United States Navy's famous Task Force 58 head to Iwo Jima, British and American carriers and capital ships close in on the Ryukyu Islands, stepping stones to Japan and the China coast. Carrier planes go aloft to begin the nine-day bombardment preceding the invasion of Okinawa. Large strategic island in the Ryukyu chain, 325 miles from the Japanese mainland. At the range of enemy land-based aircraft, the fleet is covered by a constant air umbrella, while fighters demolish Japanese ships. Around solitary survivors of group attempts to penetrate United States air cover, 90% are shot down. Back against Japan, enemy held China or Formosa. Air attacks leading to landings at Okinawa are unrelenting. Throughout the entire 800-mile island chain, bombers and fighters attack in constant waves. Our major airfields are shot up. Explosive rockets smash heavier targets. Phibious invasion of Okinawa, an operation that cuts in half the distance between American forces and Japan. Okinawa's airstrips and natural harbors, its central strategic position, are now well within our grasp. Piley-inflammable jelly gasoline incendiaries of this type have already been used against Japanese installations. The bomb hurtles down to scatter over a wide area and burn at 3,000 degrees Fahrenheit. United States Marines on Peleliu Island's Japanese cave positions on a vital ridge are proving one of the war's effective agents of military destruction. Representatives of the French and United States Navy and Merchant Marine officiate. With French crews under French command, the ships will operate within the inter-allied supply pool. The American flag comes down. This ship joins the mighty Allied shipping fleet. Fort News, Virginia, the largest warship in the world, the new Navy aircraft carrier USS Midway. Naval aviator hero Lieutenant George Gay watches the Midway's sponsor christen the mighty carrier. The Midway, named after the first great Pacific victory of this war, can base a fleet of 80 planes on its deck. It is a significant addition to the Navy. City of Svi-Brookin in the Tsar is pounded by United States 7th Army weapons, including rocket-firing tanks. On the road to Berlin, heads with his advanced units. City of Svi-Brookin, largest in the Tsar Basin, is taken by men of the 7th Division of the 7th Army. Lashing out through the Siegfried Line, the 7th Army struck north for a juncture with the 3rd Army and the complete destruction of German forces in the Tsar. In this building in 1935, the result of the Tsar plebiscite, bringing the Tsar under German control, was announced by Adolf Hitler, which is near Ramaggen. Seen of the dramatic first crossing of the Rhine, carry increasing loads of troops, trucks and tanks. From their bridgehead east of the mighty river, Allied armies have now slashed into the heart of Germany. Ten miles north, the 9th Division of the 1st Army enters Bad Godesburg, decked in white flags of surrender. Here, Hitler and Chamberlain met in 1938, preceding the Munich Pact. Now, German civilians take orders from the Allied military government. Ones must be turned in. Just north of Godesburg, the 1st Army's 1st Division drives to the wreckage of the bridge. Now, they have advanced far beyond, from just one more objective taken. Just one more collecting point for thousands of beaten soldiers of the Reich, now prisoners of the Allied.


Hello everyone, welcome to the second lecture on non-linear dynamical systems. Today, we will continue with first order systems. First order meaning the variable x has only one component. This is what we just begun last week. We will continue with this now. So, we were dealing with the case that we have the differential equation x dot equal to f of x in which x at any time instant had only one component. In other words, it was a real number. In this situation, f is a map from real numbers to real numbers. More generally, it was a map from Rn to Rn, but because x has only one component, we have a map f from R to R. So, for this differential equation, we saw that the variable x lives only on this line. So, at any time instant, suppose this is the origin, at every point there is an arrow which is either towards a positive direction of x or negative direction of x depending on whether x dot at this point is positive or x dot is negative. So, whether it is positive or negative can be very conveniently be seen by plotting f here. This is incidentally possible only when x has only one component. So, we can draw a graph and suppose the graph of f versus x is like this. So, at this point, we see that f is positive and since f of x is exactly x dot, we can see that because the value of f is positive at this value of x, we see that x is increasing. At this point, because f is 0, it means rate of change of x is equal to 0 when x is at this point and hence that is an equilibrium point. If x starts at that point, it remains there. On the other hand, if we are considering this point, then this point f is negative and hence this point x dot is negative at this point and hence it is decreasing. So, we will draw an arrow in this direction. So, for the single component case, for the scalar case when x has only one component, that time this line x is filled up with arrows either to the right or to the left. The special case arrow has length 0 when there is no arrow. All these can be drawn by just seeing whether f is positive or equal to 0 or negative. See, in other words, we can plot a graph of f versus x. But since we are interested in seeing how x varies as a function of time, we are interested not just in f with respect to x, but we are also interested in a graph of x versus time because x, the variable x is evolving as a function of time and in such a situation, we would like to draw a graph of f versus x and x versus time and this is one convenient way to do it. So, in our earlier situation, we had x here and f of x here that has been just rotated by an amount so that this particular plot x is here and f of x is here denotes precisely this plot that is being pointed by my finger. So, our previous example, f was some graph like this. So, we are able to see that this point is an equilibrium point and hence, suppose in this graph, this is time t equal to 0 and this is increasing direction of t. Suppose this value of x, we saw f of x is equal to 0. So, this is an equilibrium point. If x begins there, it just continues exactly at that same value. Why? Because at every point, at every time instant t, x dot is equal to 0. In other words, x is constant. It is equal to what constant? Precisely this value of x where f is equal to 0 and on the other hand, when x is slightly higher value, suppose here then we see that f is positive because f is positive, x dot is positive which means x is increasing as a function of time. So, if it is equal to this value at any time instant t, it just increases. How long it will increase? That depends on how f varies with x. On the other hand, if we start with the value of x that is slightly below this particular equilibrium point, say here, then here x dot is negative. So, x is going to decrease with time. So, please note that we have dependence of f with respect to x that tells us how x depends on t. Whether x is increasing like we saw here or whether x is decreasing as a function of time depends on whether f is positive or negative as a function of x. So, this we will see a little more in detail when we see an example. So, consider the example f of x equal to minus a x square by plus b x plus c where a, b and c are three real numbers. They are all chosen to be positive numbers. So, for this situation when we draw a graph of f versus x, so this is a little unconventional way of plotting a function. So, we have plotted with the same orientation of the independent and dependent axis. x has been plotted here and the dependent variable f has been plotted in this direction. So, this particular equation we can say has two roots, one positive and one negative because a and c have opposite signs. We know that the roots are real, distinct and opposite signs and for very large values of x, we know that f eventually becomes negative. For very large unbounded values of x both positive and negative, we see that f becomes negative. In other words, the graph of f versus x is a curve like this where these are the roots. As we see one is positive, one is negative. So, for this particular graph, if we are to plot a function of x versus time, here the vertical axis is x and here we have the independent variable time. So, we see that this equilibrium point of course is going to remain constant as a function of time. For slightly higher values of x, f is negative. To the right of the x axis here means f is negative. So, x is decreasing as a function of time. On the other hand, for values slightly below this equilibrium point, we see that x dot is positive because f is positive and in other words it is increasing. This is how the curves evolve as time increases. The time axis has a very well defined direction of increasing time which we want to say time is progressing. This is a situation in our world. This other equilibrium point corresponds to when f is equal to 0 again. That is also being an equilibrium point x is constant because x dot is 0. So, we see that all these are cutting this t axis. Why? Because x equal to 0 is not an equilibrium position. This particular line is only denoting the t axis. It is not denoting an equilibrium point or a trajectory corresponding to an equilibrium point. On the other hand, this because this value corresponds to an equilibrium point, this just remains constant as a function of time. Similarly this here again because x dot is negative, we have these curves going away from the equilibrium point. So, this is how trajectories evolve as a function of time. So, we will investigate this particular example in more detail. We will check whether these equilibrium points are stable or unstable and we will also ask some important questions. What are the questions we will ask? We will ask whether these particular trajectories which are approaching this equilibrium