 A warm welcome to the first discussion session of this course. In this session, I want to begin by telling you our idea behind these discussion sessions, the philosophy behind them and how we intend that these should actually involve all the people who are participating in this course live and actively. We thought that instead of just having expositions one after the other, concepts being presented one after the other to the participants of the course, it is very important that communication be two-way, so to speak. We of course have the online discussion forum, but that is not enough. We thought we should try and recreate at least to some extent a classroom environment in which students who are listening to the videos come up with their own doubts, their own comments, their own observations, their own questions which enrich the whole experience of learning what we are trying to understand in this course. The way we are going to go about it is, I have several of my own students here at IIT Bombay who have been participating in the creation of this course and who have also been reviewing the content that has been created. And they have several questions when they listen to these videos, so it is natural that many of you out there who are listening to these videos would have either these or similar questions. So in the discussion sessions, we will take up several questions that our own students here at IIT Bombay have raised or of course comments, observations, suggestions, inputs, whatever you might want to call them. And we would like this to be an impetus to all of you to come up with your questions on the discussion forum which we will then also take live. We will select, you know, every week we will select a few ideas that have come up in common in the discussion forums and we will try and answer them live as we are doing in this discussion session. So welcome to the first discussion session again, we are going to have discussion sessions like this every week and these discussion sessions as I said before are an impetus or catalyst for you to create new discussions. Well, with this we go on now to the actual discussion, the first discussion that we are going to have on the content of the first week. And before that I would like to introduce my very enthusiastic students who have participated in reviewing these videos and have come up with several questions, comments and observations which we are now going to take up one by one in this and the next couple of sessions. So let me now hand over to the students who are sitting with me here so that they can introduce themselves one by one. So hi, I am Siddhant Ranade, I am a second year undergraduate student here at IIT Bombay, I am in the engineering physics department. Hi, I am Anush Anand Kumar, I am an engineering second year student in electronics and telecommunication in Thadumil Shah and engineering college, Mumbai University. Hi, I am Sushrut Thakar, I am second year engineering student here at IIT Bombay in energy science and engineering department. Hi, this is Pratik Fegade, I am a third year engineering student at computer science and engineering department IIT Bombay. Thank you. Hi, I am Megasham and I just graduated this year from the electrical engineering department IIT Bombay. All right, very good. So welcome to all of you and we are now going to take up a few inputs from you. So let me now invite any of you who has the first few questions or the first question to put the question and then we will try and answer it. So, we learned about linearity, so my question is why is it so important, why is it such an important property? I mean are there some examples in real life? In fact, let us answer the question in two ways. One is why is linearity important, the other is to what extent is linearity meaningful. Now I will take a number of different examples and I am going to encourage you also to take a few examples if you can, right. Let me take the very simple example of a human system first, right. So, to what extent is a human system linear in its behavior? Let us look at the response of a human being when given nutrition, right. When I am saying human being, I also mean any higher level mammal, let us say. To a certain extent, the work output that a human can give changes linearly with the nutrition that you give it. So, you know, I mean you could say that if a person has not had food at all and you given give him an X quantity of food, which of course when I say X quantity it is not just the quantity in terms of weight, but it is calorie value, calorific value and so on. Up to a certain extent, if I give it X quantity of food in that sense, that much of energy can be generated by the human being working out in terms of either exercise or in terms of physical work that the person does. Give it 2X, the person might be able to do two times at work, give it 3X, to a certain extent it might be 3 times, it depends on what that X is again. However, the human system can at one time take in only so much of nutrition. So, if you give it 2 large or multiple of that X, the human being is not going to respond by giving that multiple of the work output. So, in that sense, I am trying to point out to you that the human being is and is not linear. In a certain range of behavior, the human being is linear. But if one exceeds that level, that range, then non-linearity sets in behavior. And what is the input-output relation that we are talking about here? The input is the quantity of nutrition that you have given it, measured in some reasonably invariant terms. It could be in terms of calorific input, the amount of calorie input that the food could give or so on. It should not, as I said, it should not just be measured in terms of the weight of what is being eaten. That may not make sense for different kinds of food. And the output is of course, the amount of energy that the person is able to generate by virtue of work, either by working out or by performing certain tasks or whatever