Welcome to everyone. My name is Alberto Square. I'm a senior lecturer in the School of Finance and Management which is one of the departments of SOAS University of London. And I welcome you to this event where I've been invited to share with you some views into the sort of stuff we do research and teach within the School of Finance and Management. It is just a small take towards what we do. We carry on in our activity but hopefully can provide some sense for you to better see who we are and what we do at SOAS and so on. I'd like to share some slides and setting up in one second. As I'm sharing one of the applications they have here open on my screen and I believe you must be able to see the slides at present. Pause for one second if there is any comment but possibly I can safely go on with the presentation if you can see the slides as I presume. Within SOAS it's a SOAS University of London. There are several departments and I belong to the School of Finance and Management so you may be familiar with SOAS. It's located in the very centre of London, just the corner of Russell Square, just a very close if you just a one-minute walk from a SNAID house and the British Museum. Within SOAS there are departments like the School of Finance and Management to which I belong which runs a number of undergrad, postgraduate programmes and PhD programmes and I specifically convene a teach in one of the MSc programmes, the MSc Public Policy Finance and Management which you can see here on the slide. There are other MSc programmes like Accounting and Finance, International Financial Management, International Business. Obviously as I convene a teach in Public Policy Finance and Management my main interest is in the area of public sector management, the broadly conceived. I'm interested in the process of making policies, implementing evaluating policies. I'm interested in the process of managing government and public sector organisations. I'm also interested in the process of managing financial resources in the public sector and that's specifically what I'd like to share with you a few thoughts about. What I do here is actually to share some of the writings I'm doing these very weeks as I decided to embark myself into a small paper which is intended to elaborate on an argument which was put forward about 10 years ago from the side of various colleagues. You can see here at the bottom of the slide the reference. So it was 2011 when in the Journal of Public Administration Research and Theory we had an interesting short piece called Why Public Financial Management Matters and the authors. There were several authors, you see Keoko, Malawi, Matsuki, Moody, Smith and Zhao. I occasionally referred to this piece of work whenever I teach because there is an interesting argument there. There may be arguments that we have in public sector management. They use areas of study and we can spot an area on public financial management or PFM and an area on public administration management or PAM. So public financial management has to do obviously with all the processes concerning the management of financial resources in the public sector. We have fiscal planning, we have budgeting, we have taxation, we have cash management, debt management, internal control system, auditing and so on. Public administration and management instead typically has to do with a way in which public sector organizations implement public policies. So issues concerning planning, staffing, resourcing, these entities carrying out the programs and projects and eventually evaluating what they deliver. Ten years ago these authors observed that there is unfortunately a sort of intellectual bifurcation between PFM and PAM. In the sense that the two areas of scholarship, they do have a fairly limited dialogue with each other. There is a focus concern on the side of PFM on the technicalities of managing financial resources, but it doesn't seem to be by and large much of a mutual change and the ways to investigate how the study of the financial, the management of the financial resource can affect the business managing and administering public organizations. It's not always the case that there are some studies specifically on budgeting for instance which typically may look at the technicalities of budgeting but they also discuss the politics around the budgetary process. But this was counted as an exception. So they completed this work about 10 years ago with a call in around the ways in which research on public financial management could be made more relevant to the way in which we manage government and public sector organizations. They suggested three specific areas where these studies could focus on as you can see in this slide. So they argued that research on public financial management could help PAM investigate the role of administrative expertise in democratic institutions. So this has to do with the relationship between typically elected politicians and representative bodies like parliaments and the government and the bureaucracy and the extent to which the technical expertise in the bureaucracy relates to the process of making decisions about managing the public sector within democratic institutions. But another area where PFM could contribute is how to improve the performance of public organizations. Public organizations typically have various dimensions of performance just to think about schools or hospitals or employment agencies. And of course much of this performance is not quite a financial one but nevertheless still pursuing the capacity of public sector entities to break even and improving efficiency, cost effectiveness of public sector entities can facilitate or can support the achievement of performance also in other dimensions. And finally the argue PFM research could also help investigate how public sector organizations can improve their adaptiveness to changing environment especially whenever they face a strategic problem about a better understanding the changing conditions of the environment and where to invest and where to find the resources in order to be better able to perform in the future. That's where we could say we stood up to the argument which was put forward from the side of these authors 10 years ago. And their concern about increasing public financial management with the generally public administration and management could make sense especially if you think about the context where they discussed these ideas. So 10 years ago it was the context still in the aftermath of the financial crisis 2008-09 which had various repercussions including increased the level of public debt in governments. And this entailed also sometimes a reduction of the level of public services together with austerity measures or some forms of restructuring of the systems for delivery of public services. I believe we can argue nowadays public financial management could be just more important than ever. The last decade has been very dense with events which affect the way in which the public sector works. So 2015 the launch of the sustainable development goals and the mini programs concerning poverty reduction to climate change and so on which affect decisions from the side of governments about dedicating resources to implement programs in these areas. Of course other initiatives like the Framework Convention on Climate Change brings the importance of energy transition diversification of energy sources at the top of the agenda of many of many countries. Over the last decade the full rise of China in the economic arena did have