solution, this equilibrium point, do they come and meet it at a particular time instant? On the other hand, we will also ask here we see that these trajectories are going away from the equilibrium point. We will ask the question, is it possible that a trajectory here cuts and comes to this or is it possible that when going along this, we can move out and go here and reach that equilibrium point? These are the questions we will analyze in more detail. Another question we will ask is, at every point, is it true that we have a trajectory that starts at every point x at let us say t equal to 0? Is it true that we have a trajectory starting from that point, going either in increasing direction, decreasing direction or remaining constant? But this is an existence question. Does there exist a solution from every point x? These questions we will answer in a little more detail. And we will stick to this example. Why? Because this particular quadratic equation with the leading term negative and the product of the constant and the leading term that coefficients having opposite signs is a situation we encounter in the Kalman filter case. In the continuous time Kalman filter in the recursive solution, we see that we encounter a differential equation that has this particular signs. Of course, for the situation that the state variable has only one component x. So, this particular example, we see that here all the arrows are directed inwards. This is a graph again of f versus x and on the right hand side, we have already plotted x versus time. And we see that here because f was negative, x dot was negative. So, all the trajectories are coming downwards. Let us just see this figure again because in the previous example, we have already drawn intersecting trajectories and the question is can trajectories intersect? For this particular question, we need to draw this figure again. So, consider this figure in which this is the time axis and x varies as a function of time. At any value of x, whether it increases or decreases, to decide that we have to go to this figure and check whether f is positive or negative. Why do we have to do that? Because we are studying this differential equation. This differential equation, let us go back to our previous example in which we had a quadratic equation with one root positive, one root negative. And as I said, this corresponds to continuous time Kalman filter differential equation in which these equilibrium points correspond to constant solutions to the differential equation. Suppose this value was equal to 5 and this one was equal to minus 2, then x of t equivalently equal to 5 is already a solution to the differential equation. Why? Because at x equal to 5, if you want to find whether x is increasing or decreasing, then we substitute x equal to 5 in this equation, we see f is equal to 0. Why? Because we see that fx graph f versus x cuts the x axis at x equal to 5 and hence x of t equivalently equal to 5 is a solution to the differential equation. But for values more than 5, we see that x dot is negative and hence it is all decreasing. So, the important question is can this solution which is, can this trajectory which is decreasing, can it meet the constant solution in finite time? Is it possible that a trajectory here only approaches this trajectory asymptotically or can it intersect at some time instant? This is one question we will ask. Also we will ask, here we see that for values slightly more than minus 2, x dot is positive and hence trajectories are increasing. Increasing and going towards this because we see that there is no other equilibrium point in the middle. These are how important theorems especially in second order systems are arrived at. So, we see that all trajectories are going away from this equilibrium solution. What is that equilibrium solution? The constant solution corresponding to x of t equivalently equal to minus 2. This constant solution or trajectories seem to be going away. We already decided that because arrows are directed away because every small perturbation takes the solution away from the point minus 2. We decided that this equilibrium point is an unstable equilibrium point and we can also see that from this particular phase curves, from these solution curves. So, we see that all solutions are going away but the equilibrium solution itself just remains constant along this. Is it possible that while it is remaining constant, it could go away from here so that at this particular point we have non uniqueness of solutions. There is one solution that remains constant, another solution that goes away. This is about intersection of trajectories. There is possibly one passed up to here but from here there are two futures. Since there are two futures for this particular autonomous differential equation in which there are no inputs, we want to ask is the equilibrium point an equilibrium point at all because we see that there is one solution that also comes out which says that perhaps this value of 2 of minus 2 was not an equilibrium point. Is that possible? This is an important question one has to ask in encountering non-linear differential equations. On the other hand, we see that this particular solution that evolves goes towards this equilibrium solution. While it goes towards, we want to ask does it go and intersect? Of course we know that once it intersects, if at all it intersects, once we are on the equilibrium solution there is a unique future because for slightly higher values of x we are coming down. For slightly lower values of x also we are going up which means we are approaching the equilibrium solution for higher and lower values of x but while we are approaching, is it possible that we intersect this equilibrium solution? This would be about non-uniqueness of the past of the solution. When we are at a particular point, of course we have one equilibrium solution coming and meeting this time instant x value but there is also this trajectory possibly. So, these are the questions that we will have to answer when studying non-linear differential equations. We will certainly answer this in more generality, in more generality when x has n components. Another important issue is that we have these trajectories. Is it possible that these two trajectories intersect here and then they crisscross and go? So, these questions, one is going to see that this non-uniqueness will be ruled out as soon as we assume a so-called Lipschitz condition on f. In the differential equation x dot is equal to f of x, we will define a particular property called Lipschitz property of a function. Once we impose the Lipschitz condition on f, we will see that solutions are unique both in the past and in the future, however only for a small interval of time. For a longer duration of time to prove uniqueness would require more advanced properties and conditions on the function f. So, the next important question we will ask is the question of continuum of equilibrium points. So, when would there be a continuum of equilibrium points if f were equal to 0 in this interval? What is this graph of? This is a graph of f versus x. On the left hand side of this, we always plot f versus x and we will use that to plot x versus t. So, we see that because f is 0 from this value, let us say 2 up to 5 because the graph of f is touching the x-axis for this entire interval. We see that from 2 up to 5, all solutions are just constant solutions. Why? Because x dot is equal to 0. At 5 also it is equal to 0. For slightly larger values of 5, we see that x dot is positive. In other words, the trajectories are going away. Also for slightly lower values of x, we see that x dot is again positive which means that the trajectories are increasing and hence approaching this particular equilibrium point. These trajectories could have existed for negative values of time also. So, this is a situation where we have a continuum. What is continuum? If this 2.5 is an equilibrium point, then very close to 2.5, let us say 2.5001 also is an equilibrium point. In other words, all the equilibrium points are sitting very close to each other. They are a connected set and they are a continuum of equilibrium points. This is the language we use to say that we have not just not isolated equilibrium points but a continuum. So, we will also quickly see what linearization has got to do with solution to a differential equation and especially in the context of close to an equilibrium point. A good amount of information about stability can be seen by linearizing. Suppose we consider our example. We see that at this particular point whether f is increasing or decreasing. At this point, we see that f is decreasing as a function of time, as a function of x, sorry. So, when we try to put a tangent to this curve at the point, at the equilibrium point, we see that this tangent has negative slope. So, for those who are having difficulty to answer this question with these axes tilted, let me just turn this page. This is how we should be imagining. So, this is a graph of f versus x and we see that at this equilibrium point, if we draw a tangent to this particular curve f and this tangent has a slope which is negative, which just means that f is decreasing as a function of x. On the other hand, here this tangent is a line with positive slope. So, we are going to differentiate the function f not at any point but at the equilibrium point. At the equilibrium point, we see that this tangent is a line with positive slope while this tangent is a line with negative slope. This positive or negative slope itself decides whether the equilibrium point is stable or unstable. So, that we can see also here. So, here we see that the arrows are all like this. So, to see that the slope already gives the information whether this equilibrium point is stable or unstable and similarly, this is stable or unstable, we can see, we can as well analyze this equation, this differential equation which corresponds to this line. So, consider the, before we see the differential equation corresponding to a line, we will just complete the trajectories for this