Today, let us turn to a topic of some importance in the study of dynamical systems and specifically I want to show you how probabilistic methods help you to discover things about complicated dynamical systems in particular systems where the dynamical behavior is perhaps chaotic or something like that. So, the specific things I want to discuss the following I want to discuss. So, Joan escape and recurrence in deterministic dynamical systems I will explain what I mean by this gradually. So, we are going to talk about classical dynamical systems for which the rule of evolution is specified there is no randomness no external imposed randomness no thermal fluctuations quantum fluctuations or no sources of fluctuations, but completely deterministic dynamical systems and the kind of systems we have in mind are either in discrete time or in continuous time it does not matter and specifically just to be very concrete about it. If you had a dynamical variable x which perhaps stands for x 1 up to x n say n dynamical variables in the context of mechanics these could be positions and moment of various particles or whatever angular momentum we do not care some dynamical variables of this kind and their evolution in time is given by a set of differential equations and the differential equations look like x dot first order differential equations. So, you have a sufficient number of variables x 1 x 2 up to x n to describe the system completely and then the idea is that these change with time according to a rule of the form f of x possibly dependent on time as well and this dot stands for the time derivative this could be a very complicated non-linear function of x and t and x is the set of dynamical variables here in particular if it is a Hamiltonian system or a Lagrangian system then I have generalized coordinates in generalized velocities or momentum and they would all come under the umbrella of these dynamical variables. If I have an explicit t dependence here the system is said to be non-autonomous that means the rule of evolution itself is changing with time and it is a little more complicated than if you did not have that we just had f of x. So, we will remove this and we will consider systems which are autonomous which means that this t dependence is removed. So the time evolution rule is translation invariant in time it is exactly the same rule now of course this is a very very general problem and in general we do not have closed form solutions for it for arbitrary non-linearities and the whole subject of dynamical systems is concerned with solving this problem and seeing to what extent you can extract the information now to cut a very long story short the way to do it is to define a phase trajectory in phase space namely the space of all these x's some phase space and for simplicity let us just take it to be n dimensional Euclidean space then the state of the system at any point is specified by giving x at any instant of time let us say x at time 0. So a point in this n dimensional space tells you all the values of the variables the way of values of all the variables at initial instant of time and then once you solve the set of equations in principle you know the values at later instance of time. Now the explicit solution in closed form may not be possible in general the function is non-linear it has a large number of variables this is impossible in most cases. What is possible however is to do numerical integration of this equation so that you can instant by instant if you have a sufficiently small time step you can actually numerically integrate and find out what happens at any later instant of time you can join all those points together and you have a phase trajectory this word phase space is borrowed from mechanics borrowed from Hamiltonian Lagrangian dynamic Hamiltonian dynamics so we call it a phase trajectory. Now if the system is autonomous in this fashion then under suitable conditions on f of x the solution is unique namely if you specify the initial value x of 0 it is a well-posed initial value problem and in principle the solution is unique okay which means that as time evolves this trajectory which is meandering around in this n dimensional phase space cannot ever do a thing like that it cannot intersect itself because if it did so and you had an intersection of this kind then you see once you reach this point the future is unique there is only one outward trajectory with arrow pointing outwards that you can get from here because I could start with that point as the initial condition and the future has to be unique but if this situation is permitted you have this and you have that so this is not possible not possible a trajectory in an autonomous dynamical system a phase trajectory cannot intersect itself nor can it intersect any other phase trajectory nor can it intersect something where the initial condition was that and this is some other trajectory this too cannot intersect that because at the point of intersection the future becomes nonunique but this is not much of a constraint because when you have an n dimensional space and n is large ball of thread winding around the space will hardly ever intersect itself will never do so in general okay there is only one case in which this is possible and that is if the trajectory goes out like this in this fashion comes and joins itself again in this way and that is perfectly alright but this means that if you start here after a certain finite amount of time it comes back here and then again because of the uniqueness theorem it is going to go in the same direction and come back here repeatedly which means that a closed phase trajectory a simple closed curve is the only thing possible means the motion is periodic because all the dynamical variables have returned to their initial values right so the two things we know right away are that for an autonomous dynamical system phase trajectories cannot intersect themselves or each other and the second thing is every periodic motion is necessarily a phase trajectory a closed curve is a periodic motion and vice versa a simple closed phase trajectory is periodic motion okay and all periodic motion every periodic motion is described by simple closed phase trajectory okay now look at the fact look at the simplest example you take a harmonic oscillator for instance then the phase space in this case the equations are trivial to write down you have x dot is p over m and p dot the rate of change of momentum is the force on this particle which is equal to minus m omega squared x where omega is the natural frequency and m is the mass of the oscillator p is the canonical momentum conjugate to the position x and these are the two equations of motion so in this case this vector x this the phase space is two dimensional this comprises x and p okay and it is part of this kind of set of equations but this is even more special than that what is so special about this right hand side here they are linear it is linear in x and p so therefore the problem is solvable in closed form no complications occur here at all and of course we know immediately that in the phase space of x and p so here is p here is x the phase trajectories are all simple closed curves they are ellipses they are ellipses corresponding to the fact the ellipses are also written down explicitly these two will imply that half m omega squared x squared plus p squared over 2 m equal to a constant e which in this case has the physical meaning of being the total energy of this oscillator okay and that is an ellipse in general so in this case all motion is periodic for a simple harmonic oscillator no matter what positive energy you specify the motion is a periodic motion and the energy determines the amplitude and different energies would correspond to different ellipses so you see in this case the phase space is a phase plane and the whole plane is laminated by these ellipses each ellipse corresponds to a fixed energy okay and there is one very special value of the energy which is 0 which corresponds to a critical point an equilibrium point because at that point the right hand sides of this vanish so both x dot and p dot are 0 and if you start with 0 value for both of them then you are going to remain at 0 value because these are first order equations so this point here is a phase trajectory all by itself it corresponds to the equilibrium point okay. So this problem is utterly trivial here now you can ask the question when does the system come back to its original state does it recur and here because the motion is periodic every initial condition recurs it is very clear that once you are here you are going to come back to this after a finite amount of time the time period and detour for the other thing and so on. So the motion is periodic in this case all motion is periodic so in that trivial sense every initial condition will recur after a fixed amount of time there is something even more special in this problem about this periodicity and what is that so let us suppose this energy is some energy E1 this is some E2 and so on all of them are positive numbers different numbers and the larger the energy the bigger the ellipse because the bigger the amplitude what is special about the recurrence time in this case the recurrence time is just the period of every orbit what is special about it in this particular pardon me it is independent of the amplitude that is not true in general it is independent of the amplitude only for simple harmonic motion for any motion other than any periodic motion other than simple harmonic the amplitude depends on the time period depends on the amplitude. If I drop a ball from this height and it is a perfectly elastic ball perfect elastic collision with the bottom of the floor it is going to come right back to this and the periodic motion but if I drop it from a greater height the time period changes. So every time you have a potential which is not the harmonic oscillator potential you have an amplitude or a time period which depends on the amplitude or the total energy whichever way. So this is a very very special problem this case is not the most general cases much more complicated than this is a very special case our focus is not on dynamical systems here but we want to look at recurrence properties in particular. So the first kind of recurrence you have is periodic motion as a very trivial kind of recurrence once you solve the dynamical equations you compute the time period that is the end of the matter here but already if you complicate this a little bit and take two simple harmonic oscillators let us say a particle moves in a plane and it is connected by a spring to the origin and it moves in the plane that is the combination of two simple harmonic oscillators at right angles to each other then already a complication starts arising in that case they you have got an X and a Y coordinate. So you have a four dimensional phase space and if I plot I cannot plot this phase space because it is four dimensional what I can do is to plot X 1 P 1 P 2 X 2 and I write down the usual total energy or Hamiltonian of this oscillator two different oscillators at right angles to each other the total Hamiltonian is P 1 squared plus P 2 squared over 2 m plus the potential energy this case and this potential energy is of the form half m omega 1 squared X squared plus omega 2 squared Y squared or so take the scale of one of them to be one so it is X 1 squared plus omega squared X 2 squared in general right. So if I took this oscillator I have set the frequency of one of them to be equal to one and the other one to be omega some number omega there is no reason why the frequency should be the same in the two orthogonal directions is this motion periodic is this motion going to be periodic it depends on omega it