Yeah, so let's go back to where we stopped the last time which was the logistic map and we discussed the idea of a generalized dimension to recall to you generalized dimension dq was defined as the limit as epsilon goes to 0 of 1 over 1-q the log of 1 to n of epsilon mu j to the power q divided by log 1 over epsilon this in the case if you took the full phase space whose measure is unity normalized to unity and you broke it up into cells of various kinds and this is the C j the j cell and mu j is the invariant measure associated with this so you begin to see that I have in mind a system which displays chaos which is certainly ergodic and which displays chaos and in which a typical phase trajectory wanders round and round in the phase space or some portion of the phase space like an attractor and I take that portion I break it up into cells C j and I associate some letter of the alphabet with each of these cells and I do the symbolic dynamics of the trajectory in the sense that I keep track of at each iteration in which cell the trajectory is to be found the typical trajectory is to be found and having done that then I compute this generalized dimension here which is some kind of weighted sum over this invariant measure of the jth cell raised to the power q and this set of numbers dq gives me some information on the way the attractor is on the nature of the attractor in general there is no reason why these dq should be integer numbers q need not be an integer by itself and dq is some function of q and the general statement I made was that dq is a non increasing function of q in other words as q increases from minus infinity to plus infinity dq decreases maybe not monotonically but certainly remain or certainly not non increasing so the statement specific statement was that if q prime is greater than q then dq prime is less than equal to q that was the statement I made about these about these generalized dimensions the next thing we saw was that in the case of one dimensional maps in which the density was uniform you could break it up into say n cells of equal size 1 over n and then it was a easy matter to show that dq was equal to 1 the topological dimension of the attractor the same the fractal dimension is the same as a topological dimension the entire interval was the attractor but if the invariant density or the measure is not uniform then this is no longer true and the question was what is it in the case of some of the maps we looked at for which we know what the invariant densities so let me take that up now and show you what it is for the logistic map at fully developed chaos so this is the next task to compute dq for the logistic map at mu equal to 4 so if you recall the map function f of x is 4 x times 1 minus x and the invariant density in this case the normalized invariant density was 1 over pi times square root of x times 1 minus x now what do we do well here is the interval 0 to 1 I break it up into n cells of equal size so this is 1 over n this is 2 over n and so on and this is cell C 1 this is cell C 2 and finally you have n minus 1 over n and this is cell C sub n and I am going to compute this measure for each of them take the limit n goes to infinity and compute what dq is so that's our program and what is mu j for the j cell is an integral over C j but C j runs from j minus 1 over n to j over n dx times row of x but what's that equal to well this is pi times square root of 1 minus x times 1 minus x and remember n is going to become very large and the size of each cell is going to become very very small so what does this become approximately equal to it's approximately equal to in this cell in any of these cells say the j cell here the value of that integral as this interval becomes smaller and smaller is the value of the integrand at the midpoint multiplied by the length of this interval the length of the interval is 1 over n and the value of the integral at the midpoint is 1 over n pi 1 over the square root of j minus a half over n times 1 minus j minus a half over j minus a half over because the cell runs from j minus 1 over n to j over n and the midpoint is j minus half over n so that's what mu j is and therefore dq can now be written down at least formally and we have dq equal to a limit n tending to infinity because I put epsilon equal to 1 over n 1 over 1 minus q the log summation j equal to 1 to the power q n that mess 1 over n pi square root j minus half over n 1 minus j minus half over n so whole thing raise to the power q divided by log n 1 over epsilon is n this is the quantity whose limit I want now this suggests very strongly you have a sum from 1 to n this suggests very strongly that you actually have an integral here you could convert this sum back to an integral once again and what would this integral be well if I put a if I forget about this factor and then I put a 1 over n outside this would be related to the following integral it would be related to 1 over pi apart from a factor pi x times 1 minus x to the power q it would be related to this integral do you agree because it's a Riemann sum I can always write this Riemann sum back as an integral provided the integral converges and when is it going to converge when is this integral going to converge it's got singularities at 0 and 1 out here and it's clear that it's going to converge because there's a square root here q over 2 should be less than 1 if it's greater than 1 this diverges when q over 2 is equal to 1 you have a dx over x that's logarithmically divergent so the integral converges for q less than 2 and diverges for q greater than or equal to 2 so it immediately means that you have to consider two cases separately you have to consider case 1 q less than 2 and case 2 q greater than or equal to 2 separately for q less than 2 so let's look at that first dq is equal to limit n tends to infinity 1 over 1 minus q log summation j equal to 1 to n let's get this n out of here and that's n to the minus q and then let me take it out of the integral so log n to the minus q comes out of the integral and then what I have here is 1 over pi square root of j minus half over n 1 minus j minus half over n to the power q and it's really saying take the function x times 1 minus x take the midpoints of those guys the value at the midpoints and consider this sum now if I divide that sum by a 1 over n this quantity in the limit n tends to infinity is exactly equal to the integral from 0 to 1 divided by the whole thing divided by so I divided by n therefore I better put it back so log n to the 1 minus q so please remember I divided by an n in order to convert this to an integral and I have to put that n back there divided by I get rid of this and this is log n to the 1 minus q because 1 minus q log n is log n to the 1 minus q that becomes very very simple to evaluate so this quantity is equal to dq is equal to well the first term is clear so log of this times that and I take this out and it gives me just unity it's very clear so this 1 plus the limit as n tends to infinity of this sum divided by this but you see this sum in the limit is simply the remand is simply the integral from 0 to 1 dx over x times 1 minus x pi times square root of this to the power q that's all it is because that's how we got this to start with divided by log n to the 1 minus q and recall that q is less than 2 so what happens there what happens to this thing here so I put a 1 over 1 minus q outside and I have a log n this integral is finite this integral is finite and therefore when n tends to infinity this log n in the denominator ensures that that term goes to 0 completely because this integral is finite it converges so it's immediately clear this is equal to 1 for q less than 2 d0 had better be equal to 1 because that's exactly what the fractal dimensionality of the attractor was and the attractor was the unit interval the full interval so d0 was in any case equal to 1 now we discover that dq for all q less than 2 is equal to 1 if you had a uniform density like in the Bernoulli shift or in the tent map at fully developed chaos then dq is equal to 1 for all q but here for q less than 2 we can assert it's 1 but we have to re-examine the problem for q greater than 2 so let's now do the next step that is to go back and ask what happens if q is greater than 2 we have to go back to this and then we end up with so this is the second part q greater than or equal to 2 when we can't convert this summation into an integration because the integral doesn't converge but the sum is finite completely finite so what do we do in this case this case dq equal to limit n tends to infinity in the denominator of course you're going to get log n to the 1-q so let's put that here minus q but in the numerator you have an n to the q which comes out out there but then you can't convert the rest of it into an integral so there's not much use doing that on the other hand let's simply multiply through by n and this n goes away because you get an n squared with a square root that cancels this and in the numerator if I give myself a little more space this thing becomes limit n tends to infinity log summation j equal to 1 to n let's multiply through by n so we have a 1 over pi times a square root of j minus a half n minus j plus a half this whole bracket to the power q and the n squared canceled out with this n here divided by log n to the 1-q have to deal with this now what sort of sum is this it's clear that for small j j near 1 near this end the lower end these terms the denominator is going like 1 over n to the power q over 2 so those terms go to 0 the terms for small j now we're going to have jn very very large so the terms for small j are going to go to 0 like 1 over n to the q over 2 because this factor is going to dominate these are finite small numbers on the other hand at the other end of the sum for j near n this is going to be some small number and you're going to have an n here so once again the terms are going to go like 1 over n to the q over 2 the terms in the middle near n over 2 for example both these factors are going to be of order n because this is going to be like n over 2 that's going to be like an n over 2 and therefore those terms are going to disappear like 1 over n to the q because there's a square root of n squared so we the conclusion is you have an infinite sum a big very large number of terms for very large n the terms at the lower end are going to disappear like 1 over n to the q over 2 terms at the upper end are also going to disappear like 1 over n to the q over 2 and the terms in the middle are going to disappear faster like 1 over n to the q itself so this whole thing could be written in the following form the whole thing could be written as equal to limit n tends to infinity one can make this much more rigorous but this is as simple a way of understanding it as any log of n to the power minus q over 2 summation j equal to 1 to n times something or the other which will give you in the limit a finite sum because this is the rate at which every term vanishes and the ones in the middle are going to go to 0 even faster but certainly there are lots of terms of this kind contributions of this kind divided by log n to the 1 minus q where this sum tends to a finite not infinite finite non negative quantity finite limit as n tends to infinity having pulled out the dominant n this is what the rest of it is and that immediately implies that this implies dq is equal to since in this case is an n in the denominator that cancels against this this term is finite so that limit will be 0 then I take the log of this sum it will go to some 0 because of the log n in the denominator and this becomes equal to q over twice q minus 1 because it is a minus sign so it is minus q over 2 over 1 minus q that is all and this is the answer for q greater than or equal to 2 what happens at 2 this 2 and this cancels and you have dq equal to 1 so there is continuity in this dq and then it decreases beyond that so if you sketch this dq this is what it is going to look like the generalized dimension for this problem here is q and here is dq it started with the value unity and this is