example. So, we see that the equilibrium point remains constant. Similarly, this one also remains constant and here the trajectories are all decreasing and approaching this equilibrium point. Here they are all increasing. So, I can be a little fast here because we have already seen this in more detail. This is an equilibrium point. This was an equilibrium point. This is a graph of x versus time t. So, the fact that this equilibrium point is stable, why? Because trajectories close by are coming towards this. We would like to define this as asymptotically stable in the sense of Lyapunov which we will do in a few, after a few lectures. On the other hand, this one is unstable because there are small perturbations near it which are going to take the trajectory away from this equilibrium point and this much information we claim, we can see already from the slope of the tangent of the function f with respect to x at the point, tangent drawn at which point on the curve, tangent drawn at the equilibrium point. Because this tangent has positive slope, this particular equilibrium point, trajectories are going away from it. Because this tangent has negative slope, we will see that these trajectories are coming close by. The tangent is going to decide whether the equilibrium points are stable or unstable. This we will see from studying the corresponding linear system. So, consider the example x dot is equal to 3x. So, here the line 3x versus x. On the other hand, here is a graph of x versus time. So, we see that all trajectories are going to go away. This we are able to see because the equilibrium point, we are able to see because we can draw the arrows here of this function 3x versus x. So, we see that for positive values of x, 3x is positive. This you could also consider drawing a graph of 3x versus x before this rotated axis. So, you can draw it in this particular situation and after you have drawn this, because we are actually interested in graph of x versus time, when we turn this, we see that this line with positive slope, slope 3 corresponds to this and we see that it is indeed unstable. And of course, we know the solution to this is exactly x to the power x of t is equal to e to the power 3t times the initial condition. So, we see that with this particular line has positive slope, that positive slope 3 is going to come in here, so that if the initial condition is non-zero, then we see that e to the power 3t becomes some number that is very large as t increases and hence x to the power x of t is going to become unbounded. It will become positive and unbounded if x of 0 is positive. It will become negative and unbounded if initial condition was negative and that is indeed what we see. For a negative initial condition, the trajectories are going away and for a positive initial condition, the trajectories are going to become positive and unbounded. In any case, it is going away from this equilibrium solution. Which equilibrium solution? The initial condition corresponding to 0, in which situation we see that e to the power 3t times 0 will always be equal to 0, no matter what time instant t we are interested in. So, we will quickly take an example where the slope is negative. So, we take x of t is equal to, sorry, d by dt of x is equal to minus 2 times t, in which case we know the solution x of t is equal to e to the power minus 2t times, sorry for this mistake, we are doing x dot is equal to minus 2 times x. So, for this example, we see x of t is equal to minus e to the power minus 2t times x of 0 and let us draw that same graph here. This is a line with negative slope, with slope equal to minus 2, graph of f of x versus x. In this case, f of x is equal to minus 2x. So, we see that this is a line with negative slope and hence for positive value of x, because f of x is negative, all arrows are directed towards negative direction of x. In other words, x dot is decreasing. On the other hand, for when x is negative, if you want to decide where to draw the arrows, because f of x is positive, we draw it in positive direction of x. Now, we will study how the solutions look. For linear systems, 0 has to be an equilibrium point. We see that already here. This is how the trajectories are evolving. So, this 0 is a stable equilibrium point. Why? Because close by trajectories are approaching towards it. It is asymptotically stable. That we can see because this line has slope negative, which line, the graph of f versus x at the point 0. In fact, at every point, it is a line with slope negative and that is deciding, it is deciding the exponent and because that corresponds to a line with slope negative, we see that this is e to the power negative number times t and it is going to decrease in magnitude no matter which initial condition x of 0 we start with. But then to see that when the slope is equal to 0, our test is inconclusive. To see that, we will quickly see three differential equations. x dot is equal to, let us say, 0, x dot is equal to x cube and x dot equal to minus x cube. So, for these three differential equations, we will see that in each of the cases, the tangent has slope 0 at the equilibrium point. So, we are drawing graph of f versus x. This particular example, f is equal to 0. It is a 0 function. So, it is always equal to 0. So, all the points are equilibrium points and at any point, the graph of f versus x has tangent equal to the x axis. So, the tangent also has slope 0 at every point and all the points are also equilibrium points. This is the conclusion we get for this linear system when the tangent itself is the graph f and both have slope 0. But this particular example, x dot is equal to x cube when we draw f of x, which is equal to x cube. We see that we get this curve. So, one can check that this particular curve is tangential to the x axis at this equilibrium point, which is, what is equilibrium point? We equate x cube to 0 and we see x equal to 0 is the only solution, only real solution. So, the tangent also at this point happens to have slope 0. On the other hand, we can see that the arrows for this example would always be directed away from the equilibrium point 0. However, the tangent having slope 0 is also the situation for the third example, where f of x equal to minus x cube. For this example, here all the arrows are directed towards the equilibrium point and hence this equilibrium point is a stable equilibrium point. It is asymptotically stable or trajectories are eventually going to approach 0 as t tends to infinity, but here also the tangent has slope 0. So, these three examples, one where x dot is equivalently equal to 0, second where x dot is equal to plus x cube and third example where x dot is equal to minus x cube. Each of the three examples, the tangent to the curve f of x versus x at the equilibrium point has slope 0, but in one case all the points are equilibrium points. In one case the equilibrium point is unstable, while in another case the equilibrium point is stable. So, these three different situations can easily emerge when the tangent is having a slope 0. So, this we will see is a very important situation where the general case also when the linearization has eigenvalues on the imaginary axis, the test of stability or instability using the linearization will end up being inconclusive. So, having seen this, we will proceed to the situation of second order systems after having seen the first order case. What is first order? When x has only one component, the scalar situation after having seen this, we will proceed to the second order case where x has two components. The situation when x has two components has many different possibilities and these different possibilities will also throw some light about what can happen when x has more than two components. So, consider the differential equation x dot is equal to f of x, where x at any time instant has two components, so x of t is in R2. So, we can now think of x evolving in the plane R2. So, this is relatively easy to visualize, but there are many more features that is possible. Let me emphasize, it is easier to visualize than the general case where x has n components, even though it is little more complicated than the one dimensional case where x had only one component. But this going from one component to two component introduces so many possibilities, but all these possibilities can also happen in Rn and compared to Rn, x R2 is easy to visualize. Moreover, the general situation is only a small extension of this. Why is it only a small extension? Because the polynomial of any degree with real coefficients can be written as a product of only first and second degree factors if we want the coefficients to be real, that is. So, consider the linear system x dot is equal to a of x, where a is a 2 by 2 matrix. So, we will start with linear systems because we are interested in a qualitative study of the differential equation. Qualitative study meaning we want to only extract out certain features, whether it is stable asymptotically stable, whether there are periodic orbits. For studying this situation, it turns out that linear systems have a rich set of possibilities and that is the reason that we are starting with the case a is a matrix, 2 by 2 matrix and x dot is equal to a of x. So, the eigenvalues of a, that is the roots of the determinant of S i minus a, decide the key features for this case. So, let us begin assuming that a has no eigenvalue at 0. So, for this situation, the origin in the plane is the only equilibrium point because a has no eigenvalue at 0. It is non-singular, a is non-singular and hence, if f of x is equal to 0, it means x has to be equal to 0, that is what is being said by the origin is the only equilibrium point. But this equilibrium point can be of different types depending on different a's. So, the different types we will see are, the equilibrium point could be a center, we will define what a center is, it could be a node, in which case it could be a stable or unstable node, it could be a focus, a stable or unstable focus, it could be a saddle point and for certain situations for more general, for