Welcome everyone to the sixth lecture on nonlinear dynamical systems. So, we will continue with existence and uniqueness of solutions to differential equations. In the last lecture, we already saw that under locally Lipschitz condition on F, we are assured of a solution only for an interval 0 to delta. This interval 0 to delta may be very large or may be small, but it is only finite. That is all that is assured by a locally Lipschitz property of the function f. We also saw an example, x dot is equal to x square. What is important about this? The function f of x is equal to x square is locally Lipschitz. In fact, it is differentiable at every point. At any point x is equal to 100, x is equal to minus 2000, the function is differentiable. Hence, it is locally Lipschitz at that point. Hence, at every point it is locally Lipschitz. But we also saw that for every initial condition x naught greater than 0, the solution exists only for a finite interval, for a finite duration from 0 to some delta max. So, the interval is an open, is semi open, is what is called a semi open. It is closed on one side, it is closed on this side and open on this side. So, for whatever x naught we choose, as long as it is positive and non-zero, it turns out that the solution exists only for a finite duration. So, this delta max could be very large. This is possible when x naught is very small, but delta max cannot be assured to be equal to infinity. It is possible to have only a finite duration unfortunately. Because this function is locally Lipschitz, we have a finite duration of time for which the solution exists and is unique, but we cannot have global existence. So, the question arises, can solutions exist for all time greater than or equal to 0? In other words, can we have conditions on f such that the solution exists for all t from 0 to infinity? After all, for linear system, this is true. Given that it is true for linear system, x dot is equal to Ax, where Ax is n by n matrix, we would like to ask the question, under what conditions on f, a little more general than linearity, can we have solutions that exist globally on the interval 0 to infinity? So, here we have a theorem, global existence and uniqueness. So, suppose f from Rn to Rn is globally Lipschitz, that is, what is globally Lipschitz? This particular inequality fx minus fy is less than or equal to some L times x minus y norm of x minus y. For all xy and Rn, there is one constant L that works for all vectors x and y in Rn. This inequality is satisfied for no matter which x and y in Rn we put, the same constant L will work. So, this is what we saw was globally Lipschitz property of the function f. If this is satisfied, then the theorem states that for every initial condition x0 in Rn, the state equation x dot is equal to f of x with x0 as the initial condition has a unique solution defined over the interval 0 to infinity. So, here we have, as soon as you assume that there is one constant L that works for this Lipschitz inequality for all x and y in Rn, that is enough to assure us that there is a solution on the interval 0 to infinity and moreover that solution is unique. So, because we have the 0 to infinity interval, we have called this theorem the global existence and uniqueness theorem. So, consider the linear system x dot is equal to f of x is equal to Ax, where Ax is a n by n constant matrix. Since all the elements of A are bounded, in fact, they are all constants. There exists a number L such that Ax is less than or equal to L times norm of Ax is at most L times norm of x. There is some constant L that will ensure that this inequality is satisfied for all x. So, what are candidates L for this particular inequality? We could take, for example, the maximum singular value of the matrix A when we are dealing with the two norm, the Euclidean norm as the norm here. We have this norm and in general L depends on the particular norm, whichever norm you take, there will be a constant L such that this inequality is satisfied for all x in Rn. So, linear systems x dot is equal to Ax, in that Ax is a globally Lipschitz function and hence we have existence and uniqueness of the solution over the interval 0 to infinity. More generally, even if we do not have a linear system, if we have a globally Lipschitz property that is sufficient to assure us existence and uniqueness of solutions on the interval 0 to infinity. So, we will have a quick summary of the various things we have seen so far. So, x naught, let x naught be an equilibrium point for the system x dot is equal to f of x, then we have seen various properties of the solutions. Local existence, we have yes for linear, for locally Lipschitz f, also we have local existence of solutions. For globally Lipschitz f, we have local existence of solutions to the differential equation. For non-Lipschitz, we are not able to say anything. What about locally unique? Of course, it is again yes, yes, yes. For global existence of solutions, of course, yes for linear, for locally Lipschitz we are not able to assure. For globally Lipschitz, yes, we just now saw that. What about finite escape time? Is it possible that solutions exist only for a finite duration of time beyond which it goes to infinity? This is what we call escape time. For linear system, this cannot happen. For locally Lipschitz, we are not able to say. For globally Lipschitz, it cannot happen because we have already showed that the solutions exist over the interval 0 to infinity. So, it cannot escape to, it cannot become unbounded in finite time. For non-Lipschitz, again we are not able to say anything. The next important question is, is it possible to come out of an equilibrium point? x naught is the equilibrium point. So, if a solution starts at an equilibrium point, is it possible that at some time instant it comes out of the equilibrium point? This is not possible for linear. This is not possible for locally Lipschitz. It is not possible for globally Lipschitz. Why? Because globally Lipschitz is also locally Lipschitz and the solution is unique for some interval and hence it cannot come out. But for non-Lipschitz, this is possible. This we have already seen. Is it also possible to come into an equilibrium point? There is a solution that is initially out of the equilibrium point. Is it possible that at some time instant it comes and merges with the solution that is always sitting at the equilibrium point? This is not possible for linear. This is not possible under locally Lipschitz property. For globally Lipschitz also it says a question mark here, but it is not possible. For non-Lipschitz this is possible. So, what is the significance about this? We might require to reach an equilibrium point in finite time. For example, the steady state, the set point, we might require to reach there in finite time. This is not possible under for linear systems for locally Lipschitz property of F. It is not possible for globally Lipschitz also. You might require a non-Lipschitz dynamical system if you want to reach the equilibrium point at any finite time. So, come out of and come into an equilibrium point, we mean here is at finite time. We have also seen some examples. These are examples of linear system non-Lipschitz unstable, non-Lipschitz stable, locally Lipschitz but not globally Lipschitz unstable, locally Lipschitz but not globally Lipschitz stable. So, we will proceed with one other theorem about global existence and uniqueness that does not assume globally Lipschitz condition on the function F. So, before we see that theorem, we will just analyze that theorem about global existence and uniqueness of solutions under the globally Lipschitz property of the function F. What were the drawbacks of that theorem? F in linear systems happens to be globally Lipschitz and hence the solution exists for all time t greater than or equal to 0. For non-linear systems globally Lipschitz condition rules out several common examples. It is too much to ask for a function to be globally Lipschitz. Locally Lipschitz of course is satisfied by several examples and this is something we would like to retain. So, it is perhaps possible to have existence and uniqueness of solution over the interval 0 to infinity but without requiring F to be globally Lipschitz. The condition we saw was only sufficient condition for existence of solution from 0 to infinity was that F is globally Lipschitz. Maybe there are some other weaker conditions on F under which we will have existence of solutions and uniqueness over the interval 0 to infinity. So, this requires us to review compact