the origin here is 1 here is 2 for example kept till there till this is 1 on the side it is not to scale the scale on the vertical axis is 1 there and then it drops at this point and asymptotically as q tends to plus infinity it tends to a half it is easy to see that this is what it will look like so this is the spectrum of generalized dimensions for the logistic map at fully developed chaos this set of numbers dq something like the moments of these invariant measures weighted appropriately so it is a infinite set of numbers it is actually a function in this case and gives you information about the distribution itself anything for which dq is a function of q and not just a constant is not just a fractal it means there are many many other dimensionalities buried in this system in this tractor and this kind of thing is called a multi fractal and dq is sometimes called a multi fractal set of exponents or whatever it is related to other exponents it is related to other functions which also characterize this distribution so the whole thing is an effort to actually find out based on this course grain dynamics what the nature of the attractor is it is like finding the moments of a distribution if you like and then reconstructing the distribution the probability distribution itself so that is really what is being done let us take another example where you have similar kinds of behavior and then we will try to draw a general lesson from it and this was the map in which you had intermittency so if you recall we had the square root cusp map which was f of x equal to 1-2 the modulus of the square root of the modulus of x where x was an element of the interval – 1 to ± 1 and the invariant density in this case because you had a marginally stable fixed point marginally unstable fixed point at – 1 the invariant measure density in this case was a linear function it was 1-x over 2 so it piled up near the left hand side and tapered off to 0 at x equal to ± 1 and you have to do exactly the same thing as before once again you expect a sort of phase transition quote unquote at a point like this at some value of q where would that be because ultimately what you are trying to do is to find an integral of rho of x to the power q when you do that you have this – 1 to 1 and when would this diverge as long as it is finite you could convert the summation into an integration and do the problem very easily when would it diverge it is clear if q becomes negative it starts diverging here less than – 1 because dx over 1-x is logarithmically divergent and any higher power in the denominator and it diverges so here you expect that at q equal to – 1 you have a funny business and after that beyond that it remains a constant essentially and indeed that's true and I request you to go through exactly the same steps of before and show this is an exercise show that in this problem dq is equal to 2q over q-1 what does this look like if I sketch this q less than – 1 and what does it become at q equal to – 1 it becomes 1 so it remains 1 thereafter strictly speaking I should have written this but so here is – 1 this point and beyond this point it remains 1 in this case and at q equal to – infinity it hits the value 2 and therefore 2 here dq here and in fact it starts like this it bends down come here and it goes on so this is the way dq looks in this case so it's immediately clear that if you have some kind of polynomial in rho of x or a rational function in rho of x then wherever you have the roots of the numerator or the roots of the denominator as in the other case you have points of this kind where the slope of dq changes abruptly in fact it's not hard to show and I would like you to show this general result if rho of x for example is proportional to apart from some constant say x – x – a to the point power alpha say divided by x – b to the power beta at the two ends for example in the interval a to b suppose it has singularities at the end points of the range in which the map is defined then dq has changes of slope at points which depend on alpha and beta on both sides and it looks like as follows goes like the following graph let me draw it in pictures and then you have by the way alpha beta greater than 0 positive integers so you have this that's dq and this is – 1 over alpha this is 1 over beta and dq starts with the value at this point there is a phase transition it remains like this and then it tapers off to a limiting value which is 1 – beta because certainly beta has got to be less than one otherwise it's not integrable and out here it's like this and this is 1 plus alpha so d – infinity tends to 1 plus alpha d plus infinity tends to 1 – beta and in between it hits the value 1 and it looks like that if you have more products here in between more zeros and more places where it diverges then it will show up right there all those points will in general show discontinuities in the slope of dq so this gives you some idea of what the generalized dimensions look like we will come across some other examples in more in high dimensional fractals but I urge you to complete these derivations it's very simple all you have to do is to use exactly the technique I use for the logistic map whenever you can convert things to an integral you should do so and then the limit is easy to take otherwise you must retain the sum and find out what the dominant behavior of the sum is you can do this very rigorously but what I did there was essentially right okay now let's go back a little bit and ask okay we have some kind of coarse graining but can we relate the dynamics of these coarse grained dynamical systems can we relate this symbolic dynamics to some random process itself it turns out the answer is in many cases you can we saw what the idea of a generating partition was I stated that if you divided up the phase space into cells and kept track of the symbolic dynamics of the trajectory moving between these cells then if you can reconstruct the actual dynamics by looking at these sequences of letters then it's a generating partition not at all clear if every system has generating partitions but there are other cases where the partition has other kinds of interesting properties in other words the jump of the system between these cells looks exactly like a random process called a Markov chain and let's see what that is for that I need to introduce a few concepts in probability especially in Markov chains and not clear how much background I should assume here but let's take our usual attitude and spend a few minutes is not a digression on Markov processes or Markov chains so this is a digression Markov chains or Markov processes depending on whether continuous time or in discrete time let's let's do the following so let me assume that I have a random variable X and I it changes with time and the time could be discrete or continuous doesn't matter but let's suppose that this variable generates a time series as I go along then how do you specify the probabilities or distributions of this variable well one of them would be to ask what's the probability and if it's a continuous variable then of course I must talk about a probability density otherwise I talk about a probability itself and I'll use these two words interchangeably for the purposes of this little digression I could ask what's the probability that it's got some value X at some time T and if the time is discrete then it's N let me for the moment use continuous time and go back to discrete time a little later so this would give me the probability or the density probability density that the function is either that the variable has the value X at a time T or is in a range X to X plus DX at time T let me call it P1 it's some function non-negative function but this is not enough I also need to know what's the probability that this variable has say a value X2 at time T2 and X1 at time T1 let's stick to the standard convention and write earlier times to the right and later times to the left so you need to know this joint probability as well two time probability you also need to know the three time probability that it's X3 at time T3 X2 at time T2 and X1 at time T1 and so each of these in general different functions and to specify the random variable completely I need to know all the joint probabilities for arbitrary number of time arguments so you need an infinite number of probability distributions joint probability distributions to specify the random variable completely to find all possible correlations in the random variable however by definition Pn of Xn Tn Xn-1 Tn-1 dot dot dot all the way up to X1 T1 by definition this is equal to the probability that you have Xn at Tn given and I for conditional I put a vertical bar there given that at time Tn-1 the variable had this value Xn-2 at Tn-2 up to the last one multiplied by the probability that you had all the conditions here that you had Xn-1 at Tn-1 X1 and this of course is it has n time arguments and therefore let me call this again n and let's call this n-1 this is a different object than this probability this is a joint probability this is a conditional probability so it simply says that if you have a sequence in time T1 less than T2 less than T3 etc then the probability that you had X1 at T1 time T1 right up to Xn at time Tn is by definition equal to the probability that you had all these earlier events happening at the times indicated multiplied by the conditional probability that if these events given these events have occurred the next at the next time step the values Xn so this is almost intuitively obvious well therefore you can start reducing things because now you could reduce this to a product of an n-1 fold conditional probability that the last one occurs given all the earlier ones occur and therefore in some sense you can reduce this entire thing to a product of conditional probabilities till you hit this last one that's what you can do in general that's about all you can do in general for a Markov process the memory is short and its specific by saying first order Markov process means that this conditional probability is independent of all these earlier events and depends only on the immediately preceding event so it's like a one step memory that's it so if I call this the present state the future the next step is determined by the present and not by the past history so for a Markov process Markov Markov process this whole thing gets wiped out and you have just this multiplied of course so the statement is that the conditional probability so let me state that again precisely then we come back and write out our joint probabilities the statement is Pn of Xn Tn given that Xn-1 Tn-1 occurred X1 T1 is identically equal to some function and for want of a better notation let me just call it P once again P of Xn Tn Xn-1 Tn-1 there are only two time arguments here so let me just call it P2 it's a function with two time arguments and this is the Markov property it's called first order Markov if it depends on the preceding two steps then it's second order Markov and so on but we are going to restrict ourselves to first order Markov processes yeah this is for continuous time for continuous time these are probability densities rather than probabilities oh you are asking continuous time as opposed to continuous discrete variable here these are two different events in two different times time steps so I don't see any problem there all we need is this so at any instant that's true this is true this question is if you have discrete time then I don't have this T here this is n-1's instant and this is the nth instant then you can understand this but his question is suppose time flows continuously then what are these things how close to this time can that be arbitrarily close arbitrarily close still true that's a good question we will come back to this and see what it implies it will imply a certain renewal of the process as we go along so you can ask okay on the time axis if something happened here and something happens here then we are saying the conditional probability depends on what happened here in this point but something else could have happened