depends on omega if omega is equal to one then of course it is periodic immediately what happens of omega is 2 it is still periodic because the two frequencies are commensurate with each other. So as long as omega is some rational number the motion is periodic but the moment is irrational the motion is no longer periodic it will never come back to its initial point what will happen that will come arbitrarily close to its initial value every initial condition will come arbitrarily close but never come back to its exact initial value the motion is said to be quasi periodic in this case and you can sort of tell what is the nature of this quasi periodicity by saying that basically what I want to know is moving around the trajectories look like this the sections of the trajectories look like this. So what is relevant is once you specify some initial conditions such as the initial value of the total energy of this oscillator and that you specified these ellipses then the question is if you start at this point and you go around you start here and you go around you do not come back to the starting point there when you finish a revolution here and if the frequencies are incommensurate there is no time at which you will come back to exactly the same values for all the four dynamical variables right. What is relevant in this case is the angle so it is sort of making by changing units you can make these circles like then all you want to know is if the angle here is theta 1 and the angle here is theta 2 the state of the particle the values of the variables at any point at any time can be written in terms of two angles theta 1 and theta 2 where are you on these circles for given values of the energies of the two oscillators. So you can write that as a point on a torus in which this angle here with some reference direction with respect to some reference direction gives you theta 1 and in the cross section here of this torus you get theta 2 basically you want to know your phase space is now reduced to the space of two angles each running from 0 to 2 pi. So at every value of theta 1 you have another variable which runs 0 to 2 pi so it is like a doughnut surface of a doughnut right and this torus on this torus the phase trajectory is doing this kind of thing it cannot intersect itself it is an autonomous system and if the frequencies are incommensurate it is not hard to show that any individual initial condition will wind tightly around this torus in infinite number of times never intersecting itself but coming arbitrarily close to its initial value and there are theorems which will tell you what will be the density with which this typical point will cover this torus and the answer in this case is uniform uniformly covered. So now you could ask what about recurrence what about if I start with a small patch here set of initial conditions and I let the system evolve since it is quite regular it is not diverging or anything like that this patch will move around neighboring initial conditions will remain neighboring initial condition neighboring points as time goes along but this patch this little patch will move around this torus and so on and you could ask when does a typical trajectory come back to this region. So if I specify a small cell in this phase space I could ask a trajectory which starts at this point how long does it take to come back that is it how long on the average does it take to come back so we already have our first example of a recurrence problem in which you have quasi periodic motion and you are asking what is the mean time of return to this patch sometime it will of course depend on the size of this patch possibly it will depend also on what is the resolution with which you made this patch etc. All these details can be worked out but the question is what is the mean time over which it comes back this problem has a closed answer it has a straightforward and simple answer based on number theory which we will not talk about right now but you see the first of these recurrence problems but we want to go to a more general case I want to go to the most general case possible and say you have a phase space some complicated dynamics is going on there is a complicated phase trajectory going on and now in this n dimensional space I identify some small cell and say suppose I start in this cell the system starts in this cell initial condition specified to some resolution how long will it be before it comes back will it come back at all is a question and if so how long will it be on the average before it comes back so it is a very very general question of which these are very simple examples this is a quasi periodic recurrence and so on but we are asking an extremely general question and here is how one can make some headway in answering it so schematically let us suppose that we have a big phase space since I cannot draw an n dimensional space I just draw this blob and say that is the phase space in which the system is and typically a trajectory will do some very complicated meandering in this trajectory I have shown it to be intersecting itself it is not supposed to intersect itself it is just a projection onto the plane of this trajectory so it is like a ball of wool it is never going to intersect itself but it will come arbitrarily close to itself and I would like to know something about recurrence so what I do is to simplify the problem by saying look I have some resolution with which I can identify points so I have some coarse graining by which I identify volumes in phase space so I partition this phase space into cells and this is a typical cell this whole thing is a typical cell cell in phase space and I do not keep track of individual trajectories at all I simply ask is my representative point in the cell or is it not in the cell where is it with respect to the cell so I give a label to each cell C1 C2 C3 etc and ask at a given time is my typical trajectory is any trajectory that I am following in which cell is it so I work to that resolution and I also do this with some sampling time I generally do not look at the trajectory at all times I have no information I do not solve this problem exactly I simply look at it as a discrete in discrete time with some time step then the dynamical problem that I have becomes slightly different it becomes not a flow but a map so now what I have is a situation where the typical point X at a given time n is equal to some function some nonlinear function in general of this point at time n minus 1 so this is discrete time in other words instead of following the trajectory I am following this these are the points where it is at discrete time steps okay and then I ask where is this which cell is this point in etc so now I have essentially I have replaced a continuous trajectory by a symbolic dynamics I simply say that my point to start with is in some cell C let us call it C1 it jumps to cell C3 then it jumps to C7 then it jumps to C6 etc so I have a string of letters following a typical trajectory saying it is in this cell or that cell or this cell or that cell and the question asked is what is the mean time it takes to come back to the original cell where it started from in the most complicated situation possible neighboring trajectories will actually diverge and have very different histories because the system could be chaotic in that case you have exponentially sensitive in the system sensitive to initial conditions exponentially so initial trajectories which are very close to each other initial phase phase point will diverge typically exponentially till they become as far because of space space itself separated as much as the phase space itself so we will look at cases where the full phase space is bounded some bounded volume and we will now look at a case where we assume that this process has been going on for a long time there are no regions of phase space where the system gets stuck and the system is moving about in a portion of phase space or the whole of phase space with some invariant measure or probability with some steady state probability so the idea is that the trajectory has gone on for a long long time and now it is visiting different regions of phase space here with probability measures which do not depend on time this is the equivalent of saying the system is in thermal equilibrium in all the equilibrium problems we looked at right so there is an invariant measure just as the Maxwellian distribution of velocities is the invariant measure for velocities of an ideal classical gas similarly we assume there is some invariant distribution now what is meant by this distribution so this is equivalent to saying that which each cell C j I associate a probability that the system is a priori in that in other words I should close my eyes I put pen on the paper and ask P of C j is the probability that the point hits C j that the typical trajectory hits C j right what would this be actually well think of it this way if I toss a coin and I say the probability of a head is half what I really mean is if I toss the coin in n times the total number of times that I get heads to the total number of tosses approaches the limit half that is what is meant by the a priori probability in exactly the same way P of C j would be the actual fraction of time that a typical trajectory spends in the cell C j over an infinite amount of time you take the fraction of the time that is spends in C j and that of course is going to be the a priori probability that the system is going to be that the representative point is going to be in cell C j so we have an invariant measure it is like the stationary probability now in this case in contrast to noisy problems where we had thermal fluctuations or some source of fluctuations I have to specify for you the statistics of the noise whenever we did Brownian motion or random box etc. I had to specify for you the distributions each time it was put into the system but here the dynamics generates this P of C j because in principle it is deterministic dynamics and there is nothing outside that is coming in so from that from this equation I should be able to determine what this invariant measure actually is explicitly I should be able to do that right from the equation itself now how is that done okay and that is a little trick it is called the Frobenius Perron equation and let me tell you what it is because idea is extremely simple suppose you say that at any given time in n the density of points in this space at any given time let us suppose after when you have this invariant measure is some row to start with I specify an initial distribution of initial conditions and let us suppose that is row 0 at time 0 of x and I ask what happens after one time step what happens to this density well if you start with the point here and it is on a trajectory going there after one time step it is there if I start here and it is the trajectory going here it is here etc. So this density spreads out into some other density after one unit of time let us call that row 1 so I want to find out what is row 1 of x given row 0 of x how would one approach this well the point the point y goes to the point x in one time step where x is equal to f of y by this rule so if you give me a point y it goes to f of y and I call that the point new point x at time 1 right therefore it is clear that row 1 of x must be equal to an integral over dy row 0 of y you are integrating over the initial distribution but each y goes to an x which is given by f of y so this has got to be multiplied by a delta function of x minus f of y there is a delta function in all the variables because that is all it can be we know where each y goes deterministically completely therefore the distribution has to be a delta function and now you tell me the initial value itself