sets, open, closed and bounded sets. This is something we will quickly review. A subset S of the set X is called compact if S is both bounded and a closed subset of X. When do we call a subset S open in X? A subset S is called open in X if for every point X in S, one can find some epsilon neighborhood of X of that point X such that that whole neighborhood is contained in the set S. So, this neighborhood is defined like we have seen so far. Set of all points which are less than epsilon distance away from the point X. This is an epsilon neighborhood of the point X. The set of all points in X such that the distance is less than epsilon and this epsilon is greater than 0. So, if a set S is called open if no matter which point X you take after you have chosen the point X you are able to find some epsilon greater than 0 such that that epsilon neighborhood of the point X is contained not just in X but in S. When do we call a subset S of X closed subset of X? We call it closed in X if the complement of S in X is open. This has another way we can define this in another way by saying that all the boundary points are contained inside the subset S but that we have seen before. We will not review that part now. Finally, when do we call a set S bounded? If all the elements are bounded from some number R in the norm. So, a set S is bounded if there is some number R greater than 0 such that norm of every element X is at most R. So, all elements in S are not more than distance R away from the origin in this case. We also need the notion of an invariant set. So, a set S is said to be invariant. This invariant here is with respect to some operation. In this case, it is with respect to the dynamics of X dot is equal to f of X. So, a set S is said to be an invariant set with respect to X dot is equal to f of X if whenever the initial condition is inside, sorry, there is a small mistake here, this S should be replaced by m. So, a set m is said to be an invariant set if whenever the initial condition starts inside m, the trajectory is inside m for all t greater than or equal to 0. In other words, if the solution is in m at some time instant, then it remains in m for all future. That is the definition of a set m to be invariant. So, please replace this S with m. So, finally, we have another condition for existence and uniqueness of solutions over the interval 0 to infinity. So, let f of X be locally Lipschitz on a domain D. So, we are assuming only locally Lipschitz. Please note, let W be a compact subset of D and this initial condition, some initial condition X naught is in W. Suppose it is known that for every initial condition inside this compact subset W, we have that the whole solution lies inside the compact subset W. Suppose it is known that every solution of X dot is equal to f of X with the initial condition in X naught lies entirely in W and this is required to be true for every initial condition X naught in W. If it is known, then there is a unique solution defined over the interval 0 to infinity. The solution exists and is also unique over the interval 0 to infinity. Notice that we have only locally Lipschitz condition on f, but we have this additional property that there is some compact subset W such that whenever it starts inside W, the whole trajectory, for whatever interval it is defined, that trajectory remains inside W, inside this compact subset. In other words, this W is an invariant set. For whatever time interval the solution exists, the solution does not leave the set W. In other words, the set W is invariant under the dynamics of f. If somebody gives us this compact subset W which is invariant under the dynamics of f and f is just locally Lipschitz, then we have a solution defined not just over an interval 0 to delta, but in fact 0 to infinity. So, for whatever duration the solution exists, that is an important thing here. So, this completes existence and uniqueness of solutions. As our study about that, we saw locally Lipschitz property, globally Lipschitz property and finally we have seen that if there is a compact set that is invariant, then also the solutions can be assured to be, assured to exist and it is unique over the interval 0 to infinity. We will now move on to stability, to the notion of stability. What do we want to say about stability of an equilibrium point? We would like to say that a solution starting at equilibrium point, of course we know that solution starting at an equilibrium point remains there, but what about nearby initial conditions? Can we say that solution starting nearby also remain nearby? Stability is what? Solution starting nearby, near an equilibrium point remain nearby. So, we are going to try to quantify this nearby and this nearby. So, one should also note that the definition of stability itself has evolved, like solutions evolve for a dynamical system, even the notion of definition of stability, even the notion of stability has evolved over the last few decades and finally it has converged to what we will see in the next slide. So, this is best understood as, this particular definition is best understood as a challenge. So, it is like somebody proposes a challenge and somebody who is facing the challenge tries to answer, tries to meet the challenge. So, what is this challenge, proposer-facer definition of stability? So, consider the non-linear system given by x dot is equal to f of x, in which at any time t, x of t is an element of Rn, x has n components. Let 0 be an equilibrium point. So, for convenience, we are assuming that the origin itself is an equilibrium point. If it is not origin, that is an equilibrium point, but some other equilibrium point we are studying, we can just shift the coordinates there and be studying with new coordinates. In the new coordinates, the origin is again the equilibrium point. So, let 0 be an equilibrium point, that is f0 is equal to 0. Then the equilibrium point 0 is called stable. If for every epsilon greater than 0, there exists some delta greater than 0 such that for every initial condition x naught inside this ball, inside this ball centered at 0 and of radius delta, for every initial condition inside this, we have the property that x of t belongs to this other ball, again centered at 0 and of radius epsilon for all t greater than or equal to 0. So, when do we call the equilibrium point stable? Somebody proposes that for this epsilon, can you find a delta? The equilibrium point will be called stable. If no matter what epsilon somebody proposes, we are able to find the delta greater than 0 such that as long as you start inside this initial, as long as your initial condition is inside this delta ball, your whole trajectory lies inside this epsilon ball. So, this star condition here is a very important condition where the delta where the epsilon comes here is a very important part of the definition. This star can also be replaced by whenever 0 is inside the ball, the ball of radius delta centered at the origin and open ball. The whole trajectory is guaranteed to be inside this other epsilon ball. Epsilon is what somebody else proposes to us and delta is what we are able to calculate and find. We can also replace the star by for every initial condition x naught inside the ball b0, delta. We have x of t inside this other ball b0, epsilon for all times 0 to infinity. So, what I have said in the previous slide, this challenger is a person who gives an epsilon and tells, can you ensure that the whole trajectory remains inside b0, epsilon? After some calculation, the facer of the challenge, the person who meets the challenge says, yes, take this delta, just start inside this ball b0, delta and if you start inside this ball b0, delta, we are guaranteed to be inside this other ball b0, epsilon. So, the fact that we are allowed to do some calculation means that delta is allowed to depend on epsilon. So, if you are able to meet this challenge for every epsilon, no matter how small, that is when you will call the equilibrium point as stable. Smaller epsilon might mean a smaller delta. Hence, delta is shown to be dependent on epsilon in the previous slide in the definition. So, this is, we should be seeing a figure here. This is a time axis. For the purpose of this figure, x has only one component and 0 is the equilibrium point. So, of course, acting at 0, we have the constant solution. The solution always remains at 0, but somebody proposes us an epsilon ball. So, this minus epsilon 2 plus epsilon band is here and by our convention our epsilon ball is an open ball. In other words, this boundary epsilon we should not touch. So, somebody proposes this epsilon ball and tells, can you ensure that the trajectory remains inside this epsilon ball? So, after some calculation, we come up with this delta. So, that as long as the initial condition starts inside this, it might leave the delta ball of course, but it will remain inside this epsilon ball. So, for the trajectory to remain inside this epsilon ball, this is a 0. This is another figure with x. So, this epsilon which is being shown large could also be very small.  