in between yes indeed that's true therefore this quantity is going to obey some kind of chain condition which will propagate you from here to there and then from there to there we are going to see what that is but yes that's a good point continuous time this is actually true as long as Tn-1 is less than Tn that's true but what's the implication of this it's an enormous implication because it immediately implies that you can take the general n time probability and write it as a factor of conditional probabilities of just two time arguments right away because you can then write this quantity this n fold probability to be equal to first the joint probability P2 that you had xn at Tn given xn-1 at Tn-1 and the rest of the conditional probability this dependence is gone multiplied by the n-1 time joint probability of this set of events but that could be written once again as P2 of xn-1 Tn-1 given xn-2 Tn-2 and then the n-2 time joint probability but that can again be factored out and so on till you end up with P2 of x2 T2 x1 T1 multiplied by the single time probability of x1 T1 so for a Markov process or a Markov chain if T is discrete this factorization is still possible what does that mean in practical terms it means that if you give me the function P2 and the function P1 the job is done all joint distributions are known in terms of just two probability distributions the two time conditional distribution and the one time probability itself so that's the remarkable simplification that occurs for a Markov process now we are going to make a further assumption we are going to assume that the statistical properties of this random variable don't change with time in other words that the variable is a stationary random process that the actual statistics doesn't change as a function of time and therefore it implies that you can shift all the time arguments by the same constant and nothing is going to change what does that imply it says that this function here is independent of time because it's independent of the time argument you can subtract out the T1 completely in other words if you ask for what's the average value of x or the mean square value or any moment of x this should be independent of time if I took the nth moment or kth moment of this this by definition is dx if it's a continuous variable P of xt x to the power n if it's a continuous process if the set of values of x is continuous it's this integral otherwise it's a summation but the T dependence appears here and if this is going to be statistically all these moments are independent of time the only way that can happen is if this distribution doesn't depend on time therefore for a stationary process you have a further simplification for a stationary random process P of x, T P1 of x, T equal to P of x no T dependence and since no confusion should arise I am going to drop this one just call it P of x it's a probability if this x takes on a discrete set of values it's a probability density if x takes on continuous value it's the stationary distribution and what happens to P2 of xt x prime say x0 T0 so the probability given that you had the value x0 at time T0 the probability that you have x at time T is independent of T0 you can subtract out this T0 from both sides therefore this is equal to P2 of xt minus T0 x0 and 0 you could call that the origin of time which was simplicity of notation let me simply write this as again P using in abuse of notation P of x T minus T0 x0 in other words I won't even bother to write down the fact that this is a 0 here I have shifted the time origin and that's it I use the same symbol P here I shouldn't really I should call this P2 and call that P1 but we know what is happening here so where is the time argument here there is no time here at all in the single probability single time probability and there is just a single time argument here what therefore happens to this business here this entire density well it's clear this becomes Tn minus Tn minus 1 xn minus 1 and I can erase this in a little abuse of notation I write this as because it is really dependent only on this time difference all the way up to here P of x2 T2 minus T1 x1 and then P of x1 so the matter simplified enormously in other words if you give me two functions you give me P of x and you give me P of x T x0 the job is done there is one time dependent probability or density and there is one time independent you have one more property which you expect and that's the following you would expect you have to establish this but you would expect that if I took this conditional density or probability which says given that this happened given this as the initial condition this is the value at time T given that this happened at T equal to 0 what happens to this as T tends to infinity what would you expect what would you expect would happen to this probability as T tends to infinity I would expect if things are reasonable and the system is sufficiently random I'd expect that this probability would become independent of x0 it should forget the initial condition altogether and I would expect therefore that you have limit T tends to infinity P of x T x0 to be equal to P of x itself should become the stationary distribution independent of the starting point that's a consistency condition that I put on the system if the dynamics has enough sufficient mixing it turns out this is always going to be the case so really you have just one function which you have to determine this because once I do that I take its limit as T tends to infinity and I know this and therefore this entire for arbitrary n joint probabilities are all determined by a single function of a single time argument which is this and what's the initial condition on this if it's a density if this x is a continuous variable then it's immediately clear that in addition to this thing here you therefore have limit T tends to infinity P of x T x0 equal to P of x0 P of x and you also have limit T tends to 0 P of x T x0 equal to what it should have support only at x0 no not P of x0 this is the conditional density in x it's one if x equal to x0 in some sense but if it's a continuous variable and it's normalizable then what would it be what would this be what's the continuous analog of density which is got measure one delta function yeah there are delta function so there should be delta of x – x0 so that's the only thing that contributes and everything else is 0 so you have an initial condition you have an asymptotic limit and the entire process is determined by this single function and it's this quantity for which one generally writes equations down evolution equations or master equations or whatever you call it is written down for this because after all in any physical problem it's only conditional densities that you can model you can't model absolute probabilities you can only model conditional probabilities given there are certain event or set of events has happened you can now talk about the future but you can't talk about any absolute probability there's no way of writing equations for these things so the equation one writes for Markov processes for objects of this kind are called master equations there's several of them but depending some of them are integral equations some of them are differential equations and so on and so forth and there's a very elaborate theory of Markov processes which tells you what these things are how these things evolve but now let's go to specifics and let's see how to write this down so let me work in the context of continuous processes and after that I will talk about jump processes we need to do both there are several ways of doing this but let me do this in terms of something which I will motivate in sort of an intuitive sort of way so I would like to write an equation for p of x, p and I drop the x0 for the moment because that's incorporated in initial condition saying that p of x, 0 is going to be some delta of x-x0 wherever I start then I ask if x is continuous runs over a whole set of values then I ask what's delta p over delta t equal to on this side the two ways in which I can have contributions to the increase of the probability mass at the value x one of them would be at if the system well let me let me motivate this go back a little bit and write so I go back right p of x, t x0 at 0 could be written in the following way because of this Markov property you could say that you start at x0 at 0 and you evolve to some value x' at some time t' so on the time axis you have 0 here you have t' here and you have t here t' is less than t but greater than 0 you propagate from here to there and then you propagate from x' to x in the remaining time but x' could have been anything in between and therefore you integrate over all possibilities and you expect a condition like this to be satisfied this is called the chain condition it essentially renews this process because you are really saying that this process has only one step memory so it goes from x0 to some intermediate state any intermediate value x' and then from x' to x in the remaining time and this is true for all t' between 0 and t and there is an integration over the intermediate steps this has got a long name it is sometimes called the Chapman-Kolmogorov equation or the chain condition because it is like a chain which propagate go from step one step two step two to the next step three and so on it should really be called the Bachelier-Smolukowski-Chapman-Kolmogorov equation because several people wrote it down at the same time or around the same time but we won't go into the history of this but this equation helps us to write something down for this I should immediately hasten to add that one shouldn't be under the misconception that this equation is unique to Markov processes other processes which are not Markov also obey such a chain condition because it's in some sense a renewal process all renewal processes would obey similar equations but Markov processes certainly do what one does is the following one takes this t-t' to be infinitesimal some delta t for example very close to t and then writes instead of this a transition probability per unit time and that changes this chain equation from this kind of integral equation which is nonlinear it's an integral equation and it's nonlinear in this p to a linear equation for the rate of change of p and that equation looks like this so from this it follows and the suitable conditions delta over delta t p of xt x0 is equal to on this side an integral over intermediate states dx' the probability that you are at x' at time t having started at x0 and then you make a transition from x' to x per unit time if that transition probability is wx given x' in other words this stands for the transition probability per unit time of making a transition from the value x' to x per unit time because we're writing a rate equation but you also have a lost term it says you might well have reached x itself at time t you could jump out of x into any other state so there's a w of x' x and this looks exactly like the rate equations you write down chemical reactions or any other system which is governed by rate equations so this is like a game term you go from x0 to an intermediate point x' and then you jump from x' to x and there's a rate of jumps or you go from x0 to x and you jump out of x to some other value x' how do you write the same thing if you had a Markov chain in discrete time say so suppose the values that this variable could take and so being continuous set of values was a discrete set of values x1 x2 x3 etc x of j and let's suppose this whole thing took place in time then how do you write this set of equations what would you write what sort of equation would you write here if it's continuous if the process is discrete but the time is continuous then it's very straightforward you still have a rate equation of this kind here but you'd have a summation here right so you'd have a summation which is so let me let me say w say xi from the value xj to the value xi this transition probability per unit time let me write this as some wij so let me write it in this fashion let's call it wij by