was uncertain there was many many possibilities there was a distribution you prescribed and therefore that distribution goes to a new distribution which is equal to this given by this now this is a tall order to compute what this is because remember that you are integrating over y but the delta function is an x so you have to invert this function and in general it is non-linear so it is multiple value so this is where the complexity is buried in this doing it even in the simplest one-dimensional cases is very very non-trivial for non-linear functions okay but in principle that is what it is but now you can extrapolate this in time and ask what is row n of x given row n minus 1 of y again the same rule so at time n minus 1 if you had this then at time n you have to have this density now we want the invariant density the equilibrium density the t going to infinity limit so what would that correspond to what limit should I take here well time has been replaced by discrete time n so I need to let n go to infinity but when I let n go to infinity I call this whole thing this is identically equal to row of x row invariant I do not put a subscript this is the invariant density and that is got to be equal to integral dy row of y the same row of y delta of x minus f of y the same row because it is not supposed to change it is exactly like the Maxwellian distribution of velocities in this room if I take an instantaneous snapshot the particles are all distributed by the Maxwellian distribution of velocities and instant later the individual velocities have all changed but the distribution has not changed it is exactly the same distribution although different particles will occupy different parts of this distribution now and that is exactly what has happened so it is the same row of y which has to fold back under this time evolution to give you this guy here okay now you can see that this is now an equation for row of y determined by the dynamics and the dynamics is here so this specifies the dynamics completely the non-linear dynamics and this says under that dynamics find that row which is invariant find that function which is invariant this is an integral equation for row this is the kernel of this integral equation in n dimensions of course it is a singular kernel because there is a delta function sitting here and it is a homogenous equation because there is a row here and a row here same row so it is an homogenous integral equation but with a singular kernel and as you know for a homogenous equation you can multiply it by any constant and it will still remain a solution if you have a solution already so in that sense row is only determined up to a multiplicative constant how will you find what the constant is by normalizing it as always so you would also have say if this whole thing if this whole phase space is some gamma you would say integral dx row of x equal to 1 so that will fix the overall constant okay and this equation is called the Frobenius Perron equation not guaranteed that it always has a solution not guaranteed that always has a normalizable solution that it should be non-negative row of x cannot be negative being a density probability density but this thing here is called the invariant density when I multiply row of x at any point x by a volume element dx in this phase space I get the invariant measure of this cell so I get d mu of x equal to row of x a measure is more general than a density because there could be points delta functions there could be points where there is a finite probability that is one particular point and so on but we will use the loose language of the density itself and then what is the invariant measure of a given cell of the whole cell so what is mu of cj what is this equal to it is equal to an integral over cj dx of course and the total mu of cj summed over all the j is going to be 1 so now we have in place the machinery we are supposed to be given a deterministic evolution equation f of x we are then supposed to discover the solution to this equation by solving the Frobenius-Pirou equation it is possible getting an invariant density and then computing the invariant measure of each cell by integrating over the invariant density okay. Now we have assumed a rgodicity in the sense that we have said given enough time a typical trajectory will spend time in all the possible cells does not get stuck anywhere then the property of a rgodicity is equivalent to saying that the long time time average over a single trajectory is equal entirely equivalent to an ensemble average with this invariant density so this is the whole point of this discussion in other words if you have if you have some function let us call it something else phi of the dynamical variables x and I want to find out what is the mean value of this phi of x over a long time okay then what I do is to take phi of x n at time n the any representative point is at x sub n I take this phi of x sub n sum from n equal to 0 to capital N 1 over n and take the limit as n tends to infinity that is the long time average of this function of the dynamical variables some given any given for any function of the dynamical variables right but if the system is ergodic then this time average can be replaced by an ensemble average so this would be entirely equal to integral on the right hand side and what should I write here integral dx over the phase space row of x this is a rgodicity this is the property of a rgodicity that the time average is equal to the ensemble average this is the whole point of statistical methods since you cannot do long time time averages if the system is chaotic you cannot compute what x n is given x naught with any precision then you resort to the fact that you do not need it you use if you can discover the invariant measure that is it this integral is guaranteed to be that okay so that is the way one way of handling chaotic systems where there is so much sensitivity to initial conditions that any initial error gets exponentially amplified in time and after a while it is nonsense okay but if you find the invariant measure you can circumvent that by using their rgodicity property to rewrite it in this form exactly as in equilibrium statistical mechanics if you cannot find want to find out what is the pressure of the gas in this room on a wall you cannot find or track individual particles and find out what the force they are exerting on the wall is it in instant of time instead you say the system is in thermal equilibrium I discover the probability measure with which particles moving with various velocities and then I compute the ensemble average with some equilibrium and some density matrix okay in exactly the same philosophy you replace this time average by an ensemble average here so everything is contained in this flow now comes the question which we want to answer which is given this phase space and I partition it into cells here is a typical cell C the questions I want to ask are the following I am assuming that there is an invariant measure and I know this measure so I know it for each of the cells so I know Mu of C this by the way is the same as saying the probability that a typical particle is in the cell C it is independent of time because it is invariant and the question asked is suppose I tell you that I start at this point at t equal to 0 n equal to 0 what is the statistics of the time for which I stay in this cell without jumping out I stay for a random amount of time because if I start at one point I may jump out after one time step if I start somewhere else I may jump out after 10 times steps etc so I want to know what is the statistics of the stay probability the next question I want to ask is what is the statistics the rate at which I escaped from this cell and the recurrence problem corresponds to saying what is the statistics or probability distributions of the time at which I come back to this cell here so I have so John which is stay in some given region of a space then I have escape which is like a first passage out of the system the boundaries of the system and then a recurrence back to the system so we need to discover the statistics of these quantities in particular we want to find out the following I want to find the probability that in the case of escape for instance I want to find the probability that if you start from the cell at time 0 in our notation now I would like to find the probability that at time n I escaped out of this cell so I want C n escape so escape is the complement of this the rest of it let me call C tilde equal to complement of C that is the rest of phase space so here is my designated cell and the rest of phase space I call C tilde so I want to escape at time n so at time n I am out in this fashion but before that I was still inside the cell so C n minus 1 not C 1 given that I was in this cell at time 0 I want to find this joint probability for general values of n what would the recurrence probability look like so I want the probability let us call it r sub n this is the probability that I start with C in 0 and at time n I come back so C n but till then I was out so this is the probability I am after recurrence probability for the cell C and that is the quantity we are aiming at to compute on the way we will compute the escape the so zone etc probabilities but this is the one we want to compute what is the mean time of recurrence then remember we are in discrete time so n is time first I got to make sure this is normalized so I got to sum over all possible n I should get one because recurrence mutually exclusive events if I say I come back after for the first time after 3 steps that is a distinct event from saying I come back after 4 steps these are all distinct guys so this has to be a normalized probability by sum over n and then I multiplied by n and weight it with this P I get the mean time of recurrence to the cell C similarly if I say I stay in this cell C at t equal to 0 at n equal to 0 and I escape at n equal to 1 or I escape at n equal to 2 or 3 etc these are all distinct mutually exclusive events if you escape at 3 you could not have escaped at 2 or 1 at time 1 therefore this 2 is a set of mutually exclusive events for which you can assign probabilities so zone is not like that because if I say I am staying in this cell till time 10 it implies you already stayed till time 9 right so different so zone time probabilities are not mutually exclusive events so we will not be able to normalize zone time probabilities but we will be able to normalize escape time and recurrence time probabilities so that is worth bearing in mind so I will do this next time tomorrow we will write down all these probabilities and write it but let me start the initial step for this and that is the following and the entire trick lies there so is the target clear we assume we have a dynamical system with some n dimensional phase space some complicated dynamics going on there in general not regular not periodic not quasi periodic possibly chaotic we have a region of phase space or the whole of phase space in which the system remains forever once it is caught in and there is an invariant measure okay so preassigned probability distribution perhaps discovered from solving the Frobenes Perron equation or numerically all you have to do is to run the trajectory for a long time and find out what is the fraction of the time that it stays in different cells on the average and that gives you the a priori probabilities or invariant measure then having got that you want to answer this question what is the mean time of recurrence to a given cell okay and here there is a theorem called the Poincare recurrence theorem and I will state this theorem we will prove it the theorem says the following mean time of recurrence to cell C is equal to 1 over the