This is a 0. So, once this epsilon is given, we might know that the trajectories might have to start within this very small interval, the interval within which it should start so that it is guaranteed to remain inside this bigger ball. This is the ball. What I am showing is the diameter. The radius is this. This is the radius and this is the diameter. For it to remain inside this bigger ball, it is possible that we should ensure that the initial condition lies inside this smaller ball. As long as it begins from here, from the solutions to the differential equation, we know that it remains inside this epsilon ball. Another solution here also remains inside this epsilon ball. Of course, this solution might also remain inside this epsilon ball, but it is possible that every solution inside with this much distance does not remain inside this epsilon ball. To ensure that it remains inside this epsilon ball, we are forced to make this delta very small maybe. So, but the fact that this delta is greater than 0 is what defines this particular equilibrium point as stable. That no matter what epsilon on somebody gives us, we are able to do some calculation and propose this delta so that if the initial condition starts inside this delta ball, the whole trajectory remains inside this epsilon ball. So, once this epsilon is specified, is this delta unique? Suppose, after lot of calculation, we have found this delta and somebody else does a similar calculation but obtains a different delta. Can we say that one of the two deltas is wrong because the delta should be unique or is it possible that for the same epsilon, there are many deltas. So, this is the question that we can answer without too much effort. So, this is only a guarantee. Suppose, this is our epsilon ball and after lot of calculation, one person finds this delta ball. If we start inside this interval, then the trajectory supposed to remain is guaranteed to remain inside this epsilon ball but the same guarantee will automatically be satisfied for this smaller band also. If the initial condition start inside this smaller ball, then also it is guaranteed to remain inside this epsilon ball. Why? Because once we are sure that starting anywhere inside this initial condition band assures that the solution lies inside this epsilon ball, then we know that inside a smaller band also, if we have begun, we are guaranteed to remain inside this epsilon ball. Of course, this smaller one, the smaller initial condition ball is a more conservative one but this only tells that the delta is not unique. Once we have found a delta greater than 0, we can take another delta that is strictly smaller and positive and still that delta which comes in the definition of stability that condition is satisfied for this smaller and positive delta also. This is only to note that delta is not unique. One could ask the question, can we make this delta larger and larger and for each epsilon, there might be a unique, largest possible delta. In that sense, it might be unique. So, it is also clear that when epsilon is made smaller, then we might have to make the delta smaller. One could ask the question, in general, is delta, what is the relation between delta and epsilon? Our figure appears to show that delta is smaller than epsilon but in general, should such a relation be satisfied that delta is smaller than epsilon or delta greater than epsilon or should such a relation need not, does such a relation need not exist? So, this, please note that this is called this definition of stability in the sense of Lyapunov. This is just a definition. This is not Lyapunov's theorem on stability. So, the reason that we have emphasized, we have spent a lot of discussion on stability, on the definition of stability is because it is a difficult concept and to understand the definition properly is very important to understand the theorems on stability. So, before we proceed to the Lyapunov's theorem on stability, we note that what we have seen so far is the definition of stability in the sense of Lyapunov. After having seen stability, what do we mean by asymptotic stability? So, in the definition of stability, once we are given an epsilon, we were required to find a delta that meets a certain condition. In addition to that condition required in the definition of stability, if delta can also be chosen to satisfy this additional condition that x of t goes to 0 as t goes to infinity. So, what was 0? The equilibrium point x of t converges to the equilibrium point as t goes to infinity, then the equilibrium point 0 is said to be not just stable, but in fact asymptotically stable. So, we will call the equilibrium point asymptotically stable. If it is stable, for it to be stable, we already know that for every epsilon, we have to be able to find a delta such that all initial conditions starting inside the delta ball are guaranteed to have the entire solution inside the epsilon ball. This delta which was chosen to satisfy this condition, in addition to that, if it can also be chosen to satisfy this additional condition that the solution converges to 0, the equilibrium point as t tends to infinity, then that equilibrium point is not just stable, but also asymptotically stable. It was already stable because delta satisfied the condition that the definition of stability required. In addition to that condition, it has delta satisfies this additional condition and hence that equilibrium point is asymptotically stable. For every initial condition inside the delta ball, we also have x of t goes to 0 as t tends to infinity. Solution starting close by, not just remain close by, remain close by is what x of t is contained inside the epsilon ball mean. So, they not just remain close by, but in fact converge to the equilibrium point. We had assumed that 0 is the equilibrium point. So, the solution should also converge to the equilibrium point for every initial condition starting inside the delta ball. If this is satisfied, then we will say that the equilibrium point is asymptotically stable. So, asymptotically stable naturally means that the equilibrium point is also stable, but not vice versa. For just stability, we do not require that the solutions converge to 0. We only require the solutions to remain inside an epsilon neighborhood. So, we now come to Lyapunov's theorem on stability. After having seen that the definitions of stability and asymptotic stability in the sense of Lyapunov, we are now going to see Lyapunov's theorem on stability. Let x of 0 be an equilibrium point and let d be a domain that contains this equilibrium point. Let v be a function from d to r. So, the domain d is a subset of rn. v takes its values from this d and is scalar valued. v does not take vectors as its values, it takes only a scalar. Hence, r at any point x, v of x has only one component. Let v be a continuously differentiable function. So, it is v itself is continuous and its derivative is also continuous. That is the meaning of continuously differentiable function such that v satisfies some conditions v of 0, the 0 the equilibrium point at 0 v is equal to 0 and inside that domain at every other point v is positive. v is allowed to be 0 only at the equilibrium point. At other points, it is positive. Secondly, v dot is less than or equal to 0. We have missed a 0 here. v dot is less than or equal to 0 in the domain d. So, v was a function of x but this dot here means it is a derivative with respect to time. This I will clarify very soon. So, the rate of change of v with respect to time at every point is less than or equal to 0. If we are able to find such a v which is continuously differentiable, which is positive everywhere except at 0 where it is allowed to be equal to 0 and v dot is non-positive. It is less than or equal to 0 in d. If there is some v that satisfies these three conditions, then the equilibrium point 0 is stable, is a stable equilibrium point. So, what is important to clarify is this v was not a function of time, it was a function of x. And x took its values in Rn but how do we go