definition then what what would I write down instead of this I'd write d over dt and if pj is the probability that you are in the at the value xj at time t then this rate of change is equal to a summation over it's called it pi summation from j equal to 1 to whatever the number of set the set of values you can have could be infinite could be finite times what what's the integrand what's the summand inside in that case what would you write this remember would be you have to sum over all values so what would you write here pj of tw you have to go from i to j right from j to i so you have a ij you have that minus on this side of pi of tw ji with a prime here to show that j is not equal to i in an integration it doesn't matter because the value x prime equal to x is a set of measures 0 here but in a summit matters so you put a prime here which denotes this thing here implies j not equal to i in the sum that's very much like a rate equation write down and once you specify the matrix w you can write this transition probability matrix as a matrix once you write this matrix down the job is essentially done now what kind of solutions do you have for this matrix equation so let's see if we have a convenient way of writing this thing down let's suppose that I call p of t oh by the way what's the initial condition on this what's the initial condition it's anything at all depends on what your x0 was if instead of x0 I start with state k x sub k then of course it says that pi of 0 is equal to ?ik if it's at the state k the probability is 1 otherwise it's 0 so I'm starting with some specific value at t equal to 0 and with that initial condition you have to solve this equation but there's a very convenient way of doing this let me write column vector p1 of t up to pn of t if you have for example a finite number of states but if you don't it keeps going forever and we really don't care so let me write p1 of t p2 of t etc and let me call this column vector the script p of t and that looks very much like a matrix equation what it says is that d over dt of p of t is equal to some matrix w times p of t and what is this w what would you say is w what are the diagonal elements of w for example suppose I is equal to 1 then it says dp1 over dt is a summation over j not equal to 1 so it starts off in this fashion and it says w12 times p2 plus w13 times p3 and so on so it's clear that this w has matrix elements which are like w12 and w13 all the way in this fashion and then you have a diagonal element here what is that diagonal element equal to so you have you put i equal to 1 you have dp1 over dt there is a term here which is minus p1 but there is a summation over j wji so this is w21 w31 41 and so on so you really have a structure which looks like w is equal to minus w21 plus w31 that's this first term the second term is w21 and the term here is w23 but in the diagonal here you have minus instead of the minus w what will you have here 12 just as you had 2131 etc you have a 12 plus w32 plus and so on so you have a matrix equation of this kind which is not very difficult to solve if w is a constant matrix which is what it is in our stationary case in principle and then you have a matrix w which determines the transition probabilities in which the off diagonal elements give you the transition probabilities from one state to another this is the rate at which from state 2 you jump to state 1 or 3 to 1 this is 2 to 1 to 1 to 2 and so on but the diagonal elements are the negatives of the sum of all the rest of the elements of the same the rest of the column so the sum over each column in this matrix w is 0 so you have a matrix with the property that the elements are all real and the diagonal elements are minus the sum of the off diagonal the rest of the elements in the same column and the formal solution to this is of course p of t is e to the w of t times p of 0 if p of 0 starts at a particular state then it is a column vector with 0s everywhere except a 1 at that particular state and e to the w t acting on that initial column vector gives you the probability vector for all the states so this is how you would handle discrete valued processes in continuous time and now if you go to discrete time as well you have what is called a Markov chain that 2 is determined by precisely this matrix of probability is something analogous to w once again instead of an exponent e to the w t if you have discrete time n what do you think this will be replaced by intuitively I have discrete time what do you think this guy is going to be replaced by well each time to go from time step 0 to time step 1 you are going to apply this matrix w to go from time step 1 to 2 you are going to apply the matrix w once again it will be just w power n yeah just as the discrete version of the Laplace transform is a z transform this is just a power it is exactly the same principle so you need to know w raised to an arbitrary power which you incidentally need to know if you want to exponentiate it as well but there is an even easier way to handle the matter and that is to look at what happens to the Laplace transform of this equation this immediately implies that the Laplace transform of p of t if I call this p ? s it says s p ? s – p of 0 is equal to w p ? s which implies that p ? s equal to s times the unit matrix – w inverse on p of 0 so it implies that if you have the resolvent of this matrix w this is called the resolvent of this matrix w as a function of s you have the Laplace transform and then you can invert the Laplace transform to find p of t this resolvent will exist as long as you do not hit an eigen value as long as s is not one of the eigen values so you have to find p of t this resolvent exist as long as you do not hit an eigen value as long as s is not one of the eigen values of w where the resolvent blows up even already see from here this is very similar to what we had for dynamical systems so the entire behavior of this probability matrix will be governed by the eigen values of this transition matrix w just as it was governed by the eigen values of the Jacobian matrix for our dynamical system so in that sense these Markov processes really have reduced to a set of first order equations is very similar to a dynamical system very similar mathematics is used now what I am going to do next is to take a map like the tent map or something like that do a partition of it and show that this partition can behave like a Markov chain the discrete time dynamics would behave exactly like a Markov chain and therefore I can actually tell what happens to n time arbitrary time probabilities and we will see how that is done next time.


Good afternoon, I'm Mark up to grow the director of the LBJ library and on behalf of our co-chairman for this event Anita McBride from American University I want to welcome you today to the enduring legacies of America's first ladies Throughout our nation's history first ladies have played an influential role in American politics and in history They have served alongside of our presidents as political advisors diplomats hostesses and national role models Today's program is the last in a series of first ladies conferences that we've done at all of the Texas Presidential libraries first a year ago today at the George Bush library at Texas annem Secondly at the George W. Bush library on the campus of SMU in March of this year and now this last one We frankly we stole this idea from American University They did the first conference of this sort under the direction of Anita McBride in 2011 and I can't tell you how invaluable Anita McBride has been in Establishing this conference. She'll be on our panel today She served as the chief of staff to Laura Bush during her White House tenure and has gone on to American University To serve in this capacity It is before we get started though I just want to acknowledge a couple of entities that have also been invaluable American University as a whole has been very supportive of these conferences as has the White House Historical Association Without them we simply wouldn't have been able to do these conferences Now without further ado, I want to introduce Nancy Davenport the Interim University Librarian of American University who will introduce our first panel. Good afternoon It is my pleasure to also welcome you to the enduring legacies of America's first ladies conference I want to thank the leadership at LBJ library, especially mark up to Grove and Linda Johnson Rob for hosting us Today's conference is the latest in the series and examines the highlights and the many Contributions made by America's first ladies and as mark said it is the last of the conferences to be hosted at the presidential libraries in the state of Texas American University is proud of its role in organizing and implementing these conferences We are a university dedicated to public service Our motto is ideas into action and action into service For almost 100 years American University has been generating training and teaching leaders public servants and scholars committed to professions in public service and global activism and At the university we believe that the women who have served in the position of first lady Have too often been overlooked have received too little formal recognition for their many Contributions throughout our history actions that have impacted our nation's public policy our national politics and our global democracy and diplomacy We are committed to celebrating the service of extraordinary women like Linda Bird Johnson and the other women who shared Lady Bird Johnson, I'm sorry and the other women who shared that unique experience of being a first lady We deeply value the role they have played in our society and we are proud to help them tell their stories We thank all the notable historians former White House staff and first family members as well as the first ladies Lara Bush and Barbara Bush for sharing with us their expertise their time and most especially their memories Since our inaugural conference in March of 2011 at American University The conferences have grown in size and in scope We are grateful to our partners at the National Archives and the White House Historical Association And especially to Anita McBride an executive in residence at American University School of Public Affairs Who conceived of and who directs these conferences? I? Now invite Alan Lowe to come to the podium and introduce our first set of speakers. Thank you Good afternoon everyone. I'm Alan Lowe director of the George W. Bush Presidential Library and Museum It's such a pleasure to join you all here today for I know is going to be an amazing conference Thank you all so much for being part of this my congratulations and thanks go out to all of our conference organizers and supporters And especially to Anita McBride to my fellow library director Mark up to Grove to our good friends and partners from American University And the White House Historical Association and of course to the first ladies Barbara and Laura Bush and all the first families Who have been involved in these conferences? We give you our great. Thanks this series began as you've heard at American University And then we all agree to be great to bring it on the road down here to Texas First the bush 41 and then up to our neighborhood at bush 43 where we had our conference last March Now we're glad to be part of this conference here at the wonderful lbj library It may be the final conference in this series in Texas, but I know we all want to continue Discussing and examining the vital role that first ladies play now I am director of the George W. Bush Library So if you indulge me for just a second I'd be in big trouble if I didn't say a couple sentences about what we're doing up in Dallas We're currently located at a temporary facility just north of Dallas in Lewisville, Texas But we're starting our move down to the beautiful new facility at SMU later this month or early next And we hope to have our dedication Next April and shortly thereafter will open to the public and of course. Hope you all can come visit us then our museum is going to be Spectacular, it's