important repercussions in terms of international finance and financial flows. And then of course in the very last few years we could experience first the outbreak of the COVID-19 pandemic and the consequential effects in terms of deficits for the budgets of many governments especially with the drop of revenue because of the lockdowns and reduction of business activity and also on the other hand the surge of public spending like in healthcare. And then more recently of course we still live around the Russia-Ukraine conflict and we can just anticipate at this stage the effect this may have on international trade and domestic GDP in various countries and therefore also on public finances. So I believe the scenario is one where the code still makes sense to think back at public financial management and to better investigate the way in which public financial management could relate to and possibly contribute more generally to the studies on public administration and management. So there are basically three questions I still entertain myself to think and to write about. First just empirically to look at whether anything has been done in the last decades in order to close to fix this intellectual bifurcation which was identified about ten years ago. To think about why if this bifurcation exists is still there and how an improved theoretical dialogue between public financial management and public administration and management could take place. On the empirical side of this of this research well it is just to have a look by means of the bibliographic bibliometric assisted sorry literature review to look at what kind of studies have been done around the public financial management especially the last decade. So I went through one of our search engines we had at universities to search for articles journal articles and from one of those scopus as you can see I could spot a number of articles publications with the phrase public financial management or public sector financial management published in the area of social sciences and generally. So the phrase that the term by itself already starts to be used in the 70s but possibly you can spot from the diagram how precisely in the last decade there has been a remarkable increase of interest from the side of scholars towards public financial management as you can see the vertical line is there after 2011 so after Kyoko and other colleagues publication and clearly the number of articles which have been published in the last decade is quite considerable there is a growing interest towards public financial management. If you look at where these works have been published but first you may spot we're not quite talking about the too many works first as you can see I could spot like the 33 articles published over the last decade and a number of them are in a kind of a second tier journals in the area of the public sector management generally so top journals are roughly indicated by an indicator like scopus site score and so you can see like public administration review or public management reviews there in the bottom of this ranking they just highlight how these venues do not quite pay much attention to issues of public financial management or just the works which have been done on public financial management they do not quite find much or a lot to say in venues which are more about public financial public public public administration and management and journal like public money management which is the one on the top of the list is clearly more dedicated to the study of public financial management by itself so generally also apart from the matrix if you look at the if you if we read these works which have been studied we can find that the typically obviously referred related to issues which are at the very core of public financial management but the very weekly if at all related to concerns broader concerns of public administration and management so these could be studies about making reforms of public financial management especially introducing a cruel accounting in the public sector or international public sector accounting standards specifically there are some works concerning fiscal risk and sustainability of public finances especially in scenarios of austerity which are which are imposed to the country and there are a few studies which tend to look at public financial management systems and practices in emerging economies like or in countries which have been relatively less studied so far like Iraq or Afghanistan for instance and a few studies just concern about how to improve the curriculum what we teach in public financial management so by and large I believe we may share an impression that such an intellectual verification which was lamented that ten years ago is still there what's it the case is it just a natural state of affairs after all that you know we have scholars who tend to specialize on public financial management topics only and they may not have an interest to engage with the broader literature on public administration and management well there are a number of reasons I believe that why this state of affairs exists and can persist could be just a plain division of labor you know as academics we can't really study everything and in every field there is a natural inclination towards just to be more specialized in some topic rather than than other and other ones and can be quite challenging to to bridge different areas that could be also institutional reasons in the sense that some scholars could be intellectually more attached to certain disciplines than others so public financial management is clearly more related to accounting and finance and the public administration and management could be more related broadly if you like to political science or public law or more specific areas like public administration and management there could be like a parallel trajectory of embarking into research of course interdisciplinary research is always possible but sometimes in terms of productivity of research this may not be quite an advantageous route to take there could be however also pretty much an epistemological position so I mean what kind of approach to research is followed in these different areas and I would say the main pressure public financial management is pretty much a concern with documenting practices providing accounts of how public financial management systems have been designed how they work and providing advice on the basis of such an evidence about how next public financial management systems can be can be reproduced sometimes even just descriptive accounts of of pfm systems and practices on the other hand research on public administration management is a bit more diverse there are of course even descriptive descriptive approaches there but on the other hand epistemologically speaking research not part of research on public administration and public management has also embarked into a research oriented towards developing theories and then testing theories empirically so more of a mode of research around drawing hypothesis from theories which are then tested that means of various evidence collected for example through survey questionnaires out of a large amount of accounts of the way in which public administration and management practices are followed so for a number of reasons I believe we can spot various ways in which public financial management and public public administration and management tend to diverge from from each other and an open question then is whether is there any way to find a common ground