ahead and differentiate v with respect to time? This is one important point that requires a clarification. So, v was a map from Rn to R. Why? Because x at any time instant was in Rn. Our domain d was a subset of Rn for the time being we assume that d is equal to Rn. Hence, our function v was a map from Rn to R. If a function v was not a function of time, how do we go ahead and differentiate v with respect to time? This is something we will quickly see. So, this v we are going to evaluate at different points x but through each point we have a trajectory that evolves with respect to time. So, v actually depends on x which itself depends on time. So, because x is changing with respect to time as x moves to another point, value of v will also change. In this sense, v depends on time also. Suppose this is our phase space, this is x1, this is x2 and this is some point and this is a trajectory that is evolving with respect to time and at this particular time, at some time t1, it was here. At another time t2, it has moved to this point because x itself is changing. Along this trajectory, we can see how the function v is changing. v has some value at this point, some value at this point, some value at this point. Similarly, v has some value at this point. Immediately after this, immediately further along this trajectory, v has a different value. Similarly, as the x evolves along this trajectory, the value of v of the function v is also changing. In that sense, v is a function of time also because we are evaluating v along a trajectory x. So, we will see what it means to differentiate v with respect to time now. So, v of x and x itself depends on time t. So, d by dt, this is what we call a composite function. v depends on x while x itself is a function of time. So, d by dt of v of x of t is equal to partial derivative of v with respect to x, partial derivative of v with respect to x because v itself depends on many variables, x1 up to xn and hence this derivative is not an ordinary derivative but a partial derivative and then x dx by dt. x itself depends on only one variable time, one independent variable time and hence this dx by dt. So, if v depends on x1, x2, x3, then this is nothing but del v by del x1, del v by del x2, del v by del x3 times x1 dot x2 dot x3 dot. So, throughout this course, the dot we will reserve for derivative with respect to time. If it is derivative with respect to x or some other variable, then we will just write d by dx del del by del x of v. So, v dot which we saw in our slide is supposed to be is to be understood like this. v was a function of x, v in fact depends on x1, x2 and x3, x1 itself was a function of time, x2 was a function of time and x3 was a function of time.  This is the meaning of v dot. Thus, at each point in the phase space, for this figure, our x has only two components. At each point, v has some value. This is like v is a scalar function. At each point x, x is a vector, but the value of v at each point is a scalar and this is like temperature of a room at each point. The temperature itself has only one component and as the trajectory moves through each point, there is some trajectory. This is the direction in which the x trajectory moves. Given that these trajectories are all well defined at each point, we can associate the rate of change of v at each point. At each point, there is also not just v defined, but also rate of change of v with respect to time defined for each point. That rate of change is, it is possible to define that because we have all these trajectories defined at each point and the trajectories themselves have some rate of change defined for them. For those who are more interested in this topic, this brings in use of lead derivative techniques into dynamical systems. It is not required for this course. As far as we are concerned, we want to understand the meaning of v dot, even though v was a function of x and not time. So, here at each point x, not just v, but v dot is also defined because we have a differential equation x dot is equal to f of x. So, that one also remains to be written in this slide. x is equal to 0 is an equilibrium point of the dynamical system x dot is equal to f of x. So, with respect to that dynamical system, v dot x is defined. What is v dot of x? In general, it is del v by del x times f of x. Why? Because this is nothing but del v by del x times x dot and this is what d by dt of v of x is, which we have denoted by v dot. This dot here we will reserve only for rate of change with respect to time. So, at every point x, there is a v dot defined and if that is less than or equal to 0, 0 is also missing here. If that is less than or equal to 0, then the equilibrium point 0 is stable. Further, in addition to less than or equal to 0, if this v dot, if this v function v is such that it is continuously differentiable, it is 0, it is equal to 0 only at the equilibrium point 0 and it is positive at every other point and if it is strictly less than 0 over the domain d except 0, in the domain at all the points except the point 0, if it is strictly negative, then the equilibrium point is not just stable but it is asymptotically stable. So, this is Lyapunov's theorem on stability. So, please note that this is only a sufficient condition for stability. When do we call the equilibrium point 0 of the dynamical system x dot is equal to f of x stable? We have a definition of stability. One of the ways to prove that it is stable is if you can find some function v that is continuously differentiable whose rate of change is less than or equal to 0 and which is equal to 0 at the point and it is positive at every other point. If it satisfies these three conditions, then that v, we will call a Lyapunov function and that Lyapunov function helps us to prove that this equilibrium point is stable. If such a function v, if we pick a function v and it does not satisfy these three conditions, then we are not able to conclude that the equilibrium point is not stable. It just means that perhaps this function v should have been chosen more properly, more judiciously. There might exist another function v that satisfies these three conditions and helps to prove that the equilibrium point is stable. This is only one sufficient condition to prove that the equilibrium point is stable. We will see these things in more detail in the following lectures. We also saw a sufficient condition for proving asymptotic stability. This particular function v is called the Lyapunov function and we will see these functions in more detail and some examples in the following lecture. Thank you.


At some point in our lives, we have all been under some degree of stress, that feeling of being under intense abnormal pressure to deliver or achieve certain things. In short bursts, stress can help to stir up our alertness, making us perform better in difficult situations. However, in the long run, stress has a negative impact on our health. It can lead to major health issues like heart disease, and it can trigger mental illnesses like anxiety and depression. This is why it is important that we learn how to deal with stress when it shows up, and how to prevent it from resurfacing ever again. In today's video, we'll be talking about how to deal with stress and anxiety, and eliminate them from your life forever. But before we learn how to deal with stress, let's first understand how to know when you're stressed. There are a number of symptoms that show that you might be stressed out. Mentally, stress can leave you feeling terribly upset, worried, or anxious. This is usually accompanied by persistent mood swings and an overwhelming feeling that if not handled properly, can ultimately lead to depression. Physically, stress increases perspiration, induces a change in appetite, raises blood pressure, and increases the heart rate. To help combat this sudden change in body systems, the body releases cortisol, a hormone that causes fat and sugar to be released into your system to generate energy. While this is done with good intention, it brings with it its own plethora of symptoms which includes headaches, muscle tension, pain, indigestion, and in the long term, heart disease and stroke. It therefore goes without saying that stress is bad for both your physical and mental health. To deal with stress, we must first understand what causes it and steer clear of its triggers. As the famous saying goes, prevention is better than cure. Here are the eight main causes of stress and anxiety. Financial reasons When people don't make the amount of money they think they need