similar measure. Now, why is linearity important here? Within a certain range of operation, if I want to decide that I have to ensure that the person is able to work to this extent, I must allow him to take this much of nutrition, then linearity is important. I need to have a way to relate the kind of nutrition that I provide to the person and the kind of work output that I expect from the person. But it is equally important to understand that you cannot carry linearity beyond a point. And with this example of human beings, I am trying to explain that most of nature has this very dual character. In a certain range of operation, you might observe linearity, but beyond that range of operation, linearity is lost. Now, how does additivity and homogeneity come in? Maybe I will put the question to you. Can one of you think about what would additivity mean in this context and what would homogeneity mean in this context? Take the context of the human. Also, I would like to put one more question to all of you. Let me see if some of you can venture an answer. Can you think of other non-living systems? Here I talked about a living system. Can you think of a non-living system where you could illustrate this dual character of nature. In a certain range, there is linearity and then it is lost after a certain rate. So, let me put these two questions to you in this discussion. Let me see how you respond to them. So, we could have the system of a spring with a mass attached to it. So, if you attach a mass of say 1 kilogram to the spring. So, the spring might stretch by a certain amount, let us say 1 centimeter. But and say if you double the mass attached to it to 2 kilograms, the spring might stretch by 2 centimeters, but if you attach too large of it say 100 kilograms, the spring might just break. So, in that sense, it is not linear. That is very good. In fact, that is very accurate because here you have an example of a non-living system where there is linearity in a certain zone of operation where as you know the rules of the relation between extension of a spring and the mass connected to it hold and then that rule is violated if the mass is too large. In fact, this is typical. Either when the input becomes too large in its magnitude or also sometimes too small in its magnitude, there is a loss of linearity. In this case, of course, both in the case of the human being and also in the case of the human being taking nutrition and in the case of the spring being subjected to the action of a weight, in both of these cases, it is the too large context that is meaningful, too large and linearity is lost. Very good. Now, I also want you to answer the second question that I had posed, where is additivity and where is homogeneity here? What does additivity mean? Take both the context. Take the living context of the human and the non-living context of the spring. What does additivity mean physically then? What does homogeneity mean? Well, you all look puzzled. So, let me give you a hint. You see, for example, additivity would mean suppose I had two weights, that is a hint to you. How would two weights behave in the context of the spring? Additivity means I take the same weight and increase or decrease it, if there is a way to do that. Now, try and answer the question. So within a certain range, if one of the weights produces some amount of stretching in the spring and another weight produces another different amount of stretching, so the two weights when added together would just lead to the stretching in the spring be added. Very good. That is correct. So, that could be an example of additivity. I have weight w1 which creates an extension x1, a weight w2 which creates an extension x2. If I connect w1 and w2 together to the spring, the extension is x1 plus x2 and this holds to a certain range of operation. In contrast, I have two weights, one of which is twice the other. So I connect weight w1 and I see a certain extension. I connect the second weight w2 which is 2 times w1. I see a certain extension and of course the second extension should be twice the first in a certain range of operation. Very good indeed Siddhan. That is very nice. Now could you also extrapolate this to the human context? How would additivity and homogeneity make sense there? So in terms of nutrition, if you eat a certain amount of food and you do and you lift say 300 kilograms to the first floor and if you eat a different amount of food and you lift 500 kilograms to the first floor. So if maybe if you ate the 8 amount of food equal to the sum of both of those quantities, you could lift 800 kilograms to the first floor. That is very nice. Of course, the example is very correct in conceptual terms but I mean I must see this person who can lift 500 kilograms, you know it must be really quite a wrestler, you know. So we, I mean I wonder if it could be quite interesting to see. Not all in one go, not all in one go. Not all in one go, alright. It does not matter but the idea is perfectly correct that, you know you are talking about it, you know taking in this much of food and of course food measured in terms of its energy value in some sense and being able to give that much of energy or some fraction of that energy as an output in terms of the work that you do and then you know if you were to eat a little more then you could also do a little more. If you eat, if you actually ate both of them together that is additivity for you and what would homogeneity be then? Eating double would lead you to be able to lift double the weight. That is right. Having a more generous meal would mean being able to do twice that work very nice indeed. So now you have some idea of, you know, I mean I have answered the question in the sense of to what extent does linearity make sense and to what extent does it not? I would actually encourage a lot of people from our course participants to come out with more such examples and to query them on the discussion forum which we can then take up an answer. Thank you.