to find that some way where actually research which is done in public financial management and public administration management could actually recognize their work along similar lines this could help to exchange views on their findings this could help finding designs for doing research where both pfm and public administration and management concerns can be can be shared and an idea I'd like to elaborate is the one of looking at at least the part of what is done in public financial management and public administration and management as having to do with the tackling design questions so design questions relates to the general idea of design science and the design science brings back to ideas especially coming from Herber Simon Herber Simon I believe is reputed as the only possibly Nobel Prize laureate in the area of public administration he wrote extensively about many areas including economics but definitely he also spent a lot of attention towards processes of especially making decision within public administration within government and the part of the arguments coming from Herber Simon where that we should be clearly distinguish sciences of the nature and sciences of the artificial as they put it and the sciences of the artificial has to do with investigating everything which are eventually artifacts created by us as humans so if you like a main line it may oversimplify here is that it's not quite the same to investigate the laws of nature and to the design to devise theories to formulate hypothesis and to expect evidence to falsify these hypothesis or not we may we may argue we do not know much about what's what's at source in the design of nature as we understand it but when we deal with what humans do in the social domain economics in public administration in government in managing public finances here we're dealing with the objects artifacts which are created by by by humans part of them are created in a very deliberate effort with design how to better take corruption practices in a public administration sometimes some of what humans do just in a sense arises out of their interaction so there are habits which are formed there are routines which are acquired and eventually the way which they behave is not quite completely deliberately designed and so therefore we may have a question about investigating how is it a certain state of affairs came out when approaching the science of artificial Simon suggested that well we are doing a particular sort of intellectual investigative inquiry because part of it is about precisely how to design such such systems so if you take as a matter of analogy the work an architect does an architect is embarking in a professional discipline which consists eventually in the production of an artifact which can be a particular unique house which has been built in order to get to that final end state there are a number of artifacts which are produced like the very designs of their house which is still to be to be built the way in which the design of the house is made typically comes out of a number of factors there are requirements coming from the needs of the client and there are a set of conditions including constraints which arise from the context where the design solution is to be applied so every house to be designed by the architect is a unique case in a sense so it does have to fulfill the unique needs and requirements of the specific client and it tends to take into account the specific unique features of the location could be wind, sun, shape of the terrain and so on and so on so how does an architect proceed it does not quite start from scratch it does not need to reinvent the wheel there are a number of principles typically condensed into what can be called design references like general principles about how we can design how we can solve the solutions for the specific design problem at hand okay so how to do with the light coming specifically from one side then another one or the wind and so on and so on so eventually what is produced is a specific design solution to that specific specific problem so I digressed a bit too much probably but the main point is that there has been a tendency then over the decades to also conceive part of social sciences social science research especially having to do with the design problems so there are design problems like how to design an improved way to deliver education in schools how to design a more efficient way to deliver healthcare or how to design an improved way to to to take a corruption in administrative practices or to improve planning, fiscal planning for our administration and so on and so on recently these arguments have been especially elaborated on the side of public administration and management by some colleagues of mine I'd like to mention here Professor Michael Balsely currently working at the LSE working at the LSE just 20 minutes walk from Sousa who used to be my supervisor of my PhD which I did at the LSE years ago and in his recent works precisely he elaborated the further the idea that public administration can be conceived as a design science public management can be conceived as a design oriented professional professional discipline and my interest at present is specifically in public financial management and what they believe can be told that they are is that what counts as a lot of research in PFM which is done around descripting and explaining how public financial management systems and practice work actually could be even better conceived as well as efforts to to help take design problems and offering design solutions eventually public financial management specialists come out with a number quite a considerable repertoire of ideas concerning how the financial resource can be can be managed how can we plan for it how can we design a taxes in order to extract the resources from our our economy how can we procedurally orchestrate our planning programming process so that the needs from the society can be taken into account within the venues institutional venues for producing plans programs individually budgets in government and in public sector organizations and so on and so on all of these I believe we can conceive as offering design solutions to design problems of course again with a very similarity to what are general design problem scenarios like for instance diversity of design solution taking into consideration the specific needs requirements of a client and specific condition to deal with so for example here I should mention it becomes quite apparent how apart from general principles of designing public financial management systems and theories the local context does matter so designing a system for doing the budgeting or internal control system in one country could be quite different from another country because all the institutional social system of routines and practices in every local context can be quite different and therefore also pfm systems and practices may need to be tailored to the specific specific conditions at the very end of my argument I just put forward the idea that remained a bit skeptical that the original proposal from the side of those authors was quite quite fruitful because in a sense they argued that pfm studies could focus their attention on some topics which are of general interest to public administration management but in such a way I believe that research on pfm would be an ancillary role with respect to pmm research and would need to divert their attention from away from the focus concern of the discipline I believe