have one team. Tip number two, use data as the neutralizer. And do not get into emotional debates between the two sides of the consumer journey. And maybe tip number three, upskill the teams. So the performance side of the team knows about the brand. And the brand team understands data and what it means to drive this last click. OK, so speaking of upskilling, you've been a huge champion of training and development and performance improvement for all of your team. But what about you? What's a great career tip that you've heard or have shared in your path? So the best advice I got was 10 years ago by one of the greatest mentors in brand marketing. He said to me, come to work every day as if it is your first day at work. Don't ever get comfortable. Try to learn from the others. Continue to learn and unlearn. Learn from your teams. You're not just there to guide your teams. You're also here to receive from your teams. So I'm trying to be uncomfortable and learn and never settle for any sorts of state of school.  Well, Mary, thank you so much for joining us today and thank you for joining us on top of mind.


We know they are targeting him. We know this. It's not the former first aid they are looking for.  For someone to send some people to come and start setting a former president. Mind you, you should also realize that one day you'll be a former. Tukaku kongan. Kesa huwera. Kesa huwera. It's just because they have seen that the PF is still popular. It's still popular and will remain popular. The search which was supposed to be done today, was supposed to have been done legally, the Zambia Police Service obtained a search warrant directed at the perpetrators of the act. And the title deeds are still not having been made available. As far as we are concerned, there is a report that has been made to Zambia Police. And this report was never made by President Hakainde. It's named. Those leaders of the PF bring the name of President Hakainde in their abuse. Every Zambian has a right to report any criminal activity that has been perpetrated against them to Zambia Police or any other law enforcement agents. That is the right of the Zambian. Are we saying that the lady who was abused has no rights? She should not complain to Zambia Police for the abuse that were occasioned on her. Is that what we are saying now? We as the UPND government we are saying every Zambian has the right to report any criminal activity. I want to borrow the words of my former colleague who was a member of Parliament and Minister of Home Affairs. When they were about to arrest my current president where he indicated that there is no one in this country who is above the law. And that is the correct position. No one is above the law. If you transgress against the law the law enforcement agencies will follow. I did not expect that senior members of the party immediate former ruling party can unleash abuse on a sitting president who has nothing to do with the operations of the Zambia Police that were taking place at the former president's house. I also want to express my disappointment pertaining to the statements that have been attributed to some of my colleagues in the patriotic front wherein they are alleging that the legal search to be conducted at the former president's house was targeted at the former president. They further went to state that the former president has immunity and that no search whatsoever should be conducted at that house because the president has immunity. We as the new dawn government of the Republic of Zambia and the Zambia Police are privy to the provisions of Article 98 of the Constitution of Zambia pertaining to the immunities of a sitting and a former president. We are aware of those provisions and we are not going to deliberately violate the provisions of the law.


Russian infantry banned from using armoured vehicles to attack Nipro bridgehead. Russian forces attempting to dislodge a Ukrainian bridgehead on the Nipro river have been banned from using armoured vehicles and are now attacking on foot as suicide bombers Ukraine's military has said. Speaking to the United News telethon, South Operational Command spokesperson Natalia Humenyuk said the ban was due simply to how much material Moscow's forces have been losing in the now months long battle the Kiev Post reported. For a long time we have been observing that they have a ban on using armoured vehicles during assaults meaning that at first they were making assaults using classic tactics of fighting with armoured vehicles, she said but they are losing equipment so after a while they stopped using it. We found out that they have a ban on it meaning that assaults are carried out directly by infantry groups Humenyuk said the number of Russian attacks had dropped in recent days from 10 to 15 per day to 3 or 4 She suggested that this was because it was increasingly difficult for the enemy to assemble a group of suicide bombers and that Russian forces were losing 50 to 80 men a day. As reported last month, Ukrainian Marines are deeply dug in and backed by masses of strike drones are firmly holding ground first captured in mid-October. According to a number of independent observers, no matter what Russian forces try to do they haven't been able to dislodge the Ukrainians who will still be there in weeks and months to come.