to make, they often start to worry about their finances and become terribly stressed out by it. They have numerous bails to pay, fanciful items they desire to purchase, or possibly one or two vacations they've dreamed about. All these thoughts can quickly clouds one's mind, cause stress and anxiety. Relationships Of course, it's great to have a significant other in your life, to love and to be loved. But often times, relationships can come with their own baggage. It could be an overbearing boyfriend, a nagging wife, or a cheating spouse. Whichever one it is, an unhealthy relationship is a real pressure cooker. Children Yes, children are a bundle of joy, or at least they ought to be, but most times they're also very stubborn and can easily get on your nerves, which makes meaning a child no walk in the park. It's almost like living multiple lives where you constantly have to be worried about the well-being of two or more people, depending on how many children you have. Even when they are all grown up, you still tend to worry about them, and the different choices they make, this can be very stressful. Business and career The pressure of not hitting your targets in your business, or climbing up your career ladder as fast as you imagined, can be very crippling, resulting in high levels of stress. Health The struggling with a health condition, like cancer, can be a real source of stress and worry, both for the sick person and their friends and relatives. When you consider the fact that most of these conditions also bring with them huge financial costs, you can see why they cause a lot of stress. Environmental factors Sometimes what people perceive to be stressors in their lives are external things they have little or no control over. It can be the political events in their country, or the dismal performance of their favorite sports team. Whatever is external matter that seems to bother people a lot is a potential stressor. High workload Anyone who works in a corporate environment will be able to relate to this one.  You'll find yourself not resting properly, and not eating well as 24-hour day begins to look every more inadequate. Yet, as you seem to be clearing out your desk, more work appears out of thin air. This vicious cycle can leave you perpetually stressed out. Lack of fulfillment While lots of people still grapple with the ultimate question about the purpose of life, some have given up on this quest, only left with a sense of emptiness and a lack of fulfillment that can cause a lot of stress and anxiety. As the saying goes, accepting that you have a problem is the first step to solving that problem. Many people will identify with one or more of these eight aforementioned situations as the root cause of their stress in their lives. So now that we've gotten that out of the way, here are the ways to kick stress and anxiety out of your life forever. Most of the time, we get stressed because we feel like we are underachieving compared to societal standards, but we must stop to think.  The reality is that many of us are crushed by societal standards that we desperately try to reach. So much so that we channel all of our life's activities into attaining that standard. However, the problem begins when we don't seem to be getting anywhere close to that standard. We become stressed in a bid to close that gap, and that is usually the root cause of most of the stress in our lives. If you will overcome stress and anxiety in your life forever, then there is a fact that you must be ready to accept.  This is the first and most important step to living a totally stress-free life. Once you realize that you are not responsible for everything that happens in your life, you will be able to live under less pressure to achieve anything or to be a certain way. With this simple mindset firmly engraved into your consciousness, you can then take the other practical steps to keep yourself absolutely stress-free. Here are a few of those steps. Be grateful. Living a stress-free life is possible when you subject yourself to a disposition of constant gratitude. Count the many things that you are privileged to have and be grateful for them. Don't just allow yourself to feel grateful. Rather, act grateful and be grateful. This will draw away your focus from the things you do not have, eliminating the stress that comes with lack. Make the best of your experiences. Bearing in mind that you do not control all that happens to you in your life, you should borrow from the idea and that cliche phrase that says, when life gives you lemons, make lemonade. Everyone has had negative experiences in the past, but what we probably fail to realize often is that we won't be where we are today without those experiences. So whatever life throws at you, good or bad, resolve to always make the best out of it. Enjoy the moment. Our life is often so fast-paced that we fail to stop for a few moments to take it all in and enjoy the present. Sometimes you need to tell yourself, I don't mind what eventually happens, I'm just going to enjoy this moment and learn from it. Many times, this even places you in a better position or a more favorable outcome. So whatever you do, take some time to smell the roses as you pass through the garden of life. Take care of yourself. Being in a state of good physical and mental well-being is critical to this living stress-free.  If you take good care of your physical state, your mental state is most likely to follow suit. Talk to someone. It is often said that a problem shared is a problem half solved. When life comes hard at you, remember to speak to someone. It could be a friend, a therapist, a spiritual guide, a relative, a coach, or a mentor. Sharing your challenges with someone is a refreshing and healthy way to deal with stress. Conclusion. It is entirely possible to live a fulfilled stress-free life. You just have to be intentional about it. With the right mindset and the tips we have highlighted in this video, that dream can very well become a reality. If you learned a thing or two from this video, let us know by clicking the thumbs up button down below. For more videos like this, be sure to hit the subscribe button and turn on the bell notification to get notified when we upload a new video. Thank you very much for watching. We'll see you guys in the next video.


Why hello there. Today we're going to talk about chemistry. And you might be wondering, dude, this is a biology class. Why are we going to talk about chemistry?  Why would we spend an entire lecture and in some courses, multiple lectures on chemistry in a biology class? And hopefully your little brain is working to think, we've actually already answered that question.  Does this look familiar? Yeah, totally. This is the hierarchical... the hierarchical organization of living structures. If you remember when we started making our lists of all the hierarchical levels of organization in living systems, what? We had this huge stack of levels that were most certainly not alive. And those levels are based on chemistry. So today's task, okay, cool. Today's task will focus on these levels of organization. And hopefully you get the idea that, dude, we have to understand atoms and molecules and, dude, the little subatomic particles that make up the atoms and molecules in order to understand how organelles work or cells. And that's what we're going to talk about next. So having a sense of chemistry, first of all, is really important. So that's awesome. You know why I care about chemistry. What's an atom? There it is. That is a totally inaccurate image of an atom. Atoms are composed of subatomic particles. We are going to talk about three of those subatomic particles, but I'm not kidding you, atoms are tiny. Like, they're crazy tiny. They're tinier. Like, 602 sextillion hydrogen atoms weigh the same as a paperclip. 602 sextillion, that's not trillion, or quadrillion, or quintillion. That's sextillion with like 23 zeros after the six. What? They're tiny. And they don't weigh very much, because most of them is open space. We're going to talk about each of the subatomic particles inside our atom. And atom is the building block of all matter. Everything around us, including the air, including the table, including my fantabulous pen, everything is made out of atoms. And we're going to be looking in this class at cellular processes that enable life or based on the movement of different kinds of atoms, which is like, oh, shockingly cool. So let's get started. Let's start by looking at one flavor of subatomic particle, and that is the proton.