it can be just more fruitful to be quite deliberate explicit about the existence of problems which are not those to explain to develop theories about how is it why is it that things happen but these are more questions about design so how we can design systems which can take issues at the core of interest for public financial management but at the same time they also have implications and that can be shared as a matter of design issues also from the side of public administration and management so some examples questions like you see on the slide how fraud and corruptions and corruption can be successfully prevented, deterred, detected and sanctioned and this is the core of public financial management but clearly answer to a question about how to design to provide a solution to such an issue so could be quite of concern also for public administration and management how public money can be shown to be spent effectively and efficiently it has to do with the budget execution it also has to do with accountability and how emergencies no experience from the outbreak of the pandemic or the one of the war can be promptly promptly tackled which is something which calls for a response not just in terms of leveraging resources like foreign aid or military resources but also how we can marshal the financial resources in order to cope with an emergency so it is all around these how questions about how to design something and to create the new artifacts which can perform better where I believe both PFM and public administration and management research could find a common ground well I've been talking for about 25 minutes I believe and my intention is not to make a presentation too long after all you know 20 minutes is kind of a standard length of a presentation since we make in academic conferences and presumably also quite how much my audience could stand especially given the digital medium of interaction so just to stop here as I said at the beginning this is a short presentation was mainly intended to help convey what kind of questions what kind of studies of issues we are concerned in doing our research in the School of Finance and Management although of course I'm just opening a specific take around the public sector and management because that's where I come from that's my main idea of expertise at this stage I may ask Katie also to help coordinate because I would be very happy if you like to open a discussion around these these ideas if you have any any interest and experience in the way in which the public sector works how financial resources and problems around financial resources are taken in the public sector otherwise equally I'll be very happy to take questions more generally if you are interested in so as in studying the School of Finance and Management in particular and in any of the programs which I mentioned earlier hi Alberto yeah maybe maybe just see if anyone has any questions and then if not perhaps you can we can sort of look at doing the first option instead okay thank you Katie yes I'm very happy to still wait and pose for a few for a few seconds and otherwise if there is no further interest towards this topic or if you have any interest about any other topic you'd like to know more about what we cover in our programs I'm very very happy to talk otherwise if that's fine as I said earlier just as an overview I mentioned that we have a number of postgraduate programs within the School of Finance and Management that saw us and I'm very happy also to provide an overview concerning these these programs and just have a few slides here I will go through them very very quickly and I will leave enough time plenty of time if you have specific questions instead one of the MSc programs is the one on Accounting and Finance as the name says it has to do with this particular disciplines and you can possibly spot that they are on the slide but definitely you can have more information from the website the name of the course is the modules which are taught so there are core modules which include the dissertation which is a compulsory component for the program and you have a number of modules you can see they are the list they all have to do with Accounting and Finance of course taken at a certain advanced level because of these are possibly a number or the typical student in the MSc Accounting and Finance could already have a background in any of these disciplines you can also spot the logo of CIMA that charter the Institute of Management Accountants because part of the modules which are taken for the MSc also can be carried forward as a credit in order to get the CIMA qualification so you can get some exemptions. The second MSc is about International Financial Management so as the name says here it's about the marvelous, amazing domain we have nowadays in the world of the international financial system and flows so here we have a number of modules which are all concerned with specialist knowledge on financial management in the international setting bearing in mind the conditions of the real world business environment. Comvino of the program is Professor Victor Murinda he's the chairman of the Center for Global Finance within SOAS world known expert on such issues like financial inclusion but also very active also in other areas like for instance the rise of a fintech. In the MSc International Business we have here instead more of a focus on business and management of the company's enterprises rather than the domain of accounting and finance and here my colleague the Comvino Dr Juan Zhu she's an expert especially of Asian economies of China but apart from China this MSc International Business includes attention for a number of other geographical areas that we cover as you may well know SOAS as an institutional traditional expertise especially in Africa, Asia and the Middle East and the number of colleagues of mine have specialized knowledge on particular countries or regions in the world. And last and like also to mention the MSc public policy finance and management and the Comvino and the main trade is we're dealing with the public sector here so how public policies are made how they are implemented how public sector entities are managed how financial resources are managed how public governance works so the intertwining the relationship between governments public sector entities and entities in the private sector or a non-profit sector and hopefully in the previous presentation I could also provide a sense of a flavor of what is what is studied within public policy finance and management. So Katie and everyone it took as you could as you could see five more minutes just to provide this overview about the programs that we teach and at this stage if that's fine I pause again for for a few seconds just to wait if there are any questions. Sounds good thank you Albert so thanks for going over the overview I think that's really useful. Yeah let's just leave it a minute or two if anyone has any questions you can enter them into the chat or if you would prefer you can also unmute your microphone and speak to us that way and so yeah just leave a moment. Hello Anis I can see your line in chat I believe you may either switch on your microphone otherwise I wait if you prefer to type any question there in the chat all right so you applied to the MSc international finance and development which I believe belongs to another department if I'm correct if you're not going to try and help on the spot to see if I'm correct. Sorry Alberto yeah that is correct that's actually that's part