very interactive educational and we believe very inspiring our archives are immense We have over 70 million pages just a paper records alone around 80 terabytes of electronic information over 40,000 artifacts and 4 million photographs so it is and a truly an outstanding Spectacular archives that we have as our foundation there at the bush library And we've also started with a full slate of educational and public programs or not waiting for the building to be complete We're already out in the community working with schools with teachers and students all around Texas and the region we're very proud to be part of the SMU family and of course to be walking working alongside our partners at the Bush Foundation the Bush Institute and of course we're honored to work with president and Mrs. Bush So thank you for indulging me this for a moment there We're really proud of the Bush library and again hope you can come join us in Dallas next spring now Let me turn to our proceedings today Our first panel is going to look at first ladies through American history and we've assembled some truly excellent participants for you First we have Alita black Alita is executive director of their FDR for freedoms digital initiative It's a web-based education program dedicated to the four freedoms She serves as professor of history and international affairs at George Washington University And a founding editor and advisory board chair of the Eleanor Roosevelt papers project Alita. It's good to see you again And thank you so much for being here Next we have Lisa Caputo. She's now the executive vice president of marketing and communications for travelers. She's held Leadership positions with Citigroup Walt Disney Company and CBS Lisa served in the White House as deputy assistant to president Bill Clinton and his press secretary to First Lady Hillary Rodham Clinton Thank you very much Lisa for being part of the panel Of course we have Anita McBride Anita currently is an executive in residence at American University Center for Congressional and presidential studies and also serves as a senior advisor to the George W. Bush Institute She served in the Bush administration as assistant to the president and chief of staff to First Lady Laura Bush Prior to that she had also served in many other capacities at the White House and USIA and the Department of State and of course these conferences all flow from her vision and her hard work and we're deeply grateful to you Anita, thank you for being part of this And then we have Richard Norton Smith. Richard is a tremendous historian, biographer, commentator. He's currently a scholar in residence at George Mason University But in the past he served as director of the Hoover, Eisenhower, Ford and Reagan libraries As well as serving as the director of the Dole Institute and Abraham Lincoln Library Richard's been a longtime friend and mentor and we greatly appreciate his being here today. Thank you, Richard And finally we have our moderator Dede Myers Dede currently serves as managing director of the Glover Park Group a public relations firm And it's very active as a political analyst and commentator She of course served as White House press secretary during President Clinton's first term Dede is a great honor to have you here and thanks so much for taking part with that I'll turn the proceedings over to you. Thank you all very much. Well, thank you Alan and thank you to all of you for being here It's an honor for me to be here as part of this Conference looking at the changing role of First Ladies across history as we all know the president's role As head of state is enumerated in Article 2 of the Constitution and expanded upon in the Federalist papers and other places But there's no equivalent Document historical or otherwise that really lays out the roles and responsibilities of the First Lady And so from the moment George Washington put his hand on the Bible to swear the oath of office That role has been evolving although the First Lady has continued to serve as the president's spouse mother of his children Helpmate confidant advisor and all those things and over time Obviously that role has changed as each First Lady has had the opportunity to define it and redefine it for herself But and certainly over over time There have been big changes and Richard I want to start with you to be sure the presidency itself has become so much more complex Over two centuries and so it has there been an in an equivalent increase in complexity in the role of First Lady And how has that changed? Well, it's a great question. I think In some ways the the position of First Lady is even more Fitted to the contours of the personalities the preferences the experiences the strengths and weaknesses Of the occupant than is the presidency In in in ways that it it hasn't changed all that much in 200 years of Martha Washington presided over not only weekly receptions Known as levees every Friday night in New York, but every Thursday. She presided over a state dinner She was in effect a political hostess and for her just as for Washington The presidency was an extension in many ways of his executive role during the war for her It really wasn't all that much of a change Abigail Adams on the other hand who made no secret of the fact that she was not only her husband's partner but his political partner Was was mocked in the press as Mrs. President and in fact in some ways the story of this job over the 200 years Is that tension between the purely domestic and for lack of a better word the activist Clearly in the 20th century. I want to rouse about and other first ladies have have tipped it toward the latter Very quickly, I but I can't come here without Tipping my hat to to woman who I think in many ways is the role model for first modern first ladies someone who was both a traditionalist and an activist who was an active participant in the legislative drama of her husband's presidency And at the same time a symbolic Figure to the country and that of course is is Lady Bird Johnson It's interesting when Laura Bush was asked during the the interregnum between the election and the inauguration 2000 who are your role models? She replied her mother-in-law, which was politic And no doubt sincere But also Lady Bird Johnson, and I think probably Lady Bird Johnson is a role model for a lot of folks Both Anita and Lisa you both worked for first ladies Did they have role models that they look back through history and and have first ladies that inspired them or guided them Anita? Well, I think that all first ladies the answer is yes, of course But the all first ladies are very conscious of those that have come before them And we talked about this a little bit today in the in the teacher session as well and with Linda and with Lisa You know, you're surrounded by those that have come before you you look at their portraits Every day you you recognize that you know a lot has happened in this house before you got there and what will you do with this? temporary Privileged opportunity that you have to really make a difference So you you know they're all very cognizant of those that have come before them But they also still bring their own Interests in their their own character their own passions, you know to the role and are able to Really support the president in his work yet still be themselves. What about Hillary Clinton? Well, I think it's very public that certainly Hillary Clinton channeled But but obviously She read a great deal about Eleanor Roosevelt and it took a lot of Great inspiration from Eleanor Roosevelt. She also Would talk a lot about Edith Wilson and how Edith Wilson was was a real factor in Woodrow Wilson's presidency And in fact was was writing speeches and then obviously Mrs. Kennedy Played a very Influential role on Mrs. Clinton. They they spent a great deal of time together before Mrs. Clinton went to the White House and That was really about how do you raise children in the White House and protect their privacy? Which was something that was incredibly important to both president and mrs. Clinton and I and I know obviously to President mrs. Bush and I think that that was a very important Mentor mrs. Kennedy was in some respects a mentor to mrs. Clinton in terms of how you raise children in the White House How you protect that zone of privacy as she would say And allow the children in the White House to lead as normal a life as possible No problem, right? Right But anyway, so so Alita the then that raises, you know some history as history is marched along The culture has changed and that has changed our expectations for first ladies But at the same time first ladies have often pushed the culture as I think Eleanor Roosevelt did How do they do that? Can you talk a little bit about the effect of that? Well, I think one of the things that Eleanor did that nobody could do today It has nothing to do with skill. It has to do with the culture I mean Eleanor was the third most syndicated journalist in the United States She had her own radio show. She wrote a syndicated newspaper column that appeared six days a week That put her on par with Walter Lippman and Dorothy Thompson. She held her own press conferences She was the first first lady to hold a weekly press conferences and courted the press She also not only courted the press that covered her but the press that covered her husband So she had both flanks covered. So she understood, you know, she understood radios She understood print, but she also understood accessibility and one of the things I think that also could never happen again today to my great if I could make up a heart You know a word a heart sickness. I mean I just Eleanor refused Secret Service Protection And she traveled everywhere unescorted And she did that because she believed that that would interfere or impede with her conversations with the American public And so justice today, you know President Obama has the 10 letters Eleanor read 150 letters every night And responded to them So her ability to change culture not only by, you know, dictating her coverage in the press But also maintaining huge administration long personal confidential pen pal correspondence with people that she never met Also traveling around the country unescorted Built a reservoir of goodwill for her that really just elevated her up on a pedestal in her own right Was she conscious of that at all? Well, to some extent she was I mean what if I could circle back to a question that you guys were answering I mean Eleanor went into the White House Not wanting to be Ellen Wilson not wanting to be Edith Wilson not wanting to be Lou Hoover And so the issue was how was she going to find a role that didn't eat her? That's her word And so, you know, she has to negotiate with FDR what these roles will be because she makes three offers and he says no And so, you know being the shrewd little political operative is that she is in her home right She says well, let me get my own constituency, honey, and I will show you that I can be good press for you You know, and that's what she does I mean, Lisa I think it's also fair to say that Hilary Clinton was on the leading edge of cultural change going on in the country She was the first First Lady who came with a career really She was the first to have an office in the West Wing for example How did she change culture and was she conscious of wanting to try to do it? I don't know that she was conscious of it coming into the role And you know, she approached the role of I'm just gonna I'm just gonna do the best job I can and be true to myself and so so much as we look throughout history. I think the role of First Lady Um Culturally I think we as a as a country have tried to to create a box and put our first ladies in that box Oh, they're gonna stand for beautification. Oh, they're gonna stand for say no to drugs They're gonna stand for education and and that certainly wasn't what Hilary Clinton was about and so You know, she came into this role And we were very upfront about the fact that she was going to play a policy role that she was going to be An advisor to her husband on policy. We were upfront about