had come with a talk on the EU data protection reform which is the new data protection law we're all going to be subjected to as of May 2018 but I'm obviously not giving that talk now I still can if you want me to at your conference obviously this is sort of this I cobbled this together in the hotel last night we're all reacting to breaking news as we get it I'm going to continue to write and blog about this as more facts become clear so you may want to keep an eye on my website which is webdevlaw.uk or sign up for my monthly newsletter I don't spam the link is there's two links on the site and I'm gonna do my best to keep you all updated one thing I'll tell you of course is I am not a lawyer but neither are 99.9% of the people commenting on Brexit right now so yesterday happened it actually happened we didn't think it was gonna happen but it happened I was in Hale'i and I realized I was doing this unconsciously I didn't even realize I was doing it I used to do that to my daughter when she was a baby and she had an upset tummy to calm her down I was doing that to myself so that just tells you what an impact it's had on those of us from the UK who are here at word camp here up I'm gonna try to talk you through the implications of what Brexit could mean for our work on the web and the digital professions in future I'm not gonna get into issues of passports nationality can I continue to work in the UK can I continue to work in Europe we just don't know that at all what is actually going to happen now well the first thing you need to understand about the referendum was that there it was not legally binding it has no legal or political force the referendum had as much legal force as a pole daddy pole that is all it was what happens now when you're gonna be seeing a lot about this in the media is something called article 50 and this is part of the Treaty of EU membership which talks about leaving what happens now is the country that wants to leave has to invoke article 50 to the EU no one can force the country to do it the EU can't force the country to do it and it's been over well it's been close to 36 hours now and the UK has not invoked article 50 the EU is stamping their feet saying get on with it that is just smoke and mirrors they can't force anything and what some commentators are seeing is a stalemate when David Cameron gave his resignation speech yesterday he said he is going to put article 50 to his successor which is at least three months away so he's bought himself three months of time and even Boris Johnson has said there is no rush so this is starting to look like a big political stalemate if you were cynical you could say the entire referendum itself was a chess game between members of the Bullingdon Club that is what it is really starting to look like so we have a couple of months of absolute uncertainty where we are waiting for article 50 to be invoked and as I said some commentators are saying it may well not be ever if the UK was to invoke article 50 that would mean at least two years of negotiations to at least two to disentangle all these laws it's going to take at least ten years for a full EU exit from the European Union if it does happen so that's the first point I want to make about your business and your digital work nothing is going to change overnight nothing is going to change this year a lot of people throughout the referendum asked if the UK could have special status could we sort of be half in half out of the EU the answer is no number one we couldn't be a part of the EEA like Norway and Iceland because one of the prerequisites of EEA membership is freedom of movement number two is a lesson we in Scotland learned with the first independence referendum which is you can't retain special concessions in a system you've already left number three the rest of the EU is seriously pissed off with the UK right now a number four it's the old saying if you give a mouse a crumb he's going to want a whole cookie if the EU lets the UK stay half in half out with special status then every region not member state region in the EU that has a grievance is going to want to go out or go half in half out so no the EU is not going to be half in half out with any special status the one question everyone asked me over and over throughout this whole campaign was if we leave the EU do we still need to comply with X law whatever Brexit does not get you out of compliance with EU digital laws if you are still doing business in Europe if you still have customers in Europe you still have to retain all of these laws as if you were still in the EU what is going to happen to the EU laws that are already on the books in the UK well we're not going to have a scorched earth thing where they just start tearing up all EU laws for the sake of it things like the consumer rights directive which was the e-commerce reform that banned a lot of dark patterns is still going to stay on the books that's not going to change the one thing that really worries me in terms of what they're going to look to get rid of is ECHR that's the European Convention on Human Rights Theresa May has been gagging to get rid of that for a long time because that contains the safeguards that have prevented her from massive from ratcheting up mass digital surveillance in the UK she's got her investigatory powers bill going through the system right now which is the anti-terrorism law enforcement regulation that basically wants to hoover up all of our data and with the ECHR gone she can ratchet it up however she likes and that's not even the worst part in the past four hours she's emerged as a possible alternative to Boris Johnson as the next Prime Minister I'd suggest you join open rights group now if you're not already a member what is Brexit if it does indeed happen going to mean for laws in the pipeline nothing's going to happen with accessibility we're already okay on that