of our economics department. Thank you thank you Katie it was just very quickly googling the name of this MSc just to remind me to which other department it belongs within within source I should say of course the different departments they may have a different intellectual orientation interest prefer the set of readings there is also a certain amount of overlap in between different disciplinary expertise and orientation and also colleagues in other department departments like economics or development studies they obviously pursue very interesting lines of inquiry concerning possibly matching and putting together some concern with the finance together with more specific concern for development. So if there are no questions as I believe my just very final line is to put there in the chat my email AA144 at source.ac.uk as I will be very happy to answer any questions you have or to help you find your way in the website or to find information you need before you fully understand what source is what we can offer and make a decision to apply. Well concerning scholarship Cristiana I'm not an expert in this area I can just say that within the source website there is a web page with which colleagues maintain consuming a number of scholarship schemes also for postgraduate students so the best they can do I believe is to bring your attention within the source website to these resources and Katie I believe either you are faster than me to suggest the web page to Cristiana otherwise I try and Google it and post these resources in the chat. I believe if we just Google source scholarships these must be the web pages thank you Katie it's the same one which which I found out of googling so there are a number of schemes of course a number of them could be open they're still open as I can see a few of them could be already closed just the point is to check if you qualify for them there may be restrictions depending on mission entity or other criteria. Thank you Alberto we can wrap this up slightly early if there's no further questions so thank you very much to yourself for the presentation and also to our attendees for coming along today. We do have a live chat session going on on the 13th of April and you can book that on our website and that is with our student recruitment team who will be able to answer more questions about scholarships fees and visas and accommodation and all those kind of questions before you start with us so I will just put the link to that in the chat before and before logging off um but other than that thank you so much and we'll thank you thank you Katie and everyone for staying with us for this time thank you thank you. Thanks bye


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Just crazy great design Bears rookie auto Cleal Herbert to 99 and we've got one of one Washington auto Clinton Portis Washington fantastic hit it was a beast One of one Clinton Portis auto mag that and redemption on the back is going to the Bears rookie jumbo Prime signatures booklet Justin Fields big congrats to the Bears Andrew a Get that redemption mag for you random to do and then we'll recap after the random Chargers Bengals Dolphins triple patch Herbert burrow two out of 25 luck eight times whichever team is up top after eight is gonna get the card one Two three four five six seven eight Dolphins get in the card. That was a good mixer so we had Dolphins went to this one in the random here Patrick Sertan dual patch printing plate for the Broncos Paying manning for the Broncos Michael Carter Elijah more James Robinson RPA redemption falcons leap a Frank's got Jason McPherson dual Jersey Pysco Burris CD Lam Autos Cleal Herbert George Kittle Dwayne Eskridge Kenneth Murray trace sermon Robert Smith We had a RPA book Devontae Smith Eagles the 99 RPA redemption or jumbo Prime signatures book Bears redemption Justin Fields one of one Clinton Portis auto for Washington Field past Jets Elijah more 25 RPA Jags Travis eats the end to 99 purple power fireworks Zach Wilson Jets to 49 Hyper for the Jets Zach Wilson a 175 and silver for the Jags Trevor Lawrence God flawless coming up. Thanks everybody


In the previous video, I made a promise that I'll offer some tips on how to use integration by parts properly, and so I now want to fulfill that promise. So integration by parts should only be used in the following conditions hold in your integral. So first of all, the integrand has to be written as a product of two things. There's got to be a product of some kind.  You have to have two factors. Now, admittedly, this first condition, well, most of the times we'll be thinking integration by parts because, oh, we naturally see two factors. In the previous example, we saw e times, sorry, x times e to the 5x. So I have that natural factorization right here. Now, in other examples we'll see in this video, that the factorization will be obvious. Now, in some situations, though, if you have a function f of x and the factorization is not obvious, well, admittedly, you should always look at the dx. The dx is part of that. And sometimes that's the factorization you use. You use f of x times dx. And that might seem like, well, how will that ever going to be helpful? But we'll see some examples exactly when it is. So don't focus on the first one too much, but we do have to think of how it's going to factor. Now, because there are some options on how one factors it, how do you choose you and how do you choose DV? Well, principle B here, it should be possible. We have to be able to integrate DV. You can't choose DV to be such and such if you can't integrate it. That'll get you stuck. And we have to also differentiate you. So however you choose you and DV, we should be able to find the derivative of you. We should be able to find the anti-derivative of DV. If you're missing one of those pieces, integration by parts isn't going to get you any farther than you already done. So we have to find the derivative. We have to be able to find the integral of the appropriate parts. And then this third principle is the one that matters the most. The integral of VDU has to be found. We have to be able to compute this new integral that's gonna pop up. Generally speaking, the integral of VDU should be simpler than what we started with. It shouldn't be more complicated, but it is possible that if our integral is no more complicated than we started with, it turns out if you kick the can down the road long enough, we live on a round plan and if you kick the can down the road long enough, you'll make a big circle. And it turns out I'll show you a variation of integration by parts, which I call integration by cycles. That can show you that even if this thing is no worse than you started, that's actually still a step in the right direction. But let's look at two examples of how a typical integration by parts calculation works. We have an integral. We want to integrate the function x sine of x dx. There's a natural factorization. We have this x times sine of x. The dx is always gonna go with the dv, so that's not too much in subject here. There's no question about that. The question is where we put the x, where we put the sine. Now, we have to choose someone to integrate, which we call you. We have to choose someone to integrate, which we call dv. Now, if you look at x, if we take the integral, or if we take the integral of x, we're gonna get an x squared, you know, up to a constant multiple. But if we take the derivative, we're gonna get a one. x disappears if you take its derivative, but it gets more complicated if you take its antiderivative. That right there seems to suggest to me that I should set x equal to u. But let's see what's going on with the other side of things. What about sine? If you take the derivative of sine, you get a cosine. And if you take the antiderivative of sine, you get a negative cosine.  