some non-regular cases, it could be none of these which we will also see. So, let us quickly review what eigenvalues and eigenvectors are. So, when a is a 2 by 2 matrix, the eigenvalues can be real or complex. So, for the next few lectures, we will use this convention that we will use complex if the eigenvalues are not real. We will also say if something is purely imaginary, then we will assume that it is non-zero. Why? Because if it is 0, then we will say it is 0, even though the origin is also on the imaginary axis, even though purely imaginary includes the situation that it is equal to 0, we will follow the convention in the next few lectures that purely imaginary means non-zero. So, this is not a standard convention just for the next few lectures. So, a complex number lambda is called an eigenvalue of A if there exists a non-zero vector V such that A V is equal to lambda V. So, this equation if we are able to solve for a non-zero vector V, then that lambda is called an eigenvalue and that V is called an eigenvector. So, in general we might require the two components in V to both be complex. But when A has real eigenvalues, then we can ensure that the eigenvector has both components real. We will start with the situation that when A has real eigenvalues, A is already diagonal. This is like assuming that A has two independent eigenvectors because it has two independent eigenvectors in a new basis in which basis in the basis comprising of the eigenvectors, A looks diagonal which we also say A is diagonalizable. So, in this new basis A already looks diagonal that is the reason that we are assuming this situation. So, in this case there are different features depending on the signs of lambda 1 and lambda 2 and when A is not diagonalizable then it is forced to have repeated eigenvalues because A has A is a 2 by 2 matrix, then they are forced to be real eigenvalues. In that situation we will assume that in some basis A is of this form lambda 1, 1, 0, lambda 1. This is the so called Jordan canonical form. Here the different features depend on the sign of lambda 1 and when A has non-real eigenvalues that means it has complex eigenvalues, then for the 2 by 2 case repetition of eigenvalues is ruled out. Why? Because they have to occur in conjugate pairs. Since A has real entries, eigenvalues have to occur in complex conjugate pairs that time the sign of the real part plays a key role, the real part of the 2 complex eigenvalues. We will also need to know what is real Jordan form. So, matrix A with complex eigenvalues can at least be brought to the following form using a new basis in R2. What form? The diagonal elements have alpha, have the number alpha and the off diagonal elements have beta and minus beta. Alpha and beta are assumed to be real numbers.  So, in this form the non-diagonal terms beta correspond to rotation and the diagonal terms alpha correspond to whether the trajectories are shrinking or expanding. This is what we will like to emphasize for the next few lectures. Also, beta equal to 0 means that the 2 eigenvalues are real and alpha equal to 0 means that the 2 eigenvalues are purely imaginary. They are plus minus j beta. So, suppose A has distinct eigenvalues, distinct real eigenvalues. So, let us see these cases one by one. For example, take A equal to minus 1, 0, 0, minus 2. So, this is what we call a stable node. Why is this? We will see in more detail when we see a graph of how the trajectories evolve. So, now x has two components. Hence, the components are called x1 and x2. Both are a function of time. So, when we are on the x1 axis, we are dealing with the case when A is equal to minus 1, 0, 0, minus 2. This is the example we last saw on the slide. So, differential equation corresponding to this A because A is diagonal is a set of two decoupled differential equations. x1 dot is equal to minus x1 and x2 dot equal to minus 2 x2. So, we see that if we are along the x1 axis, to be along the x1 axis means x2 component is equal to 0, then x2 always remains 0. It continues to be 0. The origin itself of course is an equilibrium point because A is non-singular. The origin is a unique equilibrium point. Along the x1 axis, we see that all arrows are directed towards the origin when we are along the x1 axis. That is what this differential equation says. Similarly, along the x2 axis also, all arrows are directed towards the equilibrium point. Moreover, these arrows along the x1 and x2 axis are parallel to the x1 and x2 axis themselves. And if we take a general point, suppose we take the point 3, 4, this general point, how do we determine the arrow at this point? So, we take the vector 3, 4 and make matrix A act on it. When we make matrix A act on 3, 4 vector, we get exactly minus 3, minus 8. So, this point 3, 4 to determine where the arrow should go towards, we will draw vector towards minus 3, minus 8 starting from this point 3, 4. So, the vector has, it is some vector like this with 3 components in this direction along the x1 direction and 8 components along the negative x2 direction. So, when we resolve this, we get this vector. So, this is how all arrows are directed. So, at any point x1 and x2, whatever components we are given, we can decide where the arrows lie and this is how we are able to fill the x1, x2 plane, fill the plane r2 with arrows. And we see that for this example, all the arrows are directed towards the origin. Why they are directed towards the origin? At any point, how to draw the arrow? The x1 component can be determined by just the x1 component of the arrow, can be determined by just looking at this arrow here. Similarly, the x2 component of the arrow starting from this point can be determined by the arrow starting at this point and the net is just the sum of these both. This is how the arrows are drawn along the x1, x2 plane, on the x1, x2 plane. So, we see that all trajectories are directed towards the origin. While the trajectories are coming towards the origin, whether the arrows are directed towards the origin or not is an important issue. For this example, we saw that the arrow had more direction along the x2 axis compared to the x1 axis and hence it was not directed towards the origin, but we see that the solutions will all converge towards the origin. So, for the same example, for which example x1 dot is equal to minus x1, x2 dot equal to minus 2x2, when we complete, when we draw a curve that is tangent to each of the arrows, we see that the curves look like this, solution curves. So, this we will see is a stable node. It is a node and all the trajectories are directed towards it. That is why it is a stable node. The other situation is when A has both eigenvalues negative and it is diagonalizable. So, let us go back to this example. When A is again a diagonal matrix and along the diagonal there are two numbers which are both positive. This situation A is an unstable node. Let us see this example just like we saw the previous case. Because it is diagonal, because the matrix A is diagonal, x1 dot is nothing but x1 and x2 dot is equal to 2x1, 2x2. So, we have two decoupled set of equations. That was the advantage of assuming A is diagonal. Here along the x1 axis all arrows are directed away. Along the x2 axis also all arrows are directed away. And at any other point, let us say 3, 4, we can see that we can draw the arrow. How do we find the coordinates of the arrow? It is an arrow starting from the point 3, 4, directed towards which point? That is to be evaluated by making the matrix A act on the vector 3, 4 at which point we are interested in finding the arrow. So, this as we evaluated in the previous example is just 3, 8 negative of the previous situation. So, this is how we are able to draw all arrows and then upon completing all these arrows into a solution, we see that this is how the trajectories look. So, what is a trajectory? Let us make note of an important property here. A trajectory is a curve such that at each point when we draw the tangent to this curve, then we get an arrow and that arrow is precisely f of x at that point. So, suppose at this point we draw this particular tangent to this curve. It is this particular arrow. And this arrow, this vector starting at this point has some coordinates. What would the coordinates be? If this is a point x0, it is precisely f of x0 which gives the coordinates of this vector but starting at the point x0. Conversely, if somebody gives us the vector field, if they give us arrows drawn at various points. So, let us see this example. Somebody gives us arrows at various points. Then what is the solution? It is, we just go along these arrows and we complete all these arrows into a curve. That is how the solution to a differential equation looks. We can now ask, is it possible that two solutions intersect? When would they intersect? At that particular point, there are two different continuations. If there are two different continuations, at this point there should be two arrows and that is not possible for a function. It appears. Why? Because a function at every point x takes only one value. One vector valued, it could be vector valued but it can take only a unique value for each value of x and hence it is not possible that at a particular point we have two arrows emanating out of that point. This already lays to rest the situation about intersection of trajectories. However, from the origin there is no arrow, it appears but it turns out that we could have under non-lipschitz property on the function f. We could have trajectories emanating out of the equilibrium point. So, that is how we should be understanding solutions to a differential equation. That they are various curves that are completion of, that is, if you are given with arrows at different points, this curve is just connection of all these arrows to form a smooth curve. A smooth curve such that tangent to the curve at any point is precisely the arrow that was specified. So, with this we would, the situation about center, the equilibrium point being a center is an example we will study in more detail from the next lecture onwards. Thank you very much.