that moss oh boy with if you're still selling to customers in the EU you still have to comply with that moss and collect taxes on behalf of all these European customers but you lose the moss system which means you have to register with the tax authority of every country where you're doing business in Europe the EU that action campaign are meeting with the HMRC in the Treasury next week to discuss the next steps cookies you still have to comply with the cookie law if you're still doing business in Europe I'm helping to work on a consultation with a group called the web guild you can read about this on my blog because the eos already is already put the next cookie law out to public consultation we're still going to participate in this at least we get some traction on our under our belt about how we consult on digital laws in a non EU Britain you can read about that on my log by blog data protection here's the big one here's the big big big one and please give me an extra minute to discuss this if we go out of the EU but we still do business in Europe we still have to be on an adequate level of data protection with Europe both with the current 1995 regulation and with the 2018 regulation if we don't we become just like America that has inadequate data protection and does not protect the data that we're handling and if you've been if you've been following the safe harbor privacy shield data drama that now becomes safe harbor privacy shield to the one worry some people have said is that a non EU Britain then waters down privacy and data protection laws to attract inward investment from tech giants so what's immediately immediately head for us number one is be careful what you wish for anyone who has said let's leave the EU so that we get out of the cookie law or VAPMOS or whatever now you're going to get the UK's version you're going to have increased secure scrutiny from your EU clients wanting to know what is going on you may have to renegotiate some contracts you're going to have some clients asking you to get your data out of Britain and put it into the EU you're going to see a brain drain of businesses and people I have already talked to people who say they're leaving we need to organize as a profession because we've lost the big boogeyman of the EU who we blame for all our problems now and as Matt said yesterday we are the 1% we are here in Europe in a bubble in a supportive international community back home it's hell and we need to carry over those values of open source and the word press ethos more than ever thank you


The same waves that wash Bangkok also touch Malaysia and South Vietnam and the Philippines and the waves move out They wash against South Korea and Australia and New Zealand and many thousands of miles away The same Pacific waves touch my own nation at Hawaii and Alaska and California These waves speak of the deepest meanings of my journey That we are Pacific neighbors with a common destiny President Lyndon Johnson journeyed to the Pacific in 1966 because he was invited to the Manila Summit Conference of Seven Nations and Because he needed to see for himself the changing face of Asia I will go to see and to listen and to learn and to act with our partners To bring an honorable peace to Southeast Asia At the first day that it is possible Trip would cover the Pacific starting with New Zealand and Australia Then the Manila Summit Conference of Nations allied in the Vietnam War Then Thailand and Malaysia and Finally Korea on such a wide-ranging trip Most men are absorbed in the plans the events and the news of the moment But a president is concerned with generations There are many searching questions that we can ask How well have we understood the feelings and aspirations of Asia's people? How well have we understood the course of revolution or have we understood the shifting tides of nationalism in all of its form American policy must be the policy of an open mind President This is Johnson ladies and gentlemen and particularly boys and girls Can I suggest that each and every one of you follow the example of the honorable Keith Holyoke Prime Minister of New Zealand? Greeted the president and Mrs. Johnson Half of the people of New Zealand and the government of New Zealand I extend to you both Very warm a very human welcome a thousand welcomes to you both and on behalf of the Maori people Who've already welcomed you in their language Kia ora Good luck and good health Prime Minister. Mrs. Holyoke. Mr. Meach. Mr. White My friends of New Zealand I Want to thank you for coming here in this rainy weather exposing yourself to give us this neighborly welcome and If you will be good enough I Hope that you will wish me on my return to have the same kind of rainy reception at my home range in Texas as I'm getting here today On October 20th the president and Mrs. Johnson arrived in Wellington, New Zealand in my jet from Hawaii and Samoa Riding the smooth jet stream at more than 500 miles an hour There's Johnson and I are quite honored to be in New Zealand Physically, we have not entirely adjusted to the Southern Hemisphere after our long flight But you may be sure that our hearts are already in residence and it was quite a change from our last arrival in New Zealand in The spring of 1942 when both nations faced very grim problems together You are welcome was as warm then as it was today, although not as numerous But it was outgoing and it was generous New Zealanders twice left these beautiful islands to fight for the freedom and liberty of all men Ours was a unity of nations who longed to live in peace But who understood from the better lessons of two wars What the consequences of appeasement would be Every man wants peace That's something that all of you should take cognizance of now You can't separate men by those who want peace and those who don't want peace every man wants peace The president's itinerary in every country