So in some respect, it doesn't matter where the sine goes. The derivative or antiderivative, you're gonna end up with a constant multiple of a cosine. And the linearity property of antiderivatives tells you the sine, the plus or minus sine doesn't matter. And so because of that, it doesn't matter where you put the sine, but it doesn't matter where we put the x. We can simplify if we take the derivative of x, we complicate if we take the antiderivative of x. And so that's gonna guide our hand here. We're gonna set u equal to x, and hence du will equal dx. And we're gonna set dv to be sine of x dx. Don't forget the dx there. And then v is gonna equal, like I said before, negative cosine of x. And so if we use that choice for integration by parts, remember integration by parts, the formula is the integral of u dv we equal uv minus the integral of v du. So putting those parts all together here, we end up with x times negative cosine of x minus, pay attention to the minus signs here. We're gonna get integral of v, which is negative cosine. You have to be very cautious here because the derivative and antiderivative of the sign is very similar. We have to be very cautious.  In this case, since we're taking antiderivative, it should be a negative cosine. So v is negative cosine, then we get du, du is just an x. So notice here, if we simplify this, we get negative x cosine of x. That part of the antiderivative is already done. We took that part. Then we get a plus integral of cosine of x. So if we look at the last principle, are we in a better situation than we were before? Can we find the antiderivative of cosine? And the answer is, yep, the antiderivative of cosine is a sign. And so taking the antiderivative there, we get negative x cosine of x plus sine of x plus a constant. And do notice it'll be a positive sign because we're taking the antiderivative of cosine, not the derivative of cosine. And you could verify by usual derivative techniques that the derivative of negative x cosine x plus sine of x plus a constant is in fact x sine of x. And so we were able to use those guidelines for integration of parts to help us find the antiderivative. We chose u and dv to make the integral simpler and the new integral was something we could calculate. Let's look at another example. This time, let's look at the integral of 2x squared e to the negative 3x. So we see this natural factorization. There's a factor of x squared. There's a factor of e to the negative 3x. The dx will always go with the dv and then the two can go wherever we want because it's a constant multiple. So we have to choose a u and we have to choose a dv. Now, generally speaking, putting monomials for u is a good idea because when you take the derivative of a monomial, it's always gonna be a monomial and oftentimes it'll be a simpler monomial. That is the power will be smaller. On the other hand, the e to the negative 3x, kind of like sine, it doesn't really matter where you put it. If you take the derivative of e to the negative 3x, you're gonna get a negative 3 times the exponential again. If you take the antiderivative of it, so we take e to negative 3x dv for dx, for example, its antiderivative is gonna be a negative 1 third e to negative 3x.  And although we might prefer integer over a fraction here, constant multiples don't make much of a, they don't have much of a consequence on the integration. So whether you take the derivative or antiderivative of the exponential, you're gonna get the same thing. It was gonna get exponential again. But by placing the power function 2x squared in for u, when you take its derivative du, you're gonna get 4x dx. You'll notice that the power of x goes down by one. And so applying this to integration by parts, we end up with a u times v. So we're gonna get a negative 2 thirds x squared e, that doesn't look like an e, e to negative 3x. Now we're gonna subtract the integral of v, which was negative 1 third e to negative 3x times du, which is 4x dx. Like so, simplifying the integral, we end up with negative 2 thirds x squared e to negative 3x. I didn't do anything to that. We're gonna get a plus 4 thirds integral of x e to the negative 3x dx. So does this new integral put us in a better position than we were before? And I would say the answer is yes, right? Could we find the integral, the anti-derivative x e to the negative 3x? Well, maybe I don't know it off the top of my head, but it's like, oh, we could just use integration by parts again, right? Can we do it a second time? We're say u equals x, so du equals dx, and dv would equal e to the negative 3x dx, and hence v would equal again negative 1 third e to the negative 3x. Because if we were to do integration by parts a second time, the power of x will go down one more time, and eventually this process will end. Because we had an x squared to begin with, we have to do integration by parts twice. If we had like x to the 12th, we could do it 12 times. That would be a long exercise. I didn't assign one that big. But in principle, we could reduce the power of x by doing integration by parts over and over and over and over again. Eventually, the power of x will disappear. And so if we do that, we're just gonna copy the piece we had before, the negative 2 thirds x squared e to negative 3x. Now we're gonna get plus 4 thirds, and make sure this 4 thirds distributes on both parts because this integral will be replaced with two parts. We have a new unv, which is gonna be negative 1 third x e to negative 3x, and then we're gonna subtract from it the integral. We're gonna get v again, which is now negative 1 third e to negative 3x times dx, like so. So in this situation, do distribute the 4 thirds onto both pieces here. And so we have two parts of the integral calculated. We have the negative 2 thirds x squared e to negative 3x. That part's just on for the journey this time. It's not gonna change. We now get a negative 4 ninths x e to negative 3x, and we have a positive 4 ninths integral of e to negative 3x dx. And so since we're using integration by parts twice, we didn't have to ask ourselves the question the second time. Is this last integral in a better position than we were before? And the answer is yes. We've now integrated e to negative 3x twice now. We know it's anti-derivative. It's gonna be a negative 1 third e to negative 3x. And so applying that, we get all of this one more time, negative 2 thirds x squared times e to negative 3x, negative 4 ninths x e to negative 3x. And then we're gonna get a minus 4 over 27. I'm taking 4 thirds times negative 1 third e to negative 3x, and don't forget your plus c. And this gives us our anti-derivative. All right, I'm gonna put a nice box on it, maybe a bow for Christmas time, because it's just so wonderful. That's my pretty bow right there. And we get our anti-derivative. Now, integration by parts is a technique that we do by parts. Imagine you have this six foot sub-same that you have to eat. No one's gonna eat it all in one bite.  Some can be done in two bites. Some, like this one, can be done in three bites. And more complicated ones can be done in much more bites if you had to. This process can be iterated by doing it over and over again until all the powers of X are exhausted.