I want the first item of business today is general questions and our first question is number one from Ivan McKee To ask the Scottish Government how it calculates the level of homelessness Minister Kevin Stewart Presiding Officer the Scottish Government collect homelessness data from local authorities this data collection provides detailed information on homeless applications by individual households information on the number of applications under the homeless persons legislation ymgaredd dweud o adeiladau sydd yn cyfrifoligol i maes ganddechrau i ddweud agynnal cyflawni. Arddwn MacKeen i'r尔od caen sydd wedi cael llawer o wirionedd yn dewis oedol. Yn 750 o wneud o gyflogu i Glasgow last year, rhaid i ymgyrch yn bwynt i bwysigol i ddydd yr oeddwn ni i ysgoldoedd o woriodau hynny, ac roeddwn ni wedi meddwl i Glasgow City L schedules with City Council that were refused accommodation. Threatened with legal action, the council then found accommodation for 98 per cent of those individuals. What is the Scottish Government doing to ensure that local authorities meet their statutory obligation to find accommodation for rough sleepers without them being threatened with legal action? Mr Meckay pointed out that local authorities have a statutory duty to provide a minimum of temporary accommodation advice and assistance to all applicants assessed as homeless. Glasgow City Council has the duty to provide housing and homelessness services in their area. I know that the Scottish Housing Regulator has been working with Glasgow City Council to help to improve the delivery of homelessness services in the city. As a Government, we are working to increase housing supply in Glasgow and across Scotland to improve the housing options that are available. Thank you. Figures for the number of households in temporary homeless accommodation, however, show an increase between March 2015-16 by 1 per cent. March's figures showing 10,555 households in temporary accommodation is an increase of almost 2000 since 2007. Although I recognise that there will always be a need for temporary accommodation, how does the minister intend to make sure that that is not used as a long-term solution for homelessness? There has been a continuing fall in homelessness applications to 34,662 in 2015-16. That is down 1,287 on the previous year, a decrease of 4 per cent. Of those applications, there were 28,226 households assessed as homeless or potentially homeless. That is down 1,589 on the previous year, a decrease of 5 per cent. Obviously, our key action in this Parliament is to increase housing supply. We intend to deliver 50,000 affordable homes, including 35,000 for social rent during the course of this Parliament, which will help the situation greatly. As I said earlier, we have the housing options hubs that are working across Scotland to try to alleviate homelessness throughout the country. I hope that their success continues and that we continue to see decreases as we have done in the last year. To ask the Scottish Government, in light of the review of the Police Scotland estate, what impact the potential closure of police stations will have on public confidence in the police? The Scottish Crime and Justice Survey suggests that the majority of people feel that the police are doing a good or excellent job in their local area. The Police Scotland estate strategy, which was approved by the Scottish Police Authority on 24 June 2015, seeks to remodel the police estate to make it fit for the policing needs of the future. The Scottish Police Authority has made clear that local policing commanders will play a leading role in deciding whether any changes to the police estate are compatible with maintaining an effective local police presence. Engagement will be undertaken by local policing teams to ensure that future decisions are built upon local consultation with communities and partners. In many cases, the approach that has been taken is to seek alternative shared accommodation with partners in the same locality. There are already a number of positive examples of that in locations such as Livingston and Ballistan. Richard Leonard I thank the cabinet secretary for that answer. I reflect that the cabinet secretary wishes to decentralise the power to cut police services but not the power to control police services. Shots police office serves the communities of Shots, Allenton, Hart Hill and Salisbury. When the public county was closed in Shots police station in February 2014, a promise was made to the community that for reasons of public safety, as long as the prison was there, the police office would be there. Will the cabinet secretary remove Shots police station from the hit list? Cabinet secretary. Well, I think that the member should engage with Police Scotland on this matter. If he did so, he would find that there has been no decision made on any of the police stations that have been set out at the present stage of the estate review. Local commanders will consult locally on what the best approach is. With relation to Shots, the reason that they are looking at Shots is because the existing facility is too large for Police Scotland. They are presently looking for alternative accommodation in the Shots area. They intend to continue to have a presence within Shots. If that is not feasible, they will be looking to try and draw other partners into sharing the facility with them in Shots itself. If the member is keen to make sure that the views of a local community and local elected members are heard in his process, the way in which to do that is to engage with local commanders through their consultation exercise to allow local decisions to be made on what the best approach is for the matter. However, it is also worth keeping in mind that the purpose of the estate review is to make sure that it is an estate that is effective and reflects the demands on the police service. That is why, in the vast majority of cases, they are looking to relocate to shared premises. Does the cabinet secretary agree with Assistant Chief Constable Andy Cowie, who told the local government and communities committee yesterday that, and I quote, "...services are delivered by people, not buildings. The public want to see officers on the street and that, following the review, service provision will be enhanced through investment in better-located accommodation." I agree with ACC Howey in this matter, who is leading the estate review for Police Scotland. Police Scotland is very clear that this is not about removing police officers from communities, but about making sure that they have a police estate that has evolved over 100 years that it reflects the changing nature of the demand on the police service. For example, the vast majority of contact with the police service is now through the 101 call system. That reflects a changing nature in the way in which people engage with the police service. We need to make sure that we have a police estate that reflects that change, while at the same time it is able to support police officers in undertaking their role effectively. As Andy Cowie has highlighted, it is not about seeking to remove police officers from local communities. It is about making sure that we have an estate that is effective and reflects the needs of local communities. The decision-making process on that will be driven by local needs, based on the views of local commanders once they have consulted local communities. To ask the Scottish Government what impact leaving the EU will have on local government. Local government shares the same ambitions as us for stronger communities, a fairer society, a thriving economy, and we will work in partnership with them to respond to the implications of the EU referendum outcome. The Scottish Government is exploring all options to protect Scotland from a hard Brexit that economists say will cost 80,000 Scottish jobs. Our five key interests are democracy, economic prosperity, social protection, solidarity and influence. European funding is important to local government. The 2007 to 2013 programmes of the European regional development fund and the European social fund awarded £158.3 million to Scotland's local authorities, which were spent in the years 2007 to 2016. We know that the EU plays an integral part at all levels of government in helping to deliver important projects. Can the minister provide an update on the position with regard to payment of EU structural funds if the UK wants to leave the EU? Since the outcome of the referendum, we have urged the UK Government to provide clarity and certainty on the vital European funds. The UK Government guarantees offered to date for European structural funds that all contracts entered into before the point that the UK leaves the EU will be guaranteed even when those payments continue beyond the EU exit point. However, the UK Government has provided absolutely no certainty or clarity on what the replacement funding arrangements for those schemes will be once the UK has left the EU. On 2 November, my colleague finance secretary Derek Mackay announced that the Scottish Government will be passing on in full to Scottish stakeholders the EU funding guarantees that have been offered by the UK Government. That will protect all spending commitments from those schemes that are entered into in the period between now and the point that the UK proposes to leave the EU. That provides certainty for over £700 million of EU funding for Scotland. To ask the Scottish Government what assessment has made of the impact of the autumn statement on the oil and gas sector in Scotland? I was bitterly disappointed to learn that the chancellor has provided no substantive measures to support the oil and gas sector, a sentiment that I am sure is shared by the hundreds of thousands of people supported by the industry, and in particular those in the north-east of Scotland who have been hardest hit during the downturn. The cabinet secretary for finance and the constitution wrote to the chancellor outlining urgent measures that should be considered