Included private talks with government leaders and whenever possible visits to rural areas Westward in the late afternoon The president and mrs. Johnson flew 1300 miles from New Zealand to Australia They were met by Prime Minister and mrs. Harold Holt back to Australia since I left here 25 years ago And here I am and I'm happy and I'm enjoying it and I like it then and I like it better now I need not to convey to you the admiration and affection that I have for the Australian people I have so much in my heart that I would like to tell you that I don't trust myself Evocally that our common engagement in Vietnam is morally the view of a minority Because we have put our trust in democracy We are bound to preserve the minorities right to express its opinion But it is exactly because we are Democracies that we cannot be turned aside from policies that the great majority of our public support And for which they have made profound sacrifices Going to Manila to try to find a formula for peace To try to review our military operations To try to exchange views With the the leaders of seven countries We don't expect any magic wonders. We don't expect any miracles But we do think that each nation who has man committed to die Their leaders ought to get around the table and get the best thinking of The best man those nations can send On behalf of my people I extend Warm and affectionate welcome to eat and everyone President Ferdinand Marcos of the Republic of the Philippines proposed and presided over the Manila summit conference I agree President Park Jung-in of the Republic of Korea I agree Prime Minister Tanum Kittikathorn of Thailand of the Republic of Vietnam Kauke of the Republic of Vietnam Minister Harold Holt of Australia Holi of New Zealand President Lyndon B Johnson of the United States of America That sweeps over the face of Asia and Asia can no longer accept Western formulas without questioning nor participation At the Malacan Yang Palace the leaders went into two days of private working meetings With an every one of you again. I say greetings Mabuhay and thank you After the doors were closed a watching world was kept informed through special releases and briefings at the International Press Center Over 2,000 newsmen from more than 70 nations were covering the conference While their husbands met together the first ladies went on excursions arranged by their hostess Mrs. Marcos Order please The leaders agreed upon a declaration of four principles for peace First aggression must fail Second the bonds of hunger illiteracy and disease must be broken Third regional cooperation must be increased Finally former enemies must learn to live together in peace The leaders also issued a joint communique on Vietnam They pledged that allied forces would be withdrawn within six months Provided that quote the other side withdraws its forces to the north Ceases infiltration and the level of violence then subsides close quote Beyond that the leaders also pledged not to invade North Vietnam To grant amnesty to South Vietnamese who have been misled or coerced into the Viet Cong To give top priority to land reform and social welfare in South Vietnam and To hold elections within six months after completion of the new constitution This declaration of principles on which we base our troops for you It is my pleasure and honor now President Johnson's final day in Manila was a particularly busy one In the morning he had gone with President Marcos to visit the island of Clarigador Which Philippine and American fighting men had held together during World War two until their food ran out Then the president went to the Philippine and American cemeteries In the afternoon he was to address the American community at the embassy And in between he visited the International Rice Research Institute to see experimental techniques of rice culture In its short four years of existence this Institute has produced Promising new strains of rice yield which are now being planted in the soil of many countries One strain developed here has been called the miracle rice I'm glad to know that the Institute is Prepared to make these seeds available to all nations to all nations Whatever their politics and ideology because if we are to win our war and The only important war that really counts if we're to win our war against poverty and against disease And against ignorance and against illiteracy and against hungry stomachs Then we have got to succeed in projects like this and you are pointing the way For all of asia to follow and I hope they're looking I hope they're listening and I hope they're following The president's next schedule stop was at the american embassy But he never got there For many days there had been speculation that the president might visit vietnam But when a visit to a war zone could not be announced in advance Most guessing centered on thursday his first day in thailand So he went from manila on wednesday when it was least expected That's something that all of you should take cognizance of now You can't separate men by those who want peace and those who don't want peace Every man wants peace every man Hates to kill thursday october 27th at bang sun thailand The president took a day of rest by the sea And had private talks with prime minister kitty catch on the next morning the president and mrs. Johnson flew to Bangkok Where they were met by king boomer paul and queen surrogate palace grounds The lord shaman and other court officials were presented to the president and mrs. Johnson King boomer paul and president johnson also exchanged gifts october 28th Their majesties to king and queen gave a state banquet in honor of the president and mrs. Johnson Steve guests in the west wing of the chakri throne hall of the grand palace They went to the national theater for a special performance of thai classical dance mrs. Johnson crews in the royal motorboat along the bang