So it seems really weird to suggest acupuncture for anxiety or panic attacks But I found that Chinese formulas as well as acupuncture is one of the single best things for treating this epidemic of anxiety and depression That we have going on in our culture today now in this video I'm gonna try my best to give a basic explanation for what with Chinese herbs and acupuncture We're really doing to help curb those anxiety and panic attacks Hey, it's Alex Hain author of the health book master of the day So I've included the free link down below this video that first link is for a free guide 5d the rituals that can help you add 10 years to your life with traditional Chinese medicine And it's a case study of this guy named Lee Chun Yun who had an unusually long life by doing certain practices Just check it out right below this video So the first way to kind of visualize and conceptualize anxiety Technically, it's your kidneys and Chinese medicine that are basically where your body's battery is And whenever we get run down or we're in actually poor health that battery you could think of it as beginning to leak Right, we use words like I'm burning out to colloquially describe this feeling when you know culturally we have this metaphor for Vitality and for the life force I can use that term loosely where when we get run down We're getting burnt out right and then we start sleeping poorly. We get anxious. We get depressed We have indigestion whatever it may be That analogy is something that Chinese medicine is used for a very long time as well And has a really clear-cut framework for being able to understand where you are in that spectrum And how to treat it with formulas and acupuncture So the way I think about it is that let's say you've had a really tough life crisis over a couple years You've been sleeping five hours a night high stress. You're watching after your mom who's in hospice care You're dealing with business stuff kids your own health stuff It's toasting you when the battery gets low one of the most common symptoms we experience is Ascending symptoms so people can develop night sweats hot flashes insomnia think of like someone who's too heady They have a hard time falling asleep staying asleep all these thoughts anxiety is one of them panic attacks is one It also be stiff neck and TMJ So it's almost like everything is going up in the body and even in Chinese medicine The bladder channel comes right up from down below the lower back all the way down to the feet all the way up to the back Of the neck and on the head here. So there's this cluster of symptoms that can happen when the battery is running low It's almost like the battery is leaking, but it doesn't go down in Chinese medicine for many people It goes up, you know for some people sleep deprivation Makes people anxious wired and a little bit weird and other people it makes them just super tired And they can fall asleep all the time and they can't think and they're foggy So it's a little bit different for each person another way to visualize this is as there's too much up and out and not enough gathering and in I've shared this video on the Chi dynamic or energy dynamic here It's gone by many different names throughout history But the general idea is that in Chinese medicine you've used symptoms and your life and your health as like Physics like vectors. So for example, we need spicy food and it like clears out your sinuses. You have too much wasabi on your sushi That's spicy flavor has movement to it and it's usually out like that it creates movement Sour, you know, it makes your face pucker. It kind of goes like this So this is an ancient concept Now these ancient doctors used to come up with objective subjective ways to decipher what herbs and practices did in the body Now the way I like to think of this is in patients with anxiety or panic attacks for a long time There's been too much up and out So that too much up and out can be too much excessive work too many active hours Working late not enough sleep. It can be too many emotions up and out like you're rushing around You're generating that fear that startle response all day long because of what you have going on or Psychologically, how you think about your life and why why you're rushing it can be to many other things But the analogy that's very important is that this person needs to focus on the in and gathering and the way I think of it Is the hot cocoa analogy? So the in and gathering hot cocoa analogy is really simple If you've ever been skiing somewhere very cold in winter and you come home You're tired you're cold and you sit on your sofa watching a nice Christmas movie drinking hot cocoa by a warm fire with a blanket on you and you feel like you could fall asleep at any moment That's the feeling you want to try your best to generate via your external habits or your life or whatever practices Internally help you feel that way because in the anxious or person with panic attacks is it's going like this all the time But frequently it was going like that for years and now even though you don't have that lifestyle The body still does this and it can be for a long time before it rains itself back in without treatment So you want to try to find that hot cocoa analogy a practice in your life Whatever that may be whether it's working less hours whether it's a sabbatical whether it's getting a massage try to generate that down and in Cozy blanket hot cocoa by the fire feeling now I don't want to leave you in this video, you know high and dry without some practices You can actually do now. There are three things I recommend for people with anxiety or depression the first one is Time in the woods Nature or outside now one of the most interesting things I observed personally was that you can feel intensely lonely in An apartment or in a city filled with people if you're living in your own apartment with no interaction But it's uncommon to feel super lonely or alone or isolated when you're out in the woods alone So I think humans have developed this kind of protective Quality in nature because that's where we've come from and where we've evolved So I found that for many people an hour a day in the woods or even walking in the neighborhood looking at trees and Feeling the sun in your face has a very protective response. The second thing is actually physical touch and affection It's something really really interesting the emotional or the social nervous system where people are starved for connection But also really starved for physical touch So if you have your interrelationship and you can get physical touch then that's great But if you're not trying to find a way to get that in a healthy way is something I found to be one of the most therapeutic For patients people with anxiety and depression or panic attacks now Obviously acupuncture or body work is one of the ways to do that. The third way is probably the most fun And it's very simple.  You're laughing you're having fun once they're trained and in general The process of getting a dog once they're behaved is a lot of affection physical touch Companionship and you're getting out you're getting activity. You're getting movement in your life and it's it's play is really what it is So those are my three action steps in terms of you know things you can do but of course I recommend seeing someone who knows Chinese formulas and acupuncture because Experienced it in my own life. Now. I also have patients those conditions a lot So I hope that helps again, check out that first link below this video That's gonna show you exactly some practices you can do to help really build your resources and extend Your lifespan and just be healthier. So check it out below this video and before you go I have two related videos you might like here


Everybody for us to your rip in 2023 top should be baseball hobby six bucks number 32 six box case number 32 random divisions Find him anytime late sports cars calm. Let's go ahead and get started. Especially as a breaker That's like it may mainly that But but everywhere it's kind of like that. Let's do it Links in chat find them online late sports cards calm probably buy your spots there tribute time Let's see. I don't think I pulled a one-on-one. Yeah, we have I pulled some we pulled some very big cards obviously ripping a lot of tribute You gave some great hits, but I haven't pulled a one-on-one out of this yet. So let's do that here Now we haven't seen one here Warren We have not seen one yet. We've ripped We've done like 40 some odd case breaks for group breaks a bunch of purses probably around 50 casers So we've ripped we haven't seen one yet Anderson and Gehrig white socks rookie auto al central to 75 Lenin Sosa. We have not seen one yet out of finest Yachtza green Bobby witt jr. Second-year al central 68 of 99 In a le stamp of approval jersey JD Martinez This one will be to 199 be a redemption Cole Trarras all right be to receive Olympus autograph orange should be to 25 be Freeman Dodgers great hit LA That'll be awesome to get that one should be Freeman at 25 On card gonna look great. You haven't seen Olympus. I definitely look them up if we don't see one here. They're awesome We'll get that mag for you and L West. Oh, yeah, they're definitely rare for everything in it Number to 50 and the checklist is like to 100 and there's inserts the autos number to 10 and they they're they gotta be super rare Soto judge Francisco Alvarez rookie auto out of 99 and L East Great first box here Kusey with the NL East Scott what's up, buddy? Get that mag for you NL East and dual jersey Verlander AL West 102 of 199 Bob Gibson of Letty Junior Was that barrios blue jays auto to 150 AL East on the blue color match Yeah, they weren't even announced on release. They just kind of started popping up and I think I don't even know if they actually announced it But I or if that's the name that people