measure of the cell in time steps of 1 and this is called the Poincare recurrence it is remarkable how general this theorem is the only thing you need to prove it is a goddess if the system given enough time comes arbitrarily close if every representative point comes arbitrarily close to every point of this phase space the accessible part of phase space then you are guaranteed that the mean time of recurrence to cell C is 1 over the measure of this cell by the way if you have a single point the measure of that point is 0 so it will never come back and that is equivalent to saying the motion is not periodic in general but it will come arbitrarily close to it and as soon as the cell has got a finite extent this is it the smaller the cell the longer it takes to come back of course if the cell is the whole of phase space the mean time is 1 because it is there already so in that limit 2 it is okay but this is a non-trivial theorem here and it is remarkable in its generality and you will see that the proof is really very elegant and does not require anything more than ergodicity but you exploit this ergodicity property carefully the first step in the proof is the following we want to focus we want to look at recurrences and so on to this cell C so let us start with the complement of it and let us define W so any complement quantity I put with a tilde on top let us define this quantity to be equal to the probability that you have a Sojourn outside so you start outside the cell at 0 you remain outside the cell at 1 dot dot dot and you remain outside the cell at time n-1 n greater than equal to 1 define these are joint probabilities not conditional joint probabilities what is W1 it is P of C tilde at time 0 but remember that we are talking about the invariant probabilities so this time argument is irrelevant it is equal to P of C tilde or mu of C tilde so that is trivial we define a sequence of numbers W1 W2 W3 etc where W1 is just the measure of the complement of the cell C that you are interested in let us also define W0 tilde by definition equal to 1 it just makes the notation easier so let us put W0 tilde equal to 1 okay what is the Sojourn probability now that you stay in a given cell say C tilde in this case till time n what is the Sojourn probability let us call that H sub n tilde because it refers to the complement it is the probability that your C tilde at time n C tilde at time minus 1 dot dot dot C tilde at time 1 given that you are in C tilde at time 0 given that now we are talking about actual physical probabilities so there would be conditional probabilities so I tell you I start in the cell C tilde in the in the complement C tilde at 0 and I now ask what is the probability that I am still there at time n and I never went into C at all okay well can I write that in terms of this remember this is up to n minus 1 and that is up to n and it is a conditional probability so if you multiply this probability by P of C tilde then you get the joint probability right so since you have not multiplied it you have to divide this guy here so if you multiply it you are going to get P of C tilde n from C tilde of 0 but you called up to n minus 1 Wn so what is that quantity going to be Wn plus 1 so this is equal to Wn plus 1 tilde but you must divide by W of C tilde which is equal to W in other words if you compute this set of numbers you have computed that what should be the normalization condition on h sub n is there a normalization condition no because sojourn is not mutually exclusive different ends are not mutually exclusive events so there is no normalization condition on this but can we say something about the Wn's if the system is ergodic it is clear that Wn must be a decreasing function of n this is the probability that it just stays in C tilde without entering C and if it is ergodic sooner or later it is got to enter C so this is a decreasing sequence starts with the value W1 tilde or even with the value 1 if you like this is a number less than 1 and decreases till at n equal to infinity it should go to 0 right if it is ergodic it should go to 0 we know that so we are going to use that fact we will use that fact with some with a rigorous theorem in analysis to try and discover what the escape probability is and then what the recurrence probability is so the arguments are very very general they do not really involve any detailed details of the dynamics at all but they are powerful precisely because they are so general and tomorrow we will do the rest of this okay.


Hydiwch, mynd i'm Stephen, I'm the Learning Technology Support Manager, so I'll lead a small support team back at the university. I'm really interested in Bites I CPD. What I mean by that is we offer lots of workshops, you pick a point in time and people can make it and some people can't. Often that's the opportunity you lost and you might get an email that says well can I have the slides? Mae'n rhoi, mae'r slide wedi ei wneud yn ddiwang. Felly erbyn i'n ddod y pethau hyn ac mae'. Felly, rwy'n meddwl o'n meddwl ychydig siaradau o'ch cy saenidol weithio'r brosesionio mewn hollig shiftot daethau'u gemmell, ar hynny'r syng yn ddaeth i'n meddwl o'r meddwl ydy'r hollig sy'n defnyddio'n meddwl sydd ei wneud.  Yn y cwrs mae gennym o'r 200 ysgol, ac yn y moment o'r 200 ysgol, ydych chi'n gweithio'r cwmpol o'r ddweud o'r disgwysg, o'r rhaid o'r meddwl, mae'n eich dwyledig. Yn y dyma, mae gennym i'r rhywbeth, mae gennym i'n gyflwyngu'r amser o'r hynau, mae gennym i'r amser, ac... Ac yna yma yw... Ynna sy'n gyflwyngu'r cyffredin, a dyna'r cyffredin cywyr. As I'm going to cover the themes that we've based the course activities around and just going to share some reflections on how things went. So this is the course layout. We picked four themes that we thought was permanent to online learning. There's more than four, but we've only got two weeks. So we just picked four. That's the top one's there. There's a section for a final reflect of activity Roedd hi eich fath i'w dda, dwi'n unaill diwethaf Cyfwyrdd yr adeiladau, a dwi'n fawr o'r meddwl i ddatblygu, a gwtaethi'n cyhoedg i ei bod o'r cyfrifio. Roedd mynd i ddefnyddio'r cwysol. Roedd ymarfer o'r ddiwethaf spotu eich werthwyr arwe 석sio, dafodd yr adeiladau. Roedd y mae'r cwc, a dwi ddysgu'n ymgyrch ar gyfer y cwysol. Roedd yma am gyfrifio'r bobl yn yr adeiladau, cfifio'r bobl yn dangos yma. Felly, we employed the restrict access. It was a terrible decision. We got terrible feedback. Things around well. I finally got time to spend on the course and now I can't move on because Section 2 is not open till 8 o'clock tomorrow morning. So, we just left all the sections open right from the very start. You can see we're using the completion progress as well. Now, the other piece of feedback that we got was from the first time that we ran it, where staff were really keen to have some kind of synchronous learning opportunity. We did everything asynchronously last time. So, we have WebEx at the institution, so we offered four sessions and the participants could just choose one of those. And the idea here was for me to give them a tour of the facilities, but also to facilitate a bit of a discussion around how they've been using this kind of technology in their teaching practice. We also wanted to hold this. A lot of course, and a lot of staff development, of course, is around modelling good practice. One of the things we wanted to promote was the use of audio in discussion forums, particularly for welcome messages. So, in week one, the opening welcome message was an audio recording. And then in week two, my colleague Julia, she recorded an audio message, which was a summary of what had gone on in the first week picking out her personal highlights. So, we really wanted to push the use of audio because it's such a simple and effective way of really enhancing the facilitator's online presence. Okay, so then on to the themes. In terms of the structure, we tried to keep everything really consistent. So, for every theme, there was three things to do. There was a resources page where there would be something to read or to listen to or to watch. And then there would be some kind of participative activity and then some kind of sharing activity. So, it was always very consistent that way. For this number one theme, so this is online social presence. So, this theme is really all about how making students feel comfortable and making them feel part of an online learning community. It really has a big impact on their success, their performance and just generally their experience online. In terms of resources, we used a podcast that we had in one of our PG cert modules. So, in that podcast, it talks about some of the strategies that a tutor can employ to combat feelings of social isolation in a fully online course. And then the participative part, we introduced our staff to Padlet. We've got an institutional version of Padlet. If you've not used Padlet before, it's a really great tool, really informal tool. And we picked the topic of memorable journeys for people to share because, well, it's very inclusive. Everyone can share a memorable journey, whether it's a positive or a negative one. And it's a topic that really lends itself well to stories, narratives and sharing of images. So, that went down really well. And, of course, we did use the obligatory discussion forum. We used that in two of our topics, not in all of them. What we're trying to do, we knew that we could arrange a participants. Some of the academics have taught online, some of them have not. We also have participants from the professional services who've never taught at all. So, we try and make the discussions as inclusive as possible by offering people a choice of perspectives to contribute from. OK, so the second of the theme was around digital literacies. We know that there are multiple frameworks for articulating digital literacies. And so, really what we wanted participants to get was a sense of what this topic meant for them in terms of their digital capabilities and the capabilities of their students. In terms of the resources, we had a couple of videos. So, we had David White's video around this notion of visitors and residents. You're articulating how individuals can interact with the web. And also Doug Belshaw's TEDx talk where he talks about his essential components of digital literacies. And we linked out to the JISC framework as well because it's got some nice top tips. So, that was the resources part. Now, in terms of an activity around digital literacies, we used the University of Exeter's eye test. So, a shout out to the university if anybody's here. Purpose of this test is you complete 30 questions and it generates a profile. And the idea of this activity was really to get the participants to reflect on how they apply their digital skills across a range of educational contexts. We then put in a discussion forum after that and some really good discussions around what student engagement looks like online, particularly around the concept of the lurker. That is someone who is participating in a course, but it's not very visible to the tutor. And I'm going to come to that at the end as well. So, the third topic was all around the kinds of coursework assessment that lends itself well to online learning. And also about the general characteristics that can engage students with the subject matter and engage students with each other. So, we're thinking things like giving students some kind of choice or control over their assessment. Allowing them to continuously engage with a wide range of resources. And also giving them opportunities for collaboration. In terms of the participative part, we based it around peer-wise. I don't know if people are familiar with peer-wise. It's a great online tool free to use. And the idea here is that students collaborate together to generate multiple choice questions. Now, I know MCQs get a bit of a bad rep in terms of just answering them. But in terms of the learning, building an MCQ question, there's actually a lot of learning in there. You've got to come up with a good question stem. You've got to come up with convincing distractors so that will avoid or mitigate against guesswork. Then come up with good explanations. And so, in this system, the students will anonymously evaluate each other's questions. So, the idea in bringing it into this course is we want to give the participants just some feeling of collaborating online in order to produce a digital artifact. So, in this case, they all are to produce a good multiple choice question. We then route them into Padlet for the reflective part. And we just set up Padlet a little bit differently, just giving them some prompts or some headings to contribute to thoughts about peer-wise and an online assessment in general. Most of the comments came around the gamification part of peer-wise.  