that in reality if you look throughout history Practically all of the first ladies have have had a role in influencing policy It just wasn't at the forefront the way Hilary Clinton was on health care, for example On welfare reform and and and other issues As it relates to children women families So yes, she was the first First Lady to have an office in the West Wing But she also as Anita and I were talking earlier to a to a class She represented and I think still to this day in some in some respects A dialogue that goes on in our country about the role of women the role of working women She's a dialogue at every dinner table. Do I want my daughter to grow up? And have a career and be financially independent and secure and and maybe not get married Or do I want my daughter to grow up get married have a family? Or or is she supposed to be somewhere in the middle and have both and can you do both and do both well? And she represents that dialogue certainly when she came into the role of First Lady she did And I think she became a Rorschach test has that has been reported widely It's a phrase we've all seen and and she kind of represented that dialogue And I think to this day as you seen her transform Into senator into secretary of state And what happened in the 2008 campaign? I think still There's a lot of dialogue About the role of women in society working women in society That hillary clinton represents Certainly as it relates to women in corporate america, but also women in government women who are elected And I and I think that You know, fortunately we have her to to be an ambassador for for women and Who who work and who have careers but also have Children Because it's it's a question that is as old as abigail adams sarah poke Experience this she was also called mrs. President because she was very much of a partner political partner with her husband At what point and how did each of your first ladies deal with the the question Who elected you anyhow? Well, see with elinor that came up way before the campaign I mean, it was it was a 1932 campaign button The new york times did full page stories on her as who was going to be The influence I mean even f dr's close advisors said the first thing we have to do is get the pants off of elinor and on to frame You know so so And that was his feats writer, you know, so So people knew that I mean what what um You know, um elinor just knew that she was going to be a lightning ride Accepted that responsibility thought it came with the job description And said that the nation was in such turmoil that it needed all the leadership it could get And so she really didn't care as long as she felt like she brought dignity to the office You know, I think if I can add to the thanks for being a moderator No, no, no, this is terrific, but um In response, you know, uh to that question I think it is it is still a challenge for the the position this undefined Position there's no statutory authority. There's It's what you make it. There's no salary You're a volunteer You're the nation's premier volunteer to take this platform's privilege You've been given and and do something with it But I still think there is a fine line when you go from being policy advocate to a policy maker Because that's not really Your role and you're not elected to do that and there is a fine bound And I think, you know in all fairness, of course, you you you saw that in 1993 and and there wasn't country wasn't quite ready for that activist role, although they wanted the Mrs. Clinton to use her position, of course And I think, you know, and if I could just fast forward to one example of You know Laura Bush and she came in To the White House, you know, she was typecast early by the press You know when she was asked You know who you're going to be Hillary Clinton or Barbara Bush and she said well, I know Laura Bush pretty well and I think I'm going to be her and You know what because she was she had a career as a teacher She had a career as a librarian. I mean this defined her passion And in education and she was a champion and an advocate for no child left behind Which was a signature domestic policy initiative of for the president So yes, you know as I as perceived as a conservative traditional woman She was extremely comfortable in who she was and when she went to travel overseas I'll leave you this last example particularly in the Middle East Where she took a groundbreaking partnership on breast cancer Research and awareness She really believes and I agree with her that she was able to go into societies closed societies like Saudi Arabia And talk about breast cancer with king Abdullah and with women there because she was perceived as a conservative woman So, um, right. So obviously each of their So their their the way they're perceived affects how what they can do right the scope of what they can do But each of you is alluded to the fact that either overtly or covertly these first leaders are influencing their husbands all the time Right, and they're sometimes they're trying to move them along and sometimes they just have a passion for something Richard talk about the way and we see him as a culture to have been more comfortable with the with the covert Influence, but it's always been there from the very beginning No, it's true. Um, you know, mrs. Washington burned all of their correspondence And god what what we would give You know, uh to to to have that because she didn't want people to know how much influence she was having No, because she I think it was her final act of rebellion Against a world that had intruded on their privacy Uh for 40 years, um, they're they're love letterers.  Uh any event? Um, you know the famous well best Truman You know the famous story harry comes downstairs one morning And she's throwing chucking papers in the in the fireplace. He said what are you doing? She said i'm burning your love letters And and he says my god think of history. She says I have But you know, so not only do um first ladies put their stamp on history, right? But they're the role of being first lady changes them So we need to talk a little bit about about the changes that you've seen in not just, uh, ellen rosevelt, but her but but also, you know, certainly Um, lady bird johnson didn't come to public life Easily she did not that was not something she wanted and yet as first lady she became a fierce advocate for her husband and a great You know was the first woman to campaign first first lady to campaign on her own And for and for controversial issues like civil rights, and I think the um, you know They really liked each other first of all ellen rosevelt lady bird johnson and um And they both took heroic stands on cultural issues that really changed the landscape of the nation I mean no people dealt with race as forthrightly really as um ellen rosevelt and lady bird johnson and certainly As much as I would throw my body in front of a train for people that are talked about up here Few people put their lives on the line as much without protection For race as those two. I mean ellen r had the largest fbi final american history There were assassination attempts on her life for her stands on civil rights, which she came through gradually one of the most unsung courageous acts I think in Post-war presidency is lady bird johnson's lady bird express You know when president johnson has decided that by god, we will have the civil rights act of 1964 He certainly knew that it was the death knell for the democratic party in the south And he and lady bird knew how to sell it And you know with her grace and charm and conviction, you know, she says No southern man is going to stand up the first lady. So put me on a train You know and she takes the train That basically follows the train that the freedom riders took and there is dynamite on that train Tracks and she knew it. Did they stop the train? No, she kept going and so every time I think that people don't vote I go furious not just for the whole intellectual civic freedoms reasons But I think they spin on those two women's graves And lady bird johnson really needs credit for that because we would not Have gotten critical votes to pass that bill if she had not done her work on that train And your mama had guts Turning to to to current politics Someday one hopes right we will have a first man How will the different gender change the role of first spouse first companion first advisor richard Have you thought about this? You know first of all, I don't think it will be I think the media will go wild My god, it's the perfect 24 seven news cycle cable tv god I just think of it, you know, they can speculate for months and then they can overanalyze for months after it happens And meanwhile history will roll along and we will all adapt Think of denise thatcher in great britain in a curious way It might actually be a throwback. We started talking about martha washington The first lady was originally envisioned 200 years ago as someone who above all Was there to provide support emotional and otherwise for for her spouse It might very well be that the first Gentleman is in a similar role He had a very heavy have a career of his own I guess what we'll be talking about independent men, uh, then I I don't know, um, maybe He he probably won't have to host a weekly reception And I think there are a lot of first ladies who would envy him that but you know what I think we'll we'll all adapt just fine But you said it's it's the expectation that the first first man will have a work outside the white house Would it be possible for a first lady to work outside the white house as jill biden has done as as the spouse of the vice president? What do you think? Yeah, I think I think that anything is possible. I do again. I believe the institution of the presidency will adapt Um to whatever is happening at the at the country at the time It just sort of you know knowing the position of watching it in the bird's eye view that I had of it of several first ladies The demands are so much greater than people really realize and the schedule is so intense And you could go from planning a state dinner one minute To planning a secret mission to kibble the next which you know laure bush did three times and so It would be a little difficult I think to have work outside the home that was full-time work outside the home Um, because I think we've also all come to this expectation that this position The president's you know partner, uh, whether it's male or female um Does something with the platform that they have in service to the nation Uh, so that that so it's really become kind of a two-person job Well too for the price of one which became a very famous line Certainly, um, but but I think one of the things I keep hearing and I and I think is becoming a very consistent thread is Because this role is not elected and because it is a platform and because it's ill defined These these women who have been in these roles each and every one of them has to have a tenacity a perseverance To define it As she sees fit in a way that's going to help the country and help her husband Um, and and I think when you look at hillary clinton as a case study, certainly she She she has grown not only From being first lady of arkansas But she grew in her tenure as first lady from day one until the day she was elected to the senate She grew as a senator and now You know will likely leave the office of of secretary of state with probably her her highest A job approval ever which to me is just fascinating because it is who she is She is a workhorse not a show horse, but what was I would like to have everybody harken back to is her A popularity would escalate when she was first lady when she was portrayed as a victim Just opposed that to today Where where she her popularity is is higher than ever because of her job performance in her own right So we've evolved as as has she and I think for the first time the country is really seeing who she really is As a person And as as the workhorse that she is but doesn't that lead doesn't that speak to attention between how we view women's roles? Well, I think america is changing I mean just