for inclusion in the autumn statement. Those proposals were focused around increasing activity in late-life assets, protecting critical pieces of infrastructure and increasing exploration. Without greater investment and activity, we risk losing vital capacity and skills that will support production and ensure that we maximise economic recovery from the North Sea. The Scottish Government will continue to do everything with the next powers to support the industry and its workforce through those challenging times. Gillian Martin Oil and gas UK have made specific requests to the UK Government on measures to allow the industry to continue with exploration in these difficult times. What impact could the lack of action in facilitating the exploration having the future supply of oil and gas and the industry as a whole? One of the major impacts will of course be the fewer number of people supporting the infrastructure that is already there. That brings into question the viability of that infrastructure, so you might have the situation in which fields are left redundant before the time when they should be. That is a vitally important thing. On that particular issue, I met the chief secretary to the Treasury some months ago who assured me that back in June that the UK Government realised that it had not acted quickly enough on this and would now do so. We have had no action whatsoever in relation to loan guarantees for those vital pieces of infrastructure. Above that, of course, the UK Government holds the tools in terms of tax and tax concessions in relation to exploration. That was their chance yesterday to pay back an industry that should put billions into the UK Treasury, which, according to the Treasury's forecast, will put more billions back into the Treasury and yet they did nothing. The Scottish Government will continue to support this industry wherever we can, unlike the UK Government. Jackie Baillie I shared the disappointment of the cabinet secretary at the autumn statement yesterday, because the industry is of huge importance to the north-east and to the economy of Scotland as a whole. Does the cabinet secretary therefore agree with Labour's proposals for a UK offshore investment limited to look at the assets to be supported with public investment, and will he make common cause with us in taking on the UK Government to try and have the proposal agreed to? We have undertaken a vast range of support measures, so the transition training fund, the money that we announced this week as part of the Aberdeen city deal for the oil and gas technology centre, the energy jobs task force, was a range of support that the Scottish Government has provided. If there was to be further investment, and of course we asked for that investment to take the form of tax concessions in relation to exploration, if there was to be further investment, the UK Government has the tools to do that. Were they to show any willingness, we would have looked to see what we could do in order to support that. However, it is quite evident from the way that things are just now, that we are not seeing that support from the UK Government. We will continue to support the ways that I have said, and we will continue to look at other ways in which we can provide that support. Tavish Scott Given the cabinet secretary's points about the importance of the UK Government changing its position from the one that he announced yesterday in the autumn statement, will the cabinet secretary seek an early meeting with either the chief secretary or the relevant UK Government minister to press the points that many of us right across the chamber want to see happening, both on decommissioning and on the late-life asset transfer that is so important for the future of the industry? Yes, we will be seeking a meeting of that type, either myself or also the cabinet secretary for finance. It will centre on those points. I know that Tavish Scott has raised the issue around tax before, and having the right tax regime to encourage exploration and the full exploitation of remaining fields. Those are the main asks, but the one that we have asked for before, which the UK Government admitted that it had not taken action on sufficiently quickly, which was the ask of the industry for loan guarantees for infrastructure, will form the centrepiece of what we ask, but we will continue with that dialogue and I am happy that we will have the support of the Liberal Democrats in doing so. 5. Murdo Fraser Thank you. To ask the Scottish Government when it last met industry and consumer stakeholders to discuss the regulation of private car parks. Minister Humza Yousaf. I am most recently met with Citizens Advice Bureau to discuss private parking practices just last month. Transport Scotland met with representatives of the parking industry, Citizens Advice Scotland and Trading Standards Scotland on 31 August to discuss how we can deliver improvements to private parking practices across the country. A further meeting with the industry and consumer stakeholders is scheduled to take place next week. Murdo Fraser I thank the minister for his response. On Monday, I was contacted by a 90-year-old lady from Cymru who has been hit with a £100 penalty notice from the inappropriately named company, Smart Parking, because when she parked in the Canillos Street car park in Perth and she keyed her number plate into the ticket machine, she inadvertently entered a capital letter 0 instead of a capital letter O instead of a zero and got a £100 fine. In such a case, it is all too typical of the hundreds of live constituency cases that I have in relation to this one car park. Does the minister agree that not only are such actions by this company an utter disgrace, but given that he has the powers to act in this matter, because it is a devolved matter under the competence of the Scottish Government, does he agree to meet with me and discuss how we can work together to try to clean up practices in this industry? Murdo Fraser Of course I would agree to meet with the member that I have written to and we have had an exchange of parliamentary questions. He probably knows the process, but I will reiterate it once again that there is a working group that is looking at this. There are some complexities, he knows that depending on which route we choose down to go, is it keep a liability, is it charters, is it education, what route is it and he knows that there is a working group established examining these issues. I should say that we are also keen to hear what the UK Government approach is on this and there is a meeting taking place today between my officials and the UK Government. Once the meeting takes place next week with the stakeholders, I will ensure that he is informed by Transport Scotland officials about that and, of course, by me on the back of that, but I will be happy to take this issue on. I know that it is one that has affected his constituents on many occasions. Christina McKelvie To ask the Scottish Government what its position is on the recent UN report on the impact of welfare reforms on disabled people in the UK. Minister Jean Freeman The UN report, published at the beginning of November, concludes that there is reliable evidence that the UK Government's treatment of disabled people has led to grave or systematic violations of the UN convention on the rights of persons with disabilities. The evidence that rests on is drawn from a variety of policies that the UK Government has pursued, including abolishing the independent living fund, the introduction of the bedroom tax, the work capability assessment and changes to personal independence payment. The Scottish Government has been consistent in our opposition to those policies, and we agree with the UN's conclusions and are pleased that the UN report acknowledged the very different approach that we are taking. Perhaps more importantly, I am pleased that the UN, in its conclusions, gives disabled people the recognition that they deserve for the considerable suffering that they have endured for so many years. It is indeed a great pity that the UK Government continues to refuse and to see and hear the real damage that it is doing to our fellow citizens. I thank the minister for that answer. Given that, as she states as well, the UN report states that the UK Government welfare reforms have evidence of grave and systematic violations. Violations are the rights of persons with disabilities. Which violation in the UN code does the minister think is worse? Imposing the bedroom tax on poor people or taking away the independent living fund for disabled people? I think that the member knows that I cannot possibly choose between two such appalling policies, which have negatively impacted on so many disabled people. Last night, I had the privilege of attending the disability agenda Scotland reception hosted by my colleague Neil Bibby on equals still not, why not? Where they point out is that we found in our own social security consultation the severe mental distress as well as real damage that is done to individuals because of the policies of the UK Government and how they are pursuing them. I am particularly disappointed, in addition to the policies that Ms McElvie mentions, that the UK Government's continued refusal to step back from their cuts to employment support allowance when they trump it so loudly to us about the benefit of helping people into work. That is a real disappointment that I would hope that they would reconsider. Thank you. Before we come to First Minister's questions, members may wish to join me in welcoming to the gallery his excellency, Mr Torbjorn Solstrom, ambassador of Sweden to the United Kingdom.


Thank you. I appreciate you guys. This is amazing. This is amazing. I really appreciate you guys. Dude, I'm just welcome, you know, walking in. Man, I just, it just makes everything I thought when I met with the group. You know, Grail, and it confirms that, you know, this is a great place and we're going to do some great things together here, man. I'm excited.