And that score is based on how you interact with a system. Your score goes up if you comment fairly on other people's questions and they agree with that comment. Or if you engage early and so that you're the first to comment, your score goes up. It's quite a difficult algorithm to get your head around and I think that's one of the difficulties when staff see it for the first time and see this notion of a reputation score and don't really understand how it comes about. WebPA, I find, is kind of similar. It produces an algorithm that's not that transparent and it does kind of throw up kind of odd opinions and kind of behaviours. Also, it comes up with badges and so on as well. So, there is a good gamification part, although it did throw up some consternation a little bit. So, our fourth theme then was around the affordances of educational media. So, here we were introducing staff to Daniel Orlard's conversational framework and particularly around the five forms of educational media that she has described. So, that's things like a narrative media, interactive, adaptive, communicative type media, productive type media as well. So, the resources were to show how you can have interactive and adaptive media using the existing Moodle tools. So, we had a link to how you can transform a YouTube video from a narrative resource into an interactive resource by overlaying an H5P on it. Also, a link to a lesson to show an example of an adaptive tool. And we gave them some access to, well, we just provided links to some open access resources or repositories so they could search. Then, we asked them to go and find a resource or provide a link to a resource or plan a resource and bring that into the course glossary which we structured around the five media forms. And then, I would use the comments tool there to provide a little bit of feedback. The only thing about the comments tool is that, as far as I'm aware, it doesn't provide any notifications. So, the participants didn't know that they had received any feedback unless they came back. So, unless I'm missing something. So, again, we're trying to showcase the tools in different ways. So, that was the four themes. And then, the final thing was a reflective activity. So, here we were asking the participants to reflect on their two weeks online and to reflect on the themes that they had covered and how they might apply these or how they are currently applying them in their professional practice. So, we just used the assignment tool for this. We didn't grade it or anything, but I did leave some feedback. And this is where you can really find out the impact of a course is when you have that reflective analysis. So, the reflections would touch on things like how difficult it is to engage in a course like this when you're trying to deal with competing demands, i.e. the same for students. They would touch on things like it's very easy to be discouraged when completing a task when you're being faced by a technology that's new to you and you don't know how it works and how you fit within it. And also, just generally putting themselves in the shoes of students and that was really the idea of the course. We wanted them to feel that discomfort. I didn't say it in the course aims, but that is really what we were after so that they'll be a little bit more thoughtful when they're coming to designing their own. And we also got some nice feedback just around being introduced to new technologies because staff don't know what they don't know. Just finally around course participation. So, we had 17 people sign up for the course which lasted two weeks. And this is the participation from the last week of the course, i.e. when it finished. So, you can see about a third engaged fully, a third engaged sporadically and a third not at all. By the end of the first week, it was all blue. There was no green on the board. I was terribly depressed. I did send out a few emails and that did seem to then bring people in. You do have to reach out to people and kind of offline as it were and kind of cajole people in a little bit. As I said, there's a lot of blue in the board and during our discussions we were talking about the kinds of course participation that are not particularly visible to facilitators. So I had a look at the course logs for this little middle group where they didn't engage from the course, from the progress bars point of view. They didn't look as though they were gauging that much. But actually when you look at the log files, so the top one there, there's someone who has completed two of the tasks, but actually they just went in for 40 minutes, did two tasks and never came back again. Whereas there was two others there who didn't really complete the tasks as far as the progress bars concerned, but actually logged in reasonably regularly and spent a lot more time. So I guess what I'm saying is the progress bar is not telling you everything and it'd be quite nice if middle could somehow surface this invisible learning that takes place. I did say to participants that the course would take about four hours. That was a figure I just picked out of the air. I had no idea how long it would take to complete it and when I had a look at the top student as it were, they took over 10 hours. I think the message is very difficult to predict and to say for certain just how long a particular course is going to take them. Okay, so just to conclude then, those of us involved in staff development or supporting staff, it's all about modelling good practice and trying where we can to show how good you can make it look in terms of nice unstructured layouts, well-sign posted materials, good facilitation skills, always responding promptly to participation, making people feel part of the community. Okay, showing tools I think I've covered. The course duration is important. We ran it for less than two weeks last time. It was too short, people couldn't engage. If you have it too long, you're giving opportunities for people to disengage and procrastinate. So you have to think quite carefully about that. Design for inclusion, give people from a wide variety of backgrounds an opportunity to contribute from their perspective. Create a community with synchronous learning opportunities and some forms of participation are less visible than others. That is a few references there. Thank you very much. That was really interesting and if you don't mind me saying, really good screenshots as well actually. Questions please. We have a couple of extra minutes as well so we can have extra questions. One at the front. Stephen really enjoyed that. It is a lovely model of CPD, particularly for those who are not new to online teaching and learning and putting the lectures and the teachers and the students who are excellent. I was also interested in how did you get those 17 participants in and then the other thing I was interested in you gave a certificate of achievement. Did you consider digital badges at all or was that enough for people? The certificate I just put up at the end, I didn't think about a certificate but one of the participants asked is there a certificate so I quickly made one. It didn't look that great but I felt fantastic when they had asked for that. It was like this is actually meany something. I was really pleased about that. In terms of getting the first time around is providence. It's all about timing, synchronicity. You can never tell. The first time around there was a learning and teaching series going on and I was asked can I come in and facilitate a three-hour face-to-face workshop on online learning? How am I going to do that? I suggested why don't we come online, we'll do a course and then we'll do a face-to-face thing as a wrap-up. The first time around that's what we did. The second time around I got the benefit because our HR department started sending around all staff emails showing here all the learning events that are taking place. I said can you put together an email for me? Here's all my learning technology events. Part of that, part through local networks as well, really exploiting your contents and calling in a few favours that people have built up over the years. Hello Stephen. I think that was fantastic as well but a great model to use for staff time and to expand their experience. I was wondering did you hit only new staff or were there more experienced staff who came along as well? Let me just think. No, there were some old staff. No, there was definitely a range of staff. Also people who, I was very surprised because they were experienced staff. So no, it wasn't just new staff. A good mixture. As I say, some from the professional services who are new, who are just doing a little bit of student support, so again, trying to be inclusive of them. But no, a good wide range. I'm going to run it again, but I'd like to try a different course. I'm thinking of doing something around Mahara, something a little bit more skills based that gives people the opportunity to. We can take our time to take them through the affordances of that technology and give them little tasks so they can build up skills over a period of time rather than come along to a three hour workshop and we'll try and cram it all in and then you'll look at it again for another six months and you've gone. So I'm not going to make it any longer. There is a little bit of debate about how these kind of courses tie into the PG cert and we've got another PG cert called Blending Online Education and whether we can take some of that out and build little two week courses that give people a bit of a taster rather than something the last 15 weeks and it's accredited because it's not accredited even though they get a great certificate. Right at the back. Thanks very much. That was really interesting. Have you asked any of the people who are already on the course if they've used it in the classrooms at all if you get any kind of feedback after when they're going back and using it for real? Well, we just had the wrap up session last week so it was all about their intentions and some of them who are already teaching a line are saying, well, Padlet looks okay but it seems very visual. I don't think it's going to work for me I'm definitely going to investigate pure wise because I need to help build up a bank of questions so can I yes or no? Mostly intentions but one of the great things about the reflection is they were talking about things like well, I realise I've designed my assessment right I don't really need to turn it all upside down so a lot of it was about validating what they already do and that's equally available as trying something new they've got it right first or second time around. Okay, lovely. Thank you very much. Thank you.