Hello everyone, we welcome you to another interesting episode of how to ace the hackathon. I are in the mandal on behalf of entire smart India hackathon team. Welcome you to another important and interesting masterclass on how to crack the hackathon. Today we have with us another participant and a winner of previous edition of smart India hackathon, Ms. Sanika. Let's have her on screen and directly speak to her about her experience about the hackathon. So, Sanika, hi, welcome to the show. Hello. How are you? I am nice, how are you? Good, good, good, good. So, thanks for taking out this time for us and you know as we want to help the first timers, the smart India hackathon aspirants to understand and have some tips from the so-called pros of the previous edition of smart India hackathon and why not because first hand experience is always the best. So, we'll start with this. What was your take about smart India hackathon? How did you hear about it first and what was the initial reaction to this entire event? In the hackathon as a network, a mail was shot in our college that says this is happening and in turn the hackathon is happening and we have to participate in it. So, we were locked down batch so we didn't have much of a exposure in college and that was our first year in our college. Being in the second year, it was our first year. So, we were a little scared to participate but our professors and all asked us to participate and give it a try. So, we made a team and we participated in the internal hackathon. Good. Did you hear about the word hackathon before this event or was it the first time thing that you were experiencing? No, no. Before, when in COVID, I have participated in various hackathons like online on other platforms. Okay. So, for the user, I mean the for the viewers who are listening to us right now, what is the one thing or what is the thing that separates smart India hackathon from the rest of the hackathons and how do you think it is a entirely different platform or ballgame together? If you can just let us know. Other hackathons were like a small exposure but smart India hackathon was a vast exposure. Like it was the first hackathon and it was so long for me. It was 36 hours long. And like other hackathons, they really asked us to participate on our problem statement this time we were challenged by SI stream as for the problem statement to work on. Great. So, viewers and my dear friends, you just heard it. Smart India hackathon is basically a platform where the real-time problems that have been faced by the country and the nation and the entire world today is being put as problem statement to you and you have been given a platform to develop a solution on that. So, basically you are in turn helping the nation grow. So, that's something that separates smart India hackathon from the rest. So, Sanika coming back to you, breaking down this entire process from initial idea submission to the training and finally to the grand finale where the main action happens. If I want to take your inputs and if you want to give our viewers that this kind of thing should be kept in mind while each phase happens, what would it be? The first and the foremost thing that everyone should keep in mind while participating in the hackathon is you have to be clear about the problem statement and you have to understand as in what the problem statement is. Our problem statement was attractiveness and user friendliness of AICP. So, we understood it as attractive as in the website being user friendly so that every user could access and attract them in an aesthetic way so that everyone finds it pleasing and everyone's attention could be caught on the side. So, understanding the problem statement, then a presentation was on the basis of what everyone is selected. So, from the first stage till the final stage until you reach the loader center, it is your presentation which presents what you are going to implement in the hackathon. So, a good presentation is a plus one for getting selected and on moreover, you have to have a strong team of good domain experience or whatever your problem statement is. So, that's all I need to go ahead with. Great. So, participants, you just as you heard that you know, we have been constantly reiterating from all the winners of the past edition that the base thing is be clear about what exactly the problem statement is demanding from you and secondly, a strong team to back it up. So, it's basically a combination of both which leads you to the final round and leads you to become a winner. Okay, Sanika, coming back to your point, any mentor was there with you and how did they contribute and how important do you think a mentor should be for a team? According to me, mentors are pretty much important in the initial stages of your hackathon. If you are participating for the first time, definitely take help of your mentor. We chose a mentor from our college professors. He assisted us with like how you have to present, how to identify your users and he also helped us with meeting a past elimination of parasites. They also guided us in a way that how all process goes on. So, I guess, mentors are pretty much important when you are participating for the first time. Great. If I talk about on-ground during the grand finale that was happening at your noodle center, what was the experience and anything that you would like to share to the viewers on the interaction between the evaluators, the mentors that happens. I mean, there are three evaluation rounds that generally happens during the session. What are the things that has to be kept in mind while you are interacting on those evaluation sessions with the evaluators? Firstly, our noodle center was Chandigarh University. So, we were very fascinated by going there and the whole process, the 36 hours were broken down into three rounds and three mentoring sessions. Like first, the mentor used to come evaluate whatever we have done and based on that, he used to mentor us as in what all could be done, what all unique could be done or what all needs to be implemented in the problem statement so that it could be presented well. So, every mentoring round was carried upon by an evaluation round. In the evaluation round, they used to judge us on the basis of what we have implemented. So, in our case, it was like first, the mentors used to come and they used to evaluate the more well we presented, our points used to go up. And that way, we carried upon with three rounds and we had very good