luang canal to the floating market watsai elementary one it is this To give each man in the world a chance to seek the highest and the deepest of the human experience as he sees fit And you are doing that today Here in thailand your educational progress is exciting and it is matched by material progress as well You have applied modern technology to agriculture making thailand the world's leading exporter of rice At chute along corn university The president received an honorary doctorate in political science From thailand the presidential plane flew south 750 miles across the gulf of syam to malaysia This was the 13th day of the president's journey At koala lampur the president and mrs. Johnson were met by his majesty The young deep or two on agong and her majesty the rasha poor my suri agong The leader Office majesty's speech Mr. President We are proud and happy indeed to welcome you The leader of the great american people to the shores of malaysia Your state visit to our country Is the first ever embarked upon By an american president Mr. President During your stay here, you will meet ordinary folk doing about their daily business You will see a small nation Visually engaged in the pursuit of peace We have had plenty of trouble after independence But by the grace of god, we have emerged none the worse for it I'm delighted to be here in malaysia I feel that i know you Because malaysia like the united states Is a federation of states Which were once colonies of great britain And because malaysia is Like the united states A nation of many diverse peoples Though i feel that i know you I have come here to learn from you I know that your nation is a model Of what may be done By determined and foresighted men in southeast asia ten years ago The city Where we spend the night Was a city of conflict You were absorbed in fighting the terrorists Your streets were filled with soldiers And your hospitals were filled with the wounded malaysia was traveling that difficult road Along which one of your great neighbors South vietnam To night toils With such sacrifice Yet here today We have seen the future And what it can hold for a troubled country We see bright and thriving and modern capital Bursting with energy We see an inspiring new mosque Symbolizing your trust in god We see a beautiful new museum We see new buildings and new industries That mark your great economic advance and progress Three of the world's great peoples Have come together here in your nation There are people who differ in many ways But who have the will to live together In both peace and harmony And with a sense of nationhood I know of your many accomplishments How you have made land available to the landless How you have improved rural health services and rural education The labu jaya village is one of 67 land development projects in malaysia 15 years ago this area was virgin jungle sheltering communist guerrillas In 1960 the government cleared the jungle and planted rubber trees Farmers who had lived in poverty Were given the opportunity to own the land they worked The government also built homes Each family pays for its home and land over a 12-year period You have demonstrated that an independent nation Can rise from long years of bitter struggle To create economic prosperity and to lead in regional cooperation However the president goes the white house goes with him His plane carries a complete office staff Press secretaries and a group of key presidential advisors He signed bills into law while traveling northeast across the china sea 2,800 miles toward the final stop of his journey korea Johnston and president chung he parked up korea Planned private talks after the welcoming ceremonies at soul city hall I thank you from the bottom of my heart For the warmth of your welcome I thank you for your courage And for your friendship And for the testimony That you are giving to the promise Of freedom in the world 16 years ago and event occurred in korea That changed the shape of asia and the world When the fighting stopped korea faced every conceivable difficulty Millions of refugees Transportation in ruins Its cities and ashes Here is one of the truly dramatic stories of our time A nation transformed within a generation It is the intelligence and the energy And the hard work and the genius of the korean people That are creating a new future for your country From the air I saw how you had turned the sector over plots Into large productive squares and thus increased the production in excess of 35% We can see great evidence of flood control Irrigation and reforestation of your hills Johnson hill journey of 17 days Seven nations and 31 000 miles A brief moment in history A tired president headed home I have seen the palaces and the universities the ordinary homes and the village schools The new land developments and the new strains of rice I have seen cabinet members and school children Farm experts and village leaders and our fighting men I've seen millions of faces friendly and well-wishing and I have been deeply encouraged


Welcome to All Hands Update, I'm Petty Officer Ian Cotter. The deputy commander for the Enlisted Women in Submarines Task Force recently led a road show to encourage E1 to E8 enlisted females to join the submarine force. CNO recently detailed the Enlisted Women Integration Plan, opening submarine force billets to enlisted women. According to the plan, all submarine ratings and Navy enlisted classification codes will open to enlisted women in fiscal year 2015. The integration will be completed in two phases. The first abort Ohio-class submarines, and later abort Virginia-class attack submarines. Female officers have already been serving aboard submarines since 2011, and introducing enlisted females is the next step in creating a fully integrated submarine force. For eligible rates, pay grades and conversions, check out the new NAV admins on the NPC website.