gave them treasure chest boxes or something like that Maybe that maybe that came out after the fact Rom Pedro rookie triple relic Riley green for the AL central nice Riley 27 of 99 Yaz Suzuki Tom Glavin NLE start out of 75 Tom Sean Sierra superhero is Oscar the Grouch, I think Ripken Acunia on the base Dale Murphy iconic for spectives auto to 10 for the NL East Now there's like some roller line like I call it damage is like scuffs Some of that might be adhesive, but I think there's some scuffs I'm gonna send a box and opera to the NL East for this one. Dale Murphy to 10 though nice hit And then Glaber stamp of approval jersey out of 99 for the AL East 23 of 99 Then Glaber's Jersey number Trying to pay attention to that franchise Who would do such a thing? Luke Voight that's to 99 NLS jock Peterson Jordan Groshan's NL East rookie out of the 199 Pete and Marta dual relic Joe Maurer AL central 23 of 99 Bregman Baez Jersey card Nesta Cortez AL East numbered 126 of 199 stamp of approval Dude Star Wars Sapphire sick I'm kind of surprised there's so many spots left because we've been pulling some very nice cards in there and Like the parallels sell well even some of the nicer base sell pretty well Got Correa and Freeman and triple relic is Tatis for the NLS game used Number 36 out of 75 two boxes in got four to go here On the five and 14 so again guys next one has Five to go optic choice basketball five boxer 54 99 a spot three to go on prism choice. I appreciate that BMJ a spot came in for the For the other one that one's out of 14 So you must have grabbed one for number 12 Number 11 is five to go guys breaking it next Corbyn burns mark a semi in Dennis Eckersley auto out of 50 for the AL West Nice egg Kyle needs to get the egg hair now Complete to look Harper. It's not TCA does and an Adler watchman rookie Nice one 18 packs. So two rookies per case Nice Adley AL East John Smoltz auto out of 75 NL East Yell at chiffon Rodriguez Jersey card Boba Shet stamp of approval for the Blue Jays out of 196 is that what I just saw at 80 of 196 Yeah, one came in for the break after the next one. So the next one's number 11 It's a got five to go. You bought in for number 12. This is one after that Appreciate you appreciate it, man Then a four to go in the next one now. Thanks to John C Appreciate it John nice nice nice Going here with a break jazz and Alcantara with a AL West rookie auto green color match shale angoliers of 99 In dual Jersey Lamae Hugh Yankees at 25 AL East Now I haven't had a buyback in Star Wars yet. I might have odds. I might have odds on the checklist I didn't see him before there's a vado. This was number 275 and Alcantara They will pull an auto tonight out of Star Wars Eddie Murray auto to 125 AL East Orioles Eddie Murray there Mike Moustakis angels hit a Homer That's uh That's interesting Vaughan. That's very interesting Maurer Murphy triple Jersey AL East Derek Jeter game used 55 of 99 Hey How long has Moustakis been in the angels Vaughan has been in the whole season? That's gonna be a great immaculate grid answer one day I can't think of any other big-name Reds Angels combo at the top of my head Rivera and Cabrera Lenin so say I'll central rookie auto out of 150. Yeah, I might not be a bad idea Scott I'm sure with a sapphire release to that those original cards also see an uptick in value because more people are like interested in them Jayrod base Greg Maddox judge stamp of approval Jersey card Yankees AL East 199 check that one out 58 of 199. Oh sold out. We got How many do you own the one after the out to vae buxton? Oh, this would be nice one Carlton Fisk on the pudge nickname auto to 99. That's white socks Fisk AL central Pudge Fisk on the auto you go. So does a Vaughan areas this evening name card. Is it say the same thing? Yeah, Goldschmidt's green the 99 Bryant's for the NLS and dual Jersey AL central is Big Hurt and Jose break Out of 199. Oh did he been okay? I was wondering so I was like Moustakis got Mountcastle big poppy Shane Bieber auto out of 125 AL central. Hey, appreciate it BMJ Wander base trout base Look at this triple patch card Sammy Sosa game used triple patch Cubs 8 of 10 and L central That's a cool card It's like a partial older logo. I would think that's sick all three colors Great hit and L central 11 to go you go ahead and tweet that out Spots rolling in. Thank you very much everybody. Then we've still got Star Wars sapphire and Chronicles draft picks football tonight outside of those few Prism choice still at three twenty seven and thirty one on Star Wars seven and fourteen on Chronicles drafts We're gonna get Star Wars filled up y'all Castillo Xander Alex Gordon auto to 25 AL central Alex Gordon AL central George Brett Albert pools Purple Pete Alonzo purple Pete to 50 NL East I appreciate it Scott an obligation. We appreciate that though 52 and 99 AL West Jersey card Alex Pregman on the stamp of approval Hopefully keep on some big cards tonight Perez O'Neill Cruz Here are and Karnaz see on rookie auto to 150 for the NL East Quan and Cabrion Hayes Gunner Henderson rookie auto green 82 of 99 Baltimore AL East Fantastic hit for Shane see awesome gunner and dual Jersey is gonna be Wander Franco and Bobby Witt jr AL East AL central 24 of 99 It is a random at the end I'm gonna also send a box trapper whoever wins this card because there's a scratch next to the wander Oh Higley, I'll do a good Donnie. Let's have Higley choir Bobby Witt's Dave Stewart AL West auto to 199 I get a haircut me a Friday Friday Thursday Friday The answer is yes. I'm trying to think what day. I don't know. I mean they all blend together now When you work on your day off at all, it's all just a mess Jackie Robinson Corey Seeger AL West to 99 Thanks, Higley. I just got I get the same thing every time now Triple relic Kirby Puckett to 199 Nice Kirby pinstripe on the one Jersey swatch AL central in the Kirby Puckett, of course D4 because I have no hair. Hey, he's on the list All these terrible puns tonight I have to expand the list You could easily do 22 Higley Dr. Peterson to James. Well, you should have thought before you typed Michael totally a rookie auto 250 for the NL West you and Donnie could be together on the trash list You can hang out with Sam and Donnie and Jake and Kyle Wade Boggs Randy Rosa Reina Got a yellows Jersey NL central stamp of approval 13 of 50 there are no benefits to be on the trash list zero Herman Marquez a rod Dual Jersey judge and Jeter numbered 36 of 50 AL East Nice dual Jersey card Mullen's a Rosa Reina to 75 AL East Tristan Casas rookie AL East and Oswaldo Cabrera rookie out of the one 99 for the AL East Ricky Henderson base Schmidt rookie triple Jersey Adley-Ruchman AL East Adley game used on those 128 of 199 That sounds brutal Scott and Botto J. Ram and tribute autograph patch guard KC my six of 50 AL central No, let's do one random here AL East AL central to 99 wander and wit dual relic You'll also get a box wrapper with it because there's a scratch next to the wander side. That's a 12 Good luck. Where was that topic for 12 gets the card? one two three four five six seven eight nine ten 11 and 12 AL central all right relics you had AL central went the dual relic here Ruchman judge and Jeter Y'all it's not gonna. I'm not gonna turn all these too much movement Kirby Puckett Bregman so says a really cool triple patch there Bray you Frank Thomas Aaron judge Derek Jeter Lemay Hugh Bischette Tatis Nesta Cortez Joe Mauric labor Torres Riley Green Verlander and JD Martinez No by-batch and tribute tonight now. I wish those are rare I don't mount hold on Romer Liste though Got Cabrera Dave Stewart Jordan Groschens. You are any kind of see on Lenin Sosa Jose Barrios Shane Bieber Eddie Murray Shea Langaliers John Smoltz Tom Glavin Lenin Sosa Michael Toglia Dennis Eckersley Alex Gordon Dale Murphy Casey Myers in the patch auto hutch Fisk on the AL central nickname auto and an Adley rookie base rookie Adley there gunner Henderson rookie green auto to 99 AL East To 99 NLD Sturkey green auto Francisco Alvarez and redemption Freddie Freeman orange tribute to Olympus autograph there And L West. Thanks again everybody moving on now to some basketball


It's theCUBE, covering the Virtual Vertica Big Data Conference 2020, brought to you by Vertica. Hello everybody, welcome back to theCUBE's coverage of the Vertica Big Data Conference, the Virtual BDC. My name is Dave Vellante and you're watching theCUBE. We're here with Joe Gonzalez, who's a Vertica DBA at MassMutual Financial. Joe, thanks so much for coming to theCUBE. I'm sorry that we can't be face to face in Boston, but at least we're being responsible. So thank you for coming on. Thank you for having me, it's nice to be here. Yeah, so let's set it up. Talk a little bit about MassMutual. Everybody knows Big Financial Firm, but what's your role there and kind of your mission? So my role is Vertica DBA. I was hired January of last year to come on and manage their Vertica cluster. They've been on Vertica for probably about a year and a half before that, started out on an on-prem cluster and then moved to AWS Enterprise in the cloud and brought me on just as they were considering transitioning over to Vertica's EON mode. And they didn't really have anybody dedicated to Vertica. Nobody who really knew and understood the product. And I've been working with Vertica for about probably six, seven years at that point. I was looking for something new and landed a really good opportunity here with a great company. Yeah, you have a lot of experience in Vertica. You've had a role as a market research, so you're a data guy, right?  