God's beauty is all around us and my goal as an artist is to capture and interpret that beauty on canvas and to take you the viewer along with me on this painting journey. Hi and thank you for joining me. This is Painting Journeys. My name is Kitty Lynn Klisch and we're not traveling very far today, but first of all I have to tell you about our last segment when we were in Assisi, Italy. I don't know if you remember that adventure or not. I lost my credit card and oh my it went on and on. But anyway I chose to do this little corner of the garden and what really moved me about it was that the there was an arrow pointing away from the garden and I thought that on a sign and I thought that was kind of cute so I that's what I decided to paint. So this is what the what it looked like the photograph that I worked from and then after we worked on it for an hour this is how it looked. It was pretty rough and then I took it back to my studio at home and I finished it and here it is. And as you can see a lot has been added. I have punched up the flowers added a few extras. I over here I brought in a some flowers up here in that in that pot that the window pot because I mean why why have that there if you're not going to have something growing in it. Okay and here's that funny sign it was metal like a copper and it you know there's that arrow pointing that the garden is this way and it seemed to me that all the garden was over here. So here's my corner garden in Assisi, Italy. And now today though I have something very exciting. We are going to stay right here in Sheboygan County where I'm where I live and we're going to go to a farm in the Kettle Moraine. The Kettle Moraine Forest is a it's a big area of Sheboygan County and there are many farms out there that are like a century maybe 150 years old and this is this is a very old farm and the farmer was so accommodating a group of us painters we went out one Saturday morning and to paint on his property and I saw this red truck okay and it's right here in my painting and I it just wasn't sitting in the right position you know so I wanted to move so can you believe that he actually brought a battery charger out a portable one and he charged the battery and he cranked this old thing up and it was like smoke was blowing out of the back of it and he was you know moving it around and then eat holler is this okay is this okay anyway it was it was very it was very he was very accommodating he and his family were very lovely people but this is how far I got now I don't know if if you understand what a plein air painting is plein air means to paint outside in the open air and some artists do that year round they will go out in the winter in the snow and the ice rain whatever and paint well I'm not that hardy of a soul but this was a rather cool day and even though it was cool it was quite sunny and so I tried to find a little shady spot where I could you know peek at my scene so I was back over here in the in the where I was standing when I was painting so I'm going to be finishing this today for you I hope that's my goal and and telling you you know some different things about the place okay I get my paint and this as usual it's just a standard palette with my my colors I like to keep all of my warm colors on this side of my palette and all of my cool colors over here don't know if you know this or not but the opposite of like the complement of a warm color will always be a cool color and vice versa so you always have to take if you want to make a color you always and it say you want to make a warmer color okay I want to make a nice warm gray so I'm going to take this blue and I'm going to add just a little tiny bit of this warm cad red light and put that in there and mix that around and that is going to make a real gray for me you see that I call that my elephant gray okay and if I take that and I put just a little bit of white in it not much I'll get a nice pretty warm color if it's not warm enough I'll add a little more of the red to it okay now maybe just a little more red okay now I've got a nice warm gray and just to show you the contrast I'm going to take over here I have Payne's gray Payne's gray is just ivory black and ultramarine blue together but if I take that gray which is a very gray color if I take that and I add a little white to that and make the gray here hopefully you can see how cool this is and how warm this is so that that is the difference between a warm gray and a cool gray so that's enough of the lessons for tonight today now let's just start painting here I want to kind of I just fell in love with this red door here I'm not going to go back into this area until I get my boards and things finished over here on the side so I think I'll just kind of I know we have some dark up in here and the dark will be kind of that cool gray that I showed you because that's in shadow and then the that's coming down here and then coming out of that we'll have these these lighter warmer boards that are coming down here and to to get those boards to look like boards we kind of have to load the brush with a lot of a lot of paint so that the different colors will show through it almost looks like I used a knife up above that day when I was out there painting anyway there's this there was this old barn there and we sat in the barn on hay bales to eat our lunch and it was just enchanting and the farmer was so nice he brought out all kinds of drinks for us and it was just it was I don't know it was just like being in another world I live I mean Sheboygan Falls the town that I live in is small it's quaint very quaint and but it isn't at all like a farm you know we moved here to this area from California oh my quite a few years ago now I'm thinking pretty close to 25 years ago and see that's that see that where the woods ends okay comes down that far and then then it has that little shadow there that's what that's for okay and I'll probably want to get some little little things in there to kind of make it look like like the boards I'm just gonna play with it a little bit you know doesn't have to be perfect that's for sure and now I'm gonna come down here underneath here oh okay and do this this was the most beautiful this area here where I have the brick I have moved this a little bit from or I must have moved when I took the photograph because there was more of that field stone base of that building there and all that field stone was so beautiful and years ago they would go out into the fields and have to pick that stone so that they could tilt till the fields and grow their crops so hence they made stone fences around their property sort of like the people in Ireland do and you can go out and into the countryside and and you'll see the remains of these stone fences that were put there by the early settlers and it's just amazing to think that how they how hard their life was I remember when I first moved here to Wisconsin from California was driving down this road one day out more out in the country and I was thoroughly enchanted with with the place because it was so different than the scenery that I'd been used to seeing but I saw this old farmhouse and across the front porch she had a clothesline and on the clothesline in the middle of the winter and I swear it was our first winter here was very hard it was like 22 below zero she had clothes hanging underwear her husband's underwear and some bloomers I couldn't believe it I thought what a hearty soul that would get out and hang their laundry in the middle of the winter time like that you know and then another thing that I couldn't get over is as I would drive along the roads looking for things to paint women wouldn't be on these great big tractor like things and they'd be mowing the lawns you know I thought oh my god I'm never gonna be mowing the lawn I don't care what my husband says that's not gonna be my job I guess you can tell right now I'm kind of a city girl that's alright though to each his own you know but I just I was impressed I was impressed with the with the their lifestyle I'm trying to do right now and I it's very difficult when you start a painting and then you come back into it a couple weeks later you you don't feel the same you're not the same person when you come to your easel you're a culmination of everything that you've experienced right up until this moment in your life and so where I was and how I felt the mood I was in everything that's all different than it was the day that I was out there painting so it's kind of hard to get this you know to recapture where I was and so that's why I'm just kind of poking around over here hoping and praying that something is gonna happen here pretty soon this is gonna look a little bit better than what it does now but anyway I forgot what I was even gonna say to you where was I it'll come to me hopefully before the hour is out that happened to you just be talking along and all of a sudden you lose what you're talking about I don't think that looks to that we'll take a little bit more dark and put in here and I'll just kind of soften that edge there and bring that down and I think that's just probably gonna have to be good enough for over here I think when we get that red Dorian that's going to to do it for us oh I remember what I was going to just tell you I don't know where you live but to me there was some of the bravest people in the world are farmers are the farmers because they have no guarantee and no guarantee of what the weather is going to be like with the crop prices are going to be you know if they're going to get sick and not be able to get their crop in I mean you talk about a risk taker you know you think oh farmer dull boring life and those people were 24-7 I mean it's just amazing I couldn't do that I couldn't live that kind of life I could never be a farmer's wife my husband grew up on a farm and he hated it he left and went to the city no it just not for everybody but then what it's that old saying different strokes for different folks I think we'll put a little more of a highlight along the top of this little roof right here this seems to be a little too yellow to me I think I want it to be a little gray or a little lighter brighter hmm stiff paint brush I wonder if that means I didn't clean it well enough that's too light kitty that's a little better sometimes you stand in front of the easel it's so easy to make magic and then other times you stand in front of the easel and it's like pulling teeth to make magic I wonder why that is I'm just gonna try a little something with my knife here see if I can get a little edge on that knife maybe not that much and come with that knife and kind of just in a couple spots here try to make a little magic for us yeah that did it all righty now I feel better something worked that's always a good feeling okay now that that door I can't wait to get to that door all right let's see here now I'm gonna put the I'm gonna you see this light that is going right around the door there okay I'm going to put that on first because if I put the red on first red will bleed into the white and it's very difficult so I always instruct my students put the white on first and then put in the red because if the white bleeds in to the red it's not that big a deal but I suppose you've all noticed that I I like to paint kind of thick impasto now I'm gonna come with a little bit of dark maybe just make this just a little bit lighter here trust you wipe out and kind of soften this down here we go and now then we have a kind of a of a shadow inside the door and right up here underneath