like our electorate is changing our cultural expectations of what the presidency is and and our responsibilities to one another changing and You know and the presidency is such An extraordinary Office, I'm just not talking about a position. I mean a president needs, you know a top flight chiefs of staff God knows they need top of the line press officials You need people also who can say to you without fear of being fired honey You cannot do this Do you don't you know you got to sit back take a breath and look at it and I think as our relationships as couples And as friends change That overshadows some the presidency because if you look at the presidential marriage At you know regardless of what people say gay straight black white, you know, wherever you live in the country It's about a relationship And your relation your personal relationships are gonna color how you see that and I think our Our country has moved so forward with you know Wanting to be leave it to beaver and loving the leave it to beaver values But realizing that times have changed and that struggles involved We understand that tension and so I am optimistic Yeah, um, you know, we just referred to the relationships with the people of the american electorate But what about the relationships of the first couples and how has that shaped the role of first ladies over time? Well, that's a great question because of all the undefined unspecified roles Assumed by a first lady I mean every marriage is unique obviously But one of the things that first ladies successful first ladies do is to compensate for their husband's weaknesses I've always thought nancy reagan frankly didn't get the credit And I think his time passes and we have more access to papers and the like I think we'll we'll realize more and more Just how significant a role behind the scenes she played for example in in personnel decisions Her husband was a man who Was without guile And who tended to think the best of everyone and we want our presidents to be like that On the other hand that can sometimes be a weakness in a president and it puts the first lady in the difficult But necessary position of compensating and we don't have to name names donald reagan comes to mind Of people we're quite frankly her judgment was superior to that of her husband Mrs. Ford, you know Was very proud of the fact that she used pill old talk to get a woman into the cabinet And if there'd been a second term, I think she would have gotten a woman on the supreme court but I think this this Out of the textbook history Goes to the heart of your point about relationships and it humanizes the white house as nothing else so Each of you what quality did your first lady Kind of was a compensate for but compliment her husband Well, there is such a strong soft softness to Certainly to laura bush that was you know, these were Incredible partners. I feel so strange saying this in front of these two girls who know their parents better than you know Any of us are certainly I can opine But it was it's a beautiful strong marriage and they have done, you know, everything together from Three months after they met each other Obviously and knew that they were partners for life and have into this day even in the in the post white house life Still supporting this credible work that they felt privileged to do on behalf of the american people and are Continuing so I think richard you want to add to that. No, I was just thinking, you know that it just comes to mind that After 9 11 Yeah, each in their own way. I mean the president in many ways ministered right to the country's needs and mrs. Bush mothered That's right in the best sense of the word right a country that was particularly its children fragile The word terrified that's right country was fragile and here she took who she was as you know as a teacher as As a mother and wrapped her arms around the nation and spoke to parents You know first and said turn off the tv and protect your children. Let them know they're safe I think the nation needed to hear that and that was part of the healing process While the president now was faced with the greatest challenge to this country In history an attack on the nation and faced with the terrible and difficult decisions to protect the american people And she was doing that too How did uh hillary clinton compliment? You know, I think that that um, if you're around both clinton's you can you can see They can the the simpatico there is is incredible it's Infectious because they can finish each other's sentences They see the world the same and they they desperately want to make the world a better place and I think you know former president clinton He'll be in a room like this and he will want to meet every single individual in the room and he'll want to hear what every individual says and hillary clinton Is very methodical and very analytical and and in a very interesting way that's such a strong compliment in the way they They integrate With with one another and I think um, you know, certainly during his presidency obviously she was A strong voice on policy, but What she became was a goodwill ambassador globally for women um for the president um and and really transformed that agenda for the clinton administration um in a way that I think is quite historic and um she became Almost the mother figure for the world for women for women's rights um as human rights and for Advocacy for young girls if you educate young girls you lift a village out of poverty so Again, I think that these first ladies in their own way become that Embracer of either the country or the world on particular issues That there's a demand for either domestically or globally You mentioned, um, hillary clinton's advocacy on behalf of women and girls around the world But each first lady develops her own portfolio of causes And how does that come together richard? Is that part personality part agenda of the president? What are the factors it's often like so much about the job? It's uh, it's circumstances It's a reaction to circumstances Betty Ford certainly never envisioned herself as an advocate on breast cancer until it happened to her And and even then it was she was still in the hospital She she turned on the radio and she heard reports That about literally thousands and thousands of women who were going to get mammograms And it struck her for the first time. I think the impact that a first lady can have Which was so open about it. She was there was that was that was characteristic of her and of the ford family I mean, well, hopefully when steve's up here a little bit later You might talk about that because it was very much a family decision There was nothing that said they had to go public with this But there was never any doubt about their doing so and just think of the reverberating, you know, the the proverbial ripples in a pond To this day the impact that that one surgical procedure has had And what about for ellen and roseville? Well, I think they're um, they're two things But I'd like to dovetail something about betty ford if I could I think one of the things that that when we think about Betty ford and breast cancer We just think about Betty ford and her impact on women You should read the letters that men Wrote her you should go to the ford library and read these letters for men who finally could go To their wives and their mothers and their sisters, you know And their neighborhood friends and go with them and the lives that it saved that to me Was more telling really than the letters that that she got for women The other thing that I would like to say about mrs. Ford that I don't think has really been Explored in a way that gives her the courage that she deserves is the nixon pardon Which is a whole nother conversation But she played a major role in that and you know will admit that in fact she did that And when she took the oath of when president ford took the oath of office, you know He held the bible and mrs. Ford repeated the oath of office with them and put her name in And when I asked your mom about that she said, you know, certainly I'm not the president But we came in in a horrific time In our country's history and I'm a citizen too and I have a patriot, you know I'm a patriot and I have my own obligations And so that's what I hope the the country can really realize About the men and women that are in this position regardless of what, you know, political affiliation you hold But to go back to ellen or you know, she was so stereotyped as f dr's eyes and ears But um, there's much more to it than that. Um FDR was very gregarious and god, he just loved being present. He'd just say I love it. I love it And and what he loved was, you know, um The politics of it and what she brought to him In a very political way was the human side of the policy Not in a way that was the feel good do good bleeding heart liberal stuff, you know But how can we make this happen? This is the knowledge that i'm getting and the second thing that she brought to him Um was the ability to make him face issues that he did not want to confront And so she would have meetings in her apartment in new york With policymakers and advocates for a specific policy and put it in this bedside table So he couldn't get away from it, you know, so they're so so so there were things like that Um First ladies, so I think that's probably pretty typical though, right? Sometimes first ladies make their husbands confront things or think about things in different ways Richard are there examples of that from your studies of first ladies? Or we were talking earlier about how betty ford helped Shape president ford's attitudes about women for example Well, yeah You know, she had some help he he he had spent a lifetime around very strong women his he was the product of a broken home His mother very courageously left an abusive father Uh with an infinite of arms left and moved to grand rapids michigan and the rest as they say is history So he was very comfortable with the idea of strong women But I do think mrs. Ford had a significant impact On his views of the international year of the woman Took place during the during the ford presidency and I think as they got older Gerald Ford unlike most of us we tend to get more conservative He did the opposite on a number of issues Um in fact, he said To someone about 10 years before he died That people had better get used to the idea of gay marriage Because it was coming And one day it would be taken taken for granted Um But on the issue obviously of abortion They were quite outspoken pro-choice And they were marooned in a party that was increasingly hostile to that position And he remained a good republican But uh, but someone who um, who believed very strongly that Individual rights entailed individual choices and that that was certainly one of them Will history reveal anything that you'd like to reveal now where laura bush changed George Bush's mind? Now I get to say that with laura bush right here Maybe you can come up here and answer that question Welcome We'll get to that later Any places where hillary clinton changed bill clinton's mind um G d d maybe you could answer that question. I I um uh Changed his mind. Um, it's easier when they're dead I actually think that works both ways So I ought to take the fifth as we say as a press secretary Yes, there were and and there remain to this day Well done now um, can I just say one thing? Yes um to to respond to um Anita's point I think that what history will show when they look at you mrs. Bush is just An unfathomable impact That you've had not only with afghan women But with pep far You have saved more lives and transformed more healthcare systems than people want to take the time to understand And I hope you get the credit And and that's a great segue. I mean, um, mrs. Bush's work on behalf of women's rights around the world continues through the library um How has You know eight years is or four years is not a particularly long time in a lifetime And so obviously first ladies go on to do many things afterwards How have the opportunities for first ladies uh to serve the country after they've served in the white