I am, I worked with the Pitney Bowes in the postage industry. I worked with healthcare auditing about seven years in market research and then I've been with MassMutual for a little over a year now. Yeah, quite so. So tell us a little bit about kind of what your objectives are at MassMutual, what you're kind of doing with the platform, what applications you're supporting, paint a picture for us if you would. Certainly. So my role is MassMutual just decided to make Vertica its enterprise data warehouse. So they've really bought into Vertica and we're moving all of our data there probably about the good 80, 90% of MassMutual's data is going to be on the Vertica platform in EON mode. And we have a wide usage of that data across the corporation. Right now we're about 50 terabytes and growing quickly and a wide variety of uses. So there's a lot of ETLs coming in overnight, loading a lot of data, transforming a lot of data and a lot of reporting tools you're using it, currently Tableau, MicroStrategy, we have all Terricks using it and we also have APIs running against it throughout the day, 24-7 with people coming in, especially now these days with some financial uncertainty going on, a lot of people coming in checking their 401ks, checking their insurance and status and whatnot. So we have to handle a lot of concurrent traffic on top of the normal big query. So it's a quite diverse cluster and I'm glad they're really investing in using Vertica as their overall solution for this. Yeah, I mean, these days you're checking your 401k like this, right? I'm afraid to look. So I wonder, Joe, if you could share with our audience, I mean, for those who might not be as familiar with the history of just Vertica and specifically, but MPP, you've had historically, you have traditional RDBMS, whether it's DB2 or Oracle, and then you had a spate of companies that came out with this notion of MPP, Vertica is the one that I think is probably one of the few, if only brands that there's survived, but what did that bring to the industry and why is that important for people to understand just in terms of whatever the scale, performance cost, can you explain that? To me, it actually brought scale at good cost and that's why I've been a big proponent of Vertica ever since I started using it. There's a number, like you said, of different platforms where you can load big data and store and house big data, but the purpose of having that big data is not just for it to sit there, for it to be used and used in a variety of ways. And that's from something smaller, like the first installation I was on was about 10 terabytes and I work with the data warehouses up to 100 terabytes and the Vertica installations with hundreds of petabytes on them, you wanna be able to use that data. So you need a platform that's gonna be able to access that data and get it to the clients, get it to the customers as quickly as possible and not paying arm and a leg for the privilege to do so and Vertica allows companies to do that, not only get their data to clients and in company users quickly, but save money while doing so. So why couldn't I just use a traditional RDBMS? Why not just throw it all into Oracle? One cost, Oracle is very expensive, where Vertica is a lot more affordable than that, but the column store structure of Vertica allows for a lot more optimized queries. Some of the queries that you can run in Vertica in two, three, four seconds will take minutes and sometimes hours in an RDBMS like Oracle, like SQL Server. They have the capability to store that amount of data, no question, but the usability really lacks when you start querying tables that are 180 billion column, 180 billion rows, rather. Tables in Vertica that are over a thousand columns, those will take hours to run on a traditional RDBMS and then running them in Vertica. I get my queries back in set. You know what's interesting to me, Joe, and I wonder if you could comment. It seems that Vertica has done a good job of embracing, you know, riding the waves. Whether it was HDFS and the big data, you know, early part of the big data era, the machine learning, machine intelligence, whether it's, you know, TensorFlow and other data science tools, it seems like Vertica somehow, and the cloud is the other one, right? A lot of times cloud is super disruptive, particularly the companies that started on-prem. It seems like Vertica somehow has been able to adopt and embrace some of these trends. Why from your standpoint? First of all, from your standpoint as a customer, is that true and why do you think that is? Is it architectural? Is it sort of mindset engineering? I wonder if you could comment on that. It's absolutely true. I've started out again on an on-prem Vertica data warehouse and we kind of, you know, rolled kind of along with them, you know, more and more people have been using data. They want to make it accessible to people on the web now and, you know, having the option to provide that data from an on-prem solution from AWS is key. And now Vertica is offering even a hybrid solution. You want to keep some of your data behind a firewall on-prem and put some in the cloud as well. So Vertica has absolutely evolved along with the industry in ways that no other company really has that I've seen. And I think the reason for it and the reason I've stayed with Vertica and specifically have remained a Vertica DBA for the last seven years is because of the way Vertica stays in touch with its customers. I've been working with the same people for the seven, eight years I've been using Vertica. They're a family. I'm part of their family. And, you know, I'm good friends with some of these people and they really are in tune, not only with the customer, but what they're doing. They really sit down with you and have those conversations about, you know, what are your needs? How can we make Vertica better? And they listen to their clients. You know, just having access to the data engineers who develop Vertica to be arranged on a phone call or whatnot. I've never had that with any other company. Vertica makes that available to their customers when they need it. So the personal touch is a huge to them. That's good. It's always good to get the confirmation from the practitioners just not here from the vendor. I want to ask you about the Eon transition. You mentioned that MassMutual brought you in to help with that. What were some of the challenges that you faced? And how did you get over them? And what is, why Eon, you know, what was the goal, the outcome and some of the challenges maybe that you had to overcome? Right. So MassMutual had an interesting setup when I first came in. They had three different Vertica clusters to accommodate three different portions of their business. The data scientists who use the data quite extensively in very large queries, very intense queries, you know, work with their predictive analytics and whatnot. It was a separate one for the API, which needed, you know, sub-second query response times and with the enterprise solution, they weren't always able to get the performance they needed because the fast queries were being overrun by the larger queries that needed more resources. And then they had a third for starting to develop this enterprise data platform and start, you know, looking into their future. The first challenge was, first of all, bringing all those three together and back into a single cluster and allowing our users to have both of the heavy queries and the API queries running at the same time on the same platform without having to completely separate them out onto different clusters. Eon really helps with that because it allows you to store that data in the history communal storage, have the main cluster set up to run the heavy queries and then you can set up sub-clusters that still point to that history data but separates out the compute so that the APIs really have their own resources to run and not be interfered with by the other processes. Okay, so I'm hearing a couple of things. One is you're sort of busting down data silos. So you're able to have a much more coherent view of your data, which I would imagine is critical. Certainly companies like MassMutual have been around for a hundred years and so you've got all kinds of data dispersed. So to the extent that you can break down those silos, that's important. But also being able to, I guess, have granular increments of compute and storage is what I'm hearing. What does that do for you? It makes it more efficient. Are there other business benefits? Maybe you could elucidate. Well, one cost is, again, a huge benefit. The cost of running three different clusters in even AWS and the enterprise solution was a little costly. You had to have your dedicated servers here and there. So you're paying for like 12, 15 different servers, for example, whereas we bring them all back into EON, I can run everything on a six node production cluster and when things are busy, I can spin up a three node sub-cluster for the APIs, only pay for when I need them and then bring them back into the main cluster when things are slowed down a bit and they can get that performance that they need. So let's say it's a ton on resource costs. You're not paying for the storage, you're paying for one S3 bucket. You're only paying for the nodes, the EC2 instances that are up and running when you need them and that is huge. And again, like you said, it gives us the ability to silo our data without having to completely separate our data into different storage areas, which is a big benefit. It gives us the ability to query everything from one single cluster without having to synchronize it to three different ones. So this one can have theirs, this one can have theirs, but everyone's still looking at the same data and replicate that in QA and Dev so that people can do it outside of production and do some testing as well. So EON obviously a very important innovation and of course, Vertica touts the difference between others who separate huge storage and they're not the only one that does that, but they are really, I think the only one that does it for on-prem and virtually across clouds. So