here and we have it on the outside right here I'm gonna come with my brush that I was making the slats with and narrow that down by coming in with what's behind it here coming in from the shape around it there we go okay now we have that window in there we have to paint the window I always like to put you know even though that's very dark in there in the window I like to put a little bit of alizarin crimson in the dark where the window is going to be so I'm just gonna paint the whole thing there we go just a nice thin coating I'm kind of like scumbling right now and this needs to come across here and be just a little more pointed right there oh boy there we go now then I'm just gonna take that little tiny brush again see about making some little cross lines across there and that really isn't all that that dark or red that door is kind of a real faded look you see it up here too and you see his dish for his television there's no way I'm putting that on I leave things like that off I don't like to put like electric poles and telephone lines and things like that into paintings I don't know why just don't okay so I have to kind of divide this into thirds there to make sure I'm going to go and then it has to be in thirds here now it'll probably it'll probably be messy but that'll be okay because I can always come back in and clean it up with the dark and that really looks crooked doesn't it so now we'll come back and clean this up we need a little more red in there I saw I see some artists they they don't worry about crooked doorways and crooked windows and and things like that but I don't know I don't feel real comfortable with that a little hint of this I'm just gonna smush in here because how do you like that word isn't that artistic smush anyway I'm just gonna get rid of that because it's all crooked and I don't like it so easy that was there we go in fact I'm gonna come with a little blue over it I think that that window needs a little bit of the feeling that we're seeing a little maybe a little sky in it or something okay now I'm gonna try to put the red crossbars in there at the right angle I just keep wanting to go up with it hmm well well just clean it up again can I say here that's okay now we'll put some red onto the door you have to get moving here because I want to get that tree in the background there I guess that doesn't look too bad I'm not gonna put those little signs he's got in the window we'll just have a little door knob right there I think I want to go in here though and kind of soften this up a little bit okay a little bit like this okay I'll knock that down just a little bit that's just a little too harsh and we'll knock this down there we go that's good enough sure now we'll just put some light on it when the light hits it everything it changes then we'll just put a little light right down here there there we go it doesn't look too bad and then underneath here we have this dark because that's kind of in a shadow maybe if I add a little bit of blue to that so the dark the black is a little harsh whoops whoa whoa girl let's come back here and lighten that up a little bit that shadow there we go that's better light that one up too that looks better and the only thing that we have left here is whoops not quite so much a little bit of dark on the bottom of this wood here coming up where it's kind of coming up from the there okay it looks okay all right let's move on this tree is really bothering me so what I'm gonna do it's got lots of rough paint on it what I'm gonna do and try not to listen is I'm gonna scrape this down and by doing this what I'm doing is I'm scraping off the clumps of paint as I was saying I paint rather with thick paint a la prima and it leaves bumps and sometimes I don't when I try to go over it then it doesn't it doesn't look like I want it to look because it you're going over these bumps underneath and it will look like you paint it over it'll look like you corrected it you know and if you don't want it to look like that why now this tree over in here that area right in there is quite dark right in my my trunk of my tree I think where I stepped out and took this picture was not in the exact same place that I was painting because I see a lot of things that are a little different but I'm not gonna worry about that I'm just gonna go with what I have here with what is what it's telling me right now this is coming down a little bit in here and we do have that lighter up there this back in here seems like it needs to be just a little bit darker too not much just a little bit it's some may perhaps just a little bluer green I'll use my cool green Viridian is a nice cool green to use when you want to have a little bluer look in your green greenery in your trees a nice Viridian will do that quite nice in fact I think I'm gonna take that right up to here I know it doesn't look that way on the painting but my goal is not to copy the photograph but to create something that I think looks like a painting all right now I've got some stuff going back there and it's starting to look I'm starting to get excited you can always tell when I get excited because I start talking faster and I am getting excited I can see I'm starting to feel it and I can see different things happening that are gonna make it come to life for me and that is a wonderful feeling it's oh my goodness it's nothing quite like being able to take this canvas and work on it and make it do what you hope it will do a little bit of dark back in here in between those trees for the light we'll get in that yellow lighten up that green now I have I'm working with a very grayed green and a grayed yellow green and that's gonna give me a whole different look and I'm trying to like dance my brush just dance that brush and make that stuff happen out there and there we go okay now we can see that there is a lot happening back there and it's so much more interesting now than it was before now I want to take a little bit of my pure sap green and add a little bit of white to it and I'm gonna come up in here and I'm just gonna try to get this to be a little more airier I mean it's it's it's on there now but I want to add a little bit of airiness to this tree to make it look maybe that's too light I think we need to go a little darker yeah I think this will be better no it's still not dark enough I'll add a little blue and I'll take a little bit of the yellow there we go that's what I'm talking about and you see now I have that dark base in here but there's light showing up in front you know so there is the dark base but then there there's branches that are coming forward and so you see the dark and through there and up in here you see the lighter I'm afraid that my trees grew but oh well you know what if I bet I bet if I waited 20 years and went back out there to the farm his trees would be that big too so nothing like helping nature huh oh dear we have so much fun I at least I hope you're having fun I'm having fun all right now right back in here I want to make a real I want to make it really light and so that it looks like it's quite distant so I'm gonna make a cool green and I'm gonna put that back there oh don't you love it now that really looks like it's popping there that's what I want maybe we'll put a little bit of that up in the tree there too maybe there's some spots for that tree isn't hanging down that that far yeah that looks good okay you can tell him having farm I'm just goofing playing so what it is is playing okay now down here looks to me like it's kind of a gold color right down in here it kind of softens up into this bright green so I'm gonna put that down there and now when I painted the truck the cast shadow was right underneath the truck and you could see through to the other side so I'm gonna keep that look but maybe not have it quite as quite as wide as it is there now I like this I like this really bright stuff that I made so I think over in here I think we'll have a little bit more of that only here we're gonna make it come up so that it looks like bushes or something you know how I like to improve upon nature all right so we're gonna make that look like it's bushes and then we're gonna come in there and we're gonna put some beautiful bright bright red that's why I wore red today because I knew I'd want to put right at red flowers in there maybe no I better do this differently better back up take that green brush get some nice dark green and I you know you have to plant those roses before they're gonna bloom kitty so let's get the planting done here the light is coming from this direction now we'll take a little bit of that light and dance around on here make a flower garden and who knows maybe they're not roses maybe they are who knows what they are can be anything we want them to be right there are flowers you know what I think I should have made those sunflowers that's what I should have done I should have made those into sunflowers what was I thinking hmm well you know what I don't think it's too late let's do that we'll wipe them out and we'll make big sunflowers and at least I think we'll wipe them out yeah we should get most of it off and I've seen sunflowers that are kind of a dark orangey red so well just or maybe the sunflowers will have some little red flowers in there and I just don't I don't think I like all that red so I'm just gonna scratch this out here and you know one thing that I tell my students is you cannot create out of fear if you don't like something don't leave it change it fix it come back in and do something different okay now I like that little bit of red that's still on there that's showing through I like that and I think I'm going to put some stalks and stuff growing around this tree here then I am going to take this brush and I'm gonna lighten that up and go back in there a little bit to soften that edge there we go and put that back a little bit I'm in here with some yellow soften it in there now all that stuff doesn't look like it's growing in front of the truck and now I'm gonna take a clean brush and I'm going to make some sunflowers is that what I want do I really want sunflowers I think so looks pretty bright doesn't it and we'll tone them down in a minute those are those great big ones and I'll put those big brown centers into it I don't know I don't even think I like these hmm maybe I just wasn't supposed to have flowers here we'll loosen them up a little bit so that they don't look all there now now they look like they're hanging down like sunflowers do there we go a couple of strokes we need a couple of big leaves on them and everything I mean what what would a farmer's farm be like without any sunflowers on there I can't imagine that we're speaking of sunflowers that reminds me when I first moved here was driving down the road again seems like I was always on the road huh I was pretty fascinated with the area actually and I found that was the first time in my life that I had ever seen a field of sunflowers I had to stop the car and get out and took them many many pictures I could not believe it I don't know what I thought I mean you know being from the city I don't know if I thought that sunflowers just you know you saw pictures of sunflowers but that really it was just a seed and a trick but there okay that works now all this leave that we are left with here that we really need to punch up is this truck and once we get this truck to look like he is a little more with the program here now he's starting to show up a little better whoops little detour here and then I think I really think that that is supposed to be more