house expanded and changed Richard Well, that's where I think mrs. Ford really stands out. I mean arguably I think she would probably agree that her greatest impact really came after she left the white house And above all in her the work with the Betty Ford center There's a wonderful story. She was coming down to the she and the president right after the inauguration after the car would inauguration He had accepted an invitation to speak at a dinner honoring Vince Lombardi And he was feeling kind of sorry for himself, you know post electoral funk And wondering whether they would really you know, they had expected to be hearing from a president Now they were going to be hearing from an ex president going on and on and on, you know And classic Betty Ford cut right to the chase. She said she leaned over and she took us here. I said, don't worry dear They're coming to see me anyway And in a lot of ways he was he he was very proud of that fact She got the gold medal. She got the medal of freedom before he did and she and he was thrilled And and every year at the alumni event at the Betty Ford center You could find him cooking hot dogs I mean, you know, he was a soldier in the ranks And it was almost it was almost as if on a very personal level It was a reversal of roles all those years when he was out on the road and she was raising the family I think there's a certain degree of guilt and this was his opportunity to pay back Yeah, yeah, and certainly elinor Roosevelt's greatest impact was in the years after the afterwards I mean with with her work with building the united nations Helping shepherd the drafting of the universal declaration of human rights Trying to work on the covenant for civil and economic rights, you know in that major diplomatic institution But also she had a profound impact in different ways. She not only lent her name But her very active presence to major or social change organizations I mean, she worked very closely with the NAACP. She worked very closely with the national consumers league She was She mentored young people and that to me I think is really a testament To her tenacity Because this is a woman like all like all the women that we've talked about But especially I think mrs. Bush and and secretary clinton was that they saw the worst the world had to offer I mean they saw Beyond feast famine war and pestilence. They saw unimaginable horror And had the opportunity and they could have given him to it or said by god I'm not going to take it and that's what I think elinor's great strength was was at the end of the war Seeing all the damage that the war had done to the united states And but also going to the camps going to the dislocation centers really seeing on a very personal way Going into battlefields flying in uninsulated military aircraft where she lost the hearing in one ear I mean she she saw it And you know, they could have given into it and said I can do fluff You know and cover myself Or by god, it's my country. It's my values and i'm going to give it all I got And they all did that that's that's so true what you say just resonates with me we when uh Secretary clinton was first lady And the global travel that we did we we would Go all over the world and we really became almost a moving public works infrastructure organization because We would come in and all of a sudden places would be transformed roads would be paved There would be infrastructure And it wouldn't stop like that. We would stay on it afterward to ensure that this would continue so yes, it was about Women's rights and poverty and famine But it was also the fundamentals of infrastructure that were completely non-existent in so many places in the world where we traveled I'd like to add one point to that because it does remind me of a lot of the Travels that got to do together with mrs. Bush and and and she would always say which is always really struck me was part of it was to go of course and shine a light on the on the work that's being done to help People that need our help because our leadership around the world is so important and is required to make change But it also was important to bring the message back to the american people of what their generosity Through their tax, you know hard-earned tax dollars, but also their generosity our compassion as a nation That we she was able to help bring that message back home and that was so important And that's what we really relied, you know on the media and the press to cover these trips So that we can bring that message home as a trusted translator of what they saw to the american people You just all talked about sort of the impact of seeing what's happening around the world on the first place in their agendas What does the what as as they travel? How are they perceived? How it would be over the course of the last, you know several many decades Um, certainly ellen roosevelt traveled around the world And what what did that mean to people in developing countries and other places in the world? Well, i'm certainly going to leave um secretary clinton to lisa and um But i have to tell you when i really I got What ellen are meant when i was in the favelas in south polo If you haven't been in the favelas of south polo, it's seven million people. They've been Revitalized somewhat, but when i was there 10 years ago. Well when she sent me there Um, the sewage was over my ankles You know to walk through and i was talking with women and children about housing in my you know 60 words of bastardized portuguese And one woman asked me why i was there You know and i just thought You know forget the policy stuff forget the stuff and i just looked at her and i said well, you know I work on ellen roosevelt and she would haunt me for the rest of my life Up i didn't come and this woman who still had a stain on her thumb from voting in a catholic country Could not had never heard of mother teresa Had never heard of princess diana Didn't know nelson mandela But knew ellen roosevelt And knew ellen roosevelt would do three things Wages for work school for her children And a safe place to live And i would say that's a kick-ass legacy Lisa you have a pretty big piece to bite off, but what what do you think well now now You know why hillary clinton channeled elinor She changed her to the next level I think that um, it's fair to say that that that's a legacy that's been an inspiration to the present secretary of state and You know for me to to travel the way uh Anita dead with mrs. Bush to All parts of the world It was incredible for me to see whether we were in Rural parts of japan women lining streets Trying to get a glimpse going to beijing for The world conference on women Which was interesting where? Hillary clinton gave the speech women's rights are human rights and human rights are women's rights Which was a very bold thing to do In a country obviously that's got issues Uh with women's rights and the I can remember going up escalators in the hotel and women Thousands of them in lobbies trying to touch her, you know on staircases and what what a Uh a rock star really she was because she embodied hope to them And when we would go to different parts of the world and we would have Either breakfasts or or lunches small very small with Influential women in the community Whether we were in Bangladesh Or in a remote part of india, which I remember quite vividly where Ilabat who founded sewa self-employed women's association women tracked 13 hours 13 hours to come to a small community meeting With hillary clinton and ilabat now I I'd never experienced anything like that before Or you take it to um a cosmopolitan city like tokyo where we would meet With a member of the parliament Women uh women who may have owned a television station And you'd hear their perspectives and their struggles as women in their own cultures And how they would look to her for advice and counsel and inspiration And I think she has carried that mantle Through her tenure as first lady obviously into the senate and now as secretary of state and I expect that she'll continue it After she leaves her role at the state department Thoughts from travels with mrs. Bush Most incredible experience that I personally have and all of us who traveled with mrs. Bush Got to see Just how important American leadership is Around the world and how by sending the first lady Who's the closest, you know person? To the president how much that mattered and how that was welcomed Only by the individuals in the communities that you would meet but also by the leaders of those countries and The relationships that the first ladies the the counterparts or each other That got to meet and work with each other and some of that work, you know can still continue When they're no longer in their official position But it it is just an incredible opportunity again a short term that they get to do something like this and They don't want to waste a minute of it. I think that was my My takeaway from that experience Closing thoughts richard was almost out of time on maybe the meaning of the first lady lady to the american people Well, let me sort of add an action. It's a nice bridge Because I mentioned someone whose name we haven't mentioned And who frankly partly because she was so effacing has never really I think gotten the recognition she deserves and that's pat nixon Um in 1970 there was a terrible earthquake in peru And mrs. Dixon. I think like somebody americans felt helpless in some ways, but She was in a position to do something to help and she wanted to and she went to her husband said I wanted, you know And and basically eventually they got a plane full of supplies From the american people and mrs. Dixon went with it on her own Um and went down there and went out into the stricken areas without a lot of photographers without an entourage She and the the wife of the the president of peru was very moving Um, and there are these wonderful photos. You know, she went to africa Um and on an extensive diplomatic mission for the president and there are these wonderful photos of her in native gerb Dancing in my very yeah And she frankly she looks more comfortable than she sometimes did on the campaign trail at home And and I remember when she died the president said something the effect that you know People may not speak the same language And they certainly don't speak the language of diplomacy But they they know love When they see it and in some ways that brings this full circle from the the social hostess thing of lady washington The role of diplomat I think is one that has been added in and in a permanent way Definitely, um, there's one other name and we are out of time, but that hasn't come up and that's michelle obama So any thoughts on how she's Put her in bad. You weren't halfway through her legacy, but you know, I it's funny I think she has walked the tightrope Extraordinarily well and when I say the tightrope one thing that has happened as our first ladies have become More and more public property Part because of the media They have higher approval ratings in their husbands in the bowls They have memoirs outsell their husbands Like a bigger fancies around And and and that's because they're not seen as partisan figures that Paradoxically makes them enormously valuable great political assets to the administration and mrs. Obama is a classic example She is well on her way to Eleanor like status. She's a rock star Uh, and she managed anyone Watched that speech that she gave to the democratic convention. She walks the tightrope very skillfully I'm just mentioning one thing because I know we're at the end But in true bipartisan fashion and we've covered the gamut of first ladies both Lisa and I are mark noted are in i'm in red. She's in blue So We wanted to convey the spirit of bipartisanship and talking about first ladies they've all And Alita's independent in purple No, it's been great But it is it is something that people regardless of party Can feel good about and that is one of the many Wonderful legacies of our first ladies and with that we really are out of time. Thank you all for this fascinating discussion about the legacy of first ladies