my question is, and I think you're doing a breakout session on the virtual BDC, we're going to be in Boston, now you're doing it online. If I'm in the audience, I'm imagining I'm a junior DBA at an organization that maybe doesn't have a Joe. I haven't been an expert for seven years. How hard is it for me to get, what do I need to do to get up speed on EON? It sounds great, I want it. I'm going to save my company money, but I'm nervous because I've only been a Vertica DBA for a year and I'm sort of not as experienced as you. What are the things that I should be thinking about? Do I need to bring in, do I need to hire somebody? Do I need to bring in a consultant? Can I learn it myself? What would you advise? It's definitely easy enough that if you have at least a little bit of Vertica experience, you can learn it yourself, okay? The concepts are still there. There's some little bits of nuances where you do need to be aware of certain changes between the enterprise and the EON edition, but I would also say consult with your Vertica account manager. Consult with, let them bring in the right people from Vertica to help you get up to speed. And if you need to, there are also resources available as far as consultants go that will help you get up to speed very quickly. And we did work together with Vertica and with one of their partners, Clarity, in helping us to understand EON better, set it up the right way. How do we pick the number of shards for our data warehouse? They help us evaluate all that and pick the right number of shards, the right number of nodes to get set up and going. And help us figure out the best ways to get our data over from the enterprise edition to EON very quickly and very efficiently. So, get yourself. I wanted to ask you about organizational issues because guys like you, practitioners always tell me, look, the technology comes and goes, that's kind of the easy part. We're good at that. It's the people, it's the process, it's the skillsets. What does your team regime look like? And do you have any sort of ideal team makeup or ideal advice? Is it two pizza teams? Is it what kind of skills? What kind of interaction and communications to senior leadership? I wonder if you could just give us some color on that. One of the things that makes me extremely proud to be working for MassMutual right now is that they do what a lot of companies have not been doing and that is invest in IT. They have put a lot of thought, a lot of money and a lot of support into setting up their enterprise data platform and putting Vertica at the center. And not only did they put the money into getting the software that they needed like Vertica, MicroStrategy and all the other tools that we're using to use that, they put the money in the people. Our managers are extremely supportive of us. We hired about 40 to 45 different people within a four month timeframe, data engineers, data analysts, data modelers, a nice mix of people across who can help shape your data and bring the data in and help the users use the data properly and allow me as the database administrator to make sure that they're doing what they're doing most efficiently and focus on my job. So you have to have that diversity among the different data skills in order to make your team successful. That's awesome. Kind of a side question and it's really not Vertica's wheelhouse, but I'm curious, in the early days of the big data movement, a lot of the data scientists would complain and they still do that 80% of my time is spent wrangling data, the tools for the data engineer, the data scientists, the database experts, they're all different and is that changing and to what degree is that changing? Kind of what inning are we in in terms of a more facile environment for all those roles? Again, I think it depends on company to company, what resources they make available to the data scientists and the data scientists, we have a lot of them at MassMutual and they're very much into doing a lot of the machine learning, model training, predictive analytics and they are used to doing it outside of Vertica too, pulling that data out into Python and Scala's bar, tools like that. And they're also now just getting into using Vertica's in database analytics and machine learning, which is a skill that definitely nobody else out there has. So being able to have one, somebody who understands Vertica like myself and being able to, pardon me, being able to train other people to use Vertica the way that is most efficient for them is key, but also just having people who understand not only the tools that you're using, but how to model data, how to architect your tables, your schemas, the interaction between your tables and schemas and whatnot, you need to have that diversity in order to make this work. And our data scientists have benefited immensely from the structure that MassMutual put in place for our data management delivery team. That's great. I think I saw somewhere in your background that you've trained about 100 people in Vertica. Did I get that right? Yes, since I started here, I've gone to our Boston location, our Springfield location and our New York City location and trained probably about this point, about 120, 140 of our Vertica users. And I'm trying to do a couple of follow-up sessions per year. So adoption obviously is a big goalie or is getting people to adopt the platform, but then more importantly, I guess deliver business value and outcomes. Absolutely. I wanted to ask you about encryption. In the perfect world, everything would be encrypted, but there are trade-offs. Are you using encryption? What are you doing in that regard? We are actually just getting into that now due to the New York and the CCPA regulations that are now in place. We do have a lot of personally identifiable information in our data store that does require encryption. So we are going through a months-long process that started in December. I think it was actually a bit earlier than that. To start identifying all the columns, not only in our Vertica database, but in the other databases that we do use. We have a Postgres database, SQL Server, Teradata for the time being until that moves into Vertica. And identify where that data sits, what downstream applications pull that data from the data sources and store it locally as well and start encrypting that data. And because of the tight relationship between voltage and Vertica, we settled on voltage as the major platform to start doing that encryption. So we're going to be implementing that in Vertica probably within the next month or two and roll it out to all the teams that have data that requires encryption. We're going to start rolling it out to the downstream application owners to make sure that they are encrypting the data as they get pulled over. And we're also using another product for several other applications that don't mesh well as well with voltage. Voltage being micro-focus is the encryption solution, correct? Correct, yes. Yeah, of course, micro-focus for the audiences is the, it owns Vertica and even though Vertica is a separate brand. So I want to ask you kind of close on what success looks like. You've been at this for a number of years, coming into MassMutual is great to hear. I've had some past experience with MassMutual, the awesome company, I've been to the Springfield facility and in Boston as well, and I have great respect for them. And they've really always been a leader. So it's great to hear that they're investing in technology as a differentiator. What does success look like for you? Let's say you're at MassMutual for a few years, you're looking back, what's success look like, Joe? Good question, it's changing every day, just with more and more applications coming on board, more and more data being pulled in, more uses being found for the data that we have. I think success for me is making sure that Vertica, first of all, is always running at its most optimal to keep our users happy. I think when I started, we had a lot of processes that were running six, seven hours, some of them were taking almost a day long because they were so complicated. We've got those running in under an hour now, some of them running in a matter of minutes. I want to keep that optimization going for all of our processes. Like I said, there's a lot of users using this data and it's been hard over the first year me being here to get to all of them. And thankfully, I'm getting a bit of help now. I have a couple of assistant DBAs and I'm training up to help out with these optimizations, fixing queries, fixing projections to make sure that queries do run as quickly as possible. So getting that to its optimal stage is one. Two, getting our data encrypted and protected so that even if for whatever reason, somehow somebody breaks into our data, they're not going to be able to get anything at all because our data is 100% protected. And I think more companies need to be focusing on that as well. And third, I want to see our data science teams using more and more of Vertica's in-database predictive analytics, in-database machine learning products and really helping make their jobs more efficient by doing so. Joe, you're awesome guest. I mean, we always, like I said, love having the practitioners on and getting the straight skinny and potent. You're welcome back anytime. As I say, I wish we could have met in Boston maybe next year at the BDC, but it's great to have you online. And thanks for coming on theCUBE. And thank you for having me and hopefully we'll meet next year. I hope so. And thank you everybody for watching. Now remember, the CUBE is running concurrent with the Vertica virtual BDC. It's vertica.com slash BDC 2020. If you want to check out all the keynotes and all the breakout sessions, I'm Dave Vellante from theCUBE. We'll be going more interviews. So keep it right there. Thanks for watching.