of a wood color on the back of the truck here and it seems to be like it's a little gray rather than the red and it's a I think it's a wooden bed on the truck that shows up a little better then of course he has his bright red see here it's tires need to be a little darker you're kind of dirty though come to think of it of course why wouldn't they be they were sitting there in the but we'll just come and try to make them show up just a little bit more that's the way up in here was quite a bit darker right under there where the engine was that was I think I want some lines coming across here to kind of show that that is wood yeah all right we're cooking take a little bit of orange let's bring him really bring him up that son is hitting him he's a happy camper sitting there let's just there really make him show make him make him the star of the show that's what we're doing there now these little red hubcaps or wheels they were kind of a reddish color going around this way and then there was dark in the center and right there and it came around I'm trying to use a really dark big brush here because we're running out of time and I wanted to get this done for you let's see here no I've never regretted moving here to Wisconsin I don't miss Los Angeles I love it here the they have all these ethnic trails and ethnic groups that moved here all the years ago and settled the the Germans and the Dutch and what they must have gone through to accomplish what they did to make this area like it is proud to be here and I know wherever you are as you're watching this that you probably love where you live too that's not to say I guess what I'm trying to say is that I'm not trying to say my place is the best I just want to assure you that I love it there we go okay he's starting to show up a little bit better maybe he's a little bit more in here and definitely needs a little touch back in here there for the other side okay now I think I need a little more green I'm seeing that I need a little more of the green coming up in here I'm running out you know what I'm running out of the green on my palate so I have to stop here pretty soon and it looks like I had another tree in here I don't know if I yeah I guess I'll put it in what the heck that was there so we'll we'll let it grow let it thrive and we'll put a little bit of color on this right in here we don't have any branches in here we need to break this up with some branches whoops that's pretty thick kitty oh well you get for not using a smaller brush but I think that does help now we just need a little highlight on the branches I'm really liking this I hope you are this has really been a lot of fun he's got a spare tire in the back we have to show it's coming up right there little spare tire inside of the truck there there we go okay now I think we're just about done with it we will I'm going to change the color of the foreground real quick I want to warm that up for you it's too cold and light and I want it to be warmer and I should be using it well don't have one handy so we'll just do it with this this is the fun part of the painting this is where you just let her go let her rip and here we go we're here it's perfectly okay if we have some thick stuff that just means that his little driveway there probably has a few rats in it maybe this needs to be softened up just a little bit right there and maybe this needs to come up into this a little bit so that there's not such a hard transgression into that and then if I were to sign this I would sign it and I'm just going to sign it real quick for you because I can't leave it unsigned but if I were going to sign this I would sign it in red days the red day all right and I would sign it right here like so and just step back take a look at it oh it looks okay to me thank you so much for joining me today once again this is painting journeys with kitty Lynn Plish I've really had fun bringing finishing this old painting that well not old painting but this painting of an old place bringing it to life for you on campus this was our journey today through the Kettle Moraine to a farm thank you so much for joining me bye bye for now


I own 11 pairs of studio headphones, but these are my favourites. The Austrian Audio Hire X65s, they're well made, they're comfortable, and they give me a sound which is close to that of my studio monitors. That's why you'll see me wearing them on all of my live shows, where I have to evaluate mixes and wear them for several hours. But they do have one drawback, they're open backed headphones, so they're not absolutely ideal for recording or tracking, because the sound bleeds from them into the performer's microphone. So I was delighted when they sent me these high X60s to try out, these are kind of closed backed versions of the same headphones, and much better for recording. I was also intrigued when they sent me these high X15s, I've heard a lot of good things about these headphones, and it got me wondering, how well can these $119 headphones compete with these $429 headphones? I mean surely for a start they would have saved money on the packaging, right? Well not really, they both come extremely well packaged, as I've found with all Austrian Audio products. Okay, so what about what's in the box? Well they both come with identical cloth bags. And whilst it's true that the more expensive X60s come with two high quality detachable cables of different lengths, and the X15s only come with one lesser quality detachable cable, they do have the same type of fitting, which means this is an upgrade option for the X15s. Perhaps a little bit of money saved here, but not a lot. So what about build quality? Well the X60s weighing in at 320 grams, sport a lot more metal in their construction than the X15s that come in at 255 grams. And whilst they both have sturdy swiveling metal hinges, the X60s go that extra mile. There's also a slightly different approach to the headbands. The X60's metal headband has a nicely padded vented design, and the X15s have a standard single strip, not quite as good a quality of padding in my opinion, but still not bad. The actual earpad design on each is not identical, but I have to say very close, and of similar quality. In other words, both great, both comfortable. So that's the physical side, but what about performance? So both of these are closed back headphones, which makes them useful for artists to wear while recording because they don't bleed so much of the sound into the microphone. Now in terms of performance, I found them somewhat similar in this regard, with the X60s having a very slight edge, but honestly, they're both good for this purpose. Comfort is such an important factor, especially during a long recording session. And we could say the X15s have a little bit of an advantage here because they're so much lighter. However, there's a little bit more to comfort than just weight.  Now in this regard, I reckon the X60s are very, very similar to my X65s, and I reckon they're the most comfortable headphones in my studio. The biggest difference between these two headphones is the sound, and I'm very pleased to be able to say that because too often with more expensive headphones, they are more expensive because of their physical properties and perhaps things like packaging. But in this case, they've really focused on the sound of them, and it is very clear. I feel like I can really trust these X60s across the board. In the low end, it's nice and solid, but it's not hyped. In the mids, they're very nice and clear with lots of separation, and the highs are crisp without being kind of fatiguing. It really is very clear to me. Now, does that mean that you shouldn't buy the X15s? We're not quite. Budget. Let's not ignore the elephant in the room. If your budget won't extend to the X60s, then buy the X15s. I don't mean over other Austrian audio headphones. I mean above any other headphone in this kind of class. I've tried a whole bunch of them, and in terms of build quality, in terms of comfort, in terms of isolation, these are definitely the best. So why even bother with the X60s? Sound quality. Okay, the overall quality is a little bit better than the X15s, but it's the sound where you can really hear the money. Now I reckon that opens these up for multiple applications, because as well as using them for recording or tracking, I reckon these can also be used for mixing. They're that good. And I know many of you are mixing on the go. You may be mixing a track on a laptop while you're in a cafe or something, and isolation is then going to be important to you. So these could very well be the choice for you. If you want high quality headphones, you can mix with, which also have great isolation properties. For recording, I will 100% be using the high X60s from now on. They've got a great build quality, they're super comfortable, and of course they've got good isolation properties as well. However, I'm in a very privileged position because I also have the high X65s, and I will still be using these for the mixing that I sometimes do on headphones and for monitoring as well. If you'd like to find out a little bit more about these, then do check out my video right here. Enjoy.


This is the VOA Special English Education Report. Many rural areas in the United States have no doctor. Some medical schools are trying different ways to treat the problem. One idea is to educate doctors in smaller communities and hope they stay. Dr. William Cathcart Rake heads a new program at the University of Kansas in the Midwest. He says, we need more docs. There's somewhere like a quarter of all of our physicians in Kansas are 60 years of age or older. So we need to be replacing physicians too. He says medical students from rural areas now typically study in Wichita or Kansas City, two of the biggest cities in Kansas. They say, you know, I really have every intention of coming back to rural Kansas, but they meet a soulmate, they get married, their soulmate happens to be from a big city, and we never see them again. The program is based in Kansas' 10th largest city, Salina, home to about 50,000 people. Salina is about a three-hour drive from Kansas City, past fields of corn, soybeans, and cattle. Student Claire Henriksen grew up in a town of about 600 people. One reason she likes the Salina program is because of the size. There are only eight students, the smallest medical school in the country. Classes are taught by professors in Salina or on a video link from Kansas City or Wichita. Students who complete the four-year program will then do their residency training in a small community in the surrounding area. One place a resident might work is the Clay Center Clinic, where Dr. Carrie Murphy is a family physician. Dr. Murphy says, this is a clinic that has currently eight doctors and four mid-level practitioners, and we cover of course this town, but we also have satellite clinics into nearby towns. We just kind of operate as what I call a cradle-to-grave operation. We deliver babies and go all the way up to doing nursing home care. Rural doctors generally serve older, poorer patients. Going into a specialty in a big city can mean better working hours and more money to pay off student loans. The Salina program will pay tuition for each year that students practice in a rural area in Kansas. Dr. Cathcart Rake hopes four years of medical school and two years of residency will be enough time to put down routes. In his words, it's going to be hopefully harder for them to break away from those routes and go to bigger cities. For VOA Special English, I'm Alex Villarreal.