It is over. There are but a few of us left, and it is good. Our purpose in this world was to protect it from things it could not understand, things that defied comprehension, things that we all secretly believed when one day overcome our meager defenses and consume everything. But then, one day, they were all gone, just like that. At first, we were relieved, but as time went by, we began to understand what it all truly meant. Without the storm clouds of the anomalous to cast their sheltering shadows over our empire, it soon began to crumble, dissipating like the remnants of some primordial night before the new light, the light of quiet days. Still, a few of us remain, flesh and blood ghost watching over what is left, over those left behind. He is, to all outward appearances, the very image of a respected academic, clean shaven, neatly dressed, infallibly polite.  He is sitting in a lecture hall, surrounded by students thirty years as junior, absent-mindedly scribbling with his pen and his notebook. The only notebook in a room full of laptops. A few years ago, he was a senior researcher, one of the best of us, the world's finest expert in the new groundbreaking field of applied hydrobiology. He had spent most of his adult life studying this field, learning all the derives to know about it. As a young man, he headed massive symposiums, traveled all around the world. His name was respected. As life had a calling, when the anomalous world disappeared, so did all that. The professor throws a question in his direction, and to his horror, he discovers that he doesn't know the answer. While he was studying a field that no longer exists, that to most never existed in the first place, the rest of the world marched forward, and he was left in the dust. He wilts under the professor's critical gaze, and mumbles something about cells. A subdued laughter echoes throughout the hall. The professor simply shakes his head and moves on to younger, more promising students. The agent researcher returns to a notebook, knowing that he will never lead again. He sighs. At least he has his pension. There are many like him, you know. Professor's destroyed, bright minds doled, curiosity riven. That's the price he pays. That's the price we pay. He sits and bombed and starts uniform, sweating like a hog as he looks out the window. Eyes squinting in the blinding light of midday. Air conditioner must be broken again. The dry expansions of the savannah pass slowly under the wheels of his armored jeep. Then the seat next to him, a subordinate drones on and on about this morning's attack. It seems like the U-Bit tribe once again assaulted the village of the Muthu, targeting, as is the way around these parts, mostly the women and the children. The man scratches at the rash slowly forming around his neck, and asks for a death toll. The subordinate says it's too early to say. They haven't finished sorting the bodies. Tomorrow, the Muthu will most likely counterattack, and the man and his score of so-of-soldiers will most likely spend the next day doing much the same as they did today. All of this for a dispute over 200 years old, a dispute over which hill the sacred salamnese crossed on the first day of her holy pilgrimage. The man remembers other days, days in which he fought for something greater, for the good of all mankind, for something he believed in. In those days, Wendy had best gazed up at him. Not only does he gaze his back, but he did so through the scope of an anti-material rifle. Things were simple back in the Coalition. Life in the peacekeeping force was anything but. It made sense, he's supposed. When one encountered a 20-ton fire-breathing sloth with 30 eyes and a nasty temper, one usually knew what to do. Not so much with a host of desperate, starving people locked in the midst of a civil war. That the GOC killed the sacred salamnese 10 years prior with an air-to-service missile only made things so much worse. There are many like him, you know. Nights left in the world with no more dragons. Nights in the world of windmills. That's the price he pays. That's the price we pay. A dark study in the middle of the night. Someone flips the light on. Slippered feet glide over the soft carpeted floor. A forum sinks into a comfortable chair stationed before a work desk. The room is lined with shelves, covering every wall. And on the shelves, toys. The soft light shines over bright red fire trucks, over plastic guns and glass chemistry sets. In large boxes where, if one looks closely, vaguely humanoid figures can be seen. Blank smiles plastered over faces no longer animate. The owner of the slippered feet rises again, approaches one of the boxes, one hand caressing the glossy cover. Introducing Dr. Wondertainment's Mr. Motion, the box exclaims in big bold letters. Multiple grip action. Make high-voltage pseudo-transmogrification, fun for all ages. Just twist his shoes and watch him move. The hand drops from the box, from the motionless figure within. Not anymore. Sighing, the owner of the slippered feet turns off the light and leaves to study, returning to the desktop computer in the living room, to the Excel sheets that needed to be filled by tomorrow. One had to make a living somehow, after all. There are many like him, you know? They tried to keep face to what they once were, but in the end, it didn't matter at all. What was once was, was, and never will be again. They will make no more toys. That's the price they pay. That's the price we pay. If he was anything other than what he was, you could say he was just a face in the crowd. Nothing about him is that memorable, unique, or interesting. If he was anything other than what he was, he might have been upset because of that. But he isn't. Instead, he just feels that it isn't enough. He knows that if he speaks to someone, they'll hear him, and remember, if for a while. That if he does something, people will notice, and they'll remember, if for a while. That for the first time ever, his actions will have repercussions. He is now a prisoner of the cruelest of all jailers, permanency, and it's a life sentence. There are many like him, though no two are quite the same. He knows that every little thing he does will last forever. He knows that every move he makes can be observed, and thus controlled. He knows that he will never truly be free again. That's the price he pays. That's the price we pay. A young woman sitting on a seemingly empty subway train. We can try and be poetic, stated the train is heading from nowhere to nowhere, but that would be a lie. She's returning from the hospital, where she visits a fertility clinic once every week. The doctors keep telling her that she shouldn't come back, that they've long given up on trying to understand exactly what's wrong with her. She won't, though. If ever up to her, she might have, but she's thinking on the man waiting for her at home. She's thinking about the warmth in his eyes every time he sees her, about how safe he makes her feel, safer than she ever remembers feeling, how the nightmares become just a little bit more bearable when he's around. She thinks about the box of old toys he keeps in the garage, about how he always wanted kids, about the disappointment he kept trying to hide every time she came back. Though she still goes, every week, just to hear the same reply over and over again. He doesn't know she's still doing it, and she's not about to tell him. There is only one like her, and for that we are thankful. Knowing that lets us sleep at night. Sometimes we had to make sure, you see. It was hard enough to make ourselves release her to begin with. The risk was too great. We had to make sure. She'll never be whole again. That's the price she pays. That's the price we all pay.


Welcome to part two of our breakdown of how banks work. In part one, we explored the simple business of banking, receiving deposits, lending money, and earning a spread. However, in reality, many of the large banks you have heard of offer a wide range of products focused on different customer types. There are a lot of different ways that banks split out their solutions, but one way to think about how banks are structured is to focus on the type of client each product serves, individuals, companies, and institutional investors. Individuals tend to interact with the retail bank, making money through the simple act of taking deposits and lending money. If you are wealthy enough, you may gain access to the wider range of services offered to high net worth individuals, where banks receive both spreads and product and advisory related fees. High quality retail banks offer brilliant customer service and a wide range of well priced retail banking products. Companies small and large tend to interact firstly with a commercial bank, which supports companies with their core business operations from payments and cash management through to trade finance and risk management. A commercial bank also offers a wide range of lending products, which forms the basis of the bank's business model. Larger companies will also interact with the investment bank, where fees are earned based on the quality of advice and speed of execution across corporate strategy and fundraising. Finally, institutional investors, also known as the buy side, will be the primary clients of a bank's global markets division. Here the bank offers a wide range of execution services relating to investing and trading activities. The best banks will win based on the quality of their execution, risk management and strategic expertise. Now in reality, the line between client types is blurred, with both retail and business customers taking advantage of global markets products. As you can see, banks can get extremely complicated. In fact, the complex jumbling of risk across retail, commercial and investment banking was one of the causes of the global financial crisis, as liabilities from one part of the bank compromised other safer divisions. This affected the UK banks to the extent that core retail banking and associated deposits were separated from investment and international banking activities.


And it goes, chance here for Calvert-Lewin, brilliantly taken. Oh, he's on the mark tonight, it's Dominic Calvert-Lewin, picked up by Davis. Davis into the penalty area for Umar Nias, oh, what a finish, that's what he can do. One more game from Evertsons, Lutman driving at the top of defence, goes for goal, Adamola Lutman, what a terrific goal by Adamola Lutman, well he's involved now. That's quality. Calvert-Lewin for Davis, Nias inside the penalty area, Umar Nias, from zero to hero, Umar Nias for Everton.