competitors, but still we managed to get the position. That's great. So, here we want to highlight another important point for all the first-timers that it's not only about the ideation that you do. It's how much you can implement it on a real time while you are at the hackathon menu. And also, is your prototype actually implementable in the real time? Is it scalable? So, these are points should be kept in mind while you present or develop your prototype during the 36 hours hackathon or in case 72 hours hackathon which happens for the hardware problem statements. So, Sanika, what exactly were evaluators, were there any suggestions that were given to you and how do you think the mentoring round during those 36 hours helped you out? The mentoring rounds were pretty much important. Like it was the clue which you have to implement. Like the mentor used to come and he used to give us a clue that this problem is very much important right now and like how it could be implemented and what could be the use case. Like what is the main problem? So based upon the problem we used to think like what more could be implemented so that that problem could be solved. So, that is what they expected in an evaluation round. And on the basis of those mentoring rounds only we got points. So, like in our case it was like first mentoring round we was pretty much excited about like we did very well but in that excitement we like we did not do the second round very good. So, you have to maintain the consistency to that also plays a very important role ahead. Exactly. So, you know as you heard from directly from the winners it's consistency. So, you know sometimes it ought to happen that you are so excited that you forget where you are actually heading towards. So, do not do that mistake. So, yes Anika I think it was pretty much insightful to conclude it. If you can just give couple of tips and hacks that would help these first timers to you know sail through this entire process what would it be? A couple of pointers. A couple of pointers to get selected is a good presentation understanding of your problem statement and knowing the implementation value is all I could say. Be confident if like there are enough sources on the internet you would study and solve the problem. So, you have to be confident is what could take you forward. Perfect. Thank you so much Anika. Thank you for taking out this time and all the very best for your all future endeavours hope to see you soon. Thank you. Thank you so much. So, viewers wasn't that interesting? I am sure you could relate to Sanika at some point of time. If not in the coming hackathon you will definitely do. With this I think you got a great set of ideas and tricks that will definitely help you during all these rounds of smart India hackathon.  Till then have a great time and stay tuned for more information and interesting videos like this. See you take care.


What's up everybody, Jason here for JasperysCaseBrace.com. We just sold out 19-20 Pinney Hoops premium stock, five box break. Number six, two random teams each. So again, 15 total spots, everybody gets two random teams, the NBA, all card ship. So again guys, premium stock hoops. Dice Roller, here's the customer names from Kevin down to Robert. I land it down to the Washington Wizards. So let's roll it and it is a two and a six for eight times. Good luck. One, two, three, four, five, and six, six, and then seven, eight. Young men, young men, six, seven, eight. On the teams now. One, two, three, four, five, six, seven, and eight. Miami Heats down to the Golden State Warriors. Young men with the heat, Rob, you have the Clippers and Thunder, John with the Mavericks, Rob with the Nuggets, James Connell, you have the Pelicans, John D with the Grizzlies, Robert with the Raptors, Matthew with the Pacers, Robert with the Magic, Matthew with the Bucks, Chad with the Kings, Robert with the 76ers Cavs, Kevin with the Lakers, James K with the Blazers, Matthew with the Tewels, Nick Stober with the Pistons, James with the Hawks, Matthew with the Wizards, Brian with the Knicks, Nick Stober with the Hornets, Brian with the Nuts, or with the Netsart, James with the Celtics, Chad with the Suns, Robert with the Jazz, Bulls, and Spurs, Kevin with the Rockets, and young men with the Warriors. Yes, so we have partnered up with this guy that does submissions and he picks up and drops off at PSA Weekly. His name is Mark Scarts. He's on Instagram. So yeah, the way it works, guys, is basically we're kind of just a hub, you know. So if you were to hit something huge and you want to get a graded ASAP, what you would do is, as it says on the post, you know, you go onto his website, you fill out the form.  So it's kind of just more of a partnership.   So yeah, there's no kind of avoiding the middleman. He's kind of helping us out to help you guys out as well. So, you know, you start to go through him basically, but the fact is that you can now just keep it here, fill out the form and then, you know, like I say, he'll come pick up from here. We do Jack, we don't have any.  I just don't know if we have got any in a while. We are Mike. We took out a couple more spots so that way it can get closer. So we have those three spots straight up and then a couple more fillers and that's it. But we are much closer now for sure. We sold another three spots today straight up. We did another filler today. So I have to do a couple more fillers. But like I said, we are getting close guys. If that doesn't happen tonight, it's gone tomorrow. It is gone tomorrow. I don't think it's passing tomorrow. So let's continue to sell those fillers. That's just really been selling us out quickly guys. Honestly. I know what wants to trade guys. So yeah, refer to this page. Like I said, three spots plus three fillers. We're gonna get another one done tonight. We only have two fillers and three spots. So let's get it done. No, it's all good. I mean, I think for people that, you know, don't maybe break with us so often, but everybody has their own different formats and different ways to break. But yeah, sometimes for us, I mean, there's no time schedule when it's gonna break. And that's whenever you guys fill it. All right, guys, I'm closing the trade window. But no worries, man. I mean, I always want to fill breaks as quick as possible. But I know, especially for a mixture like that's been a while since we had a really high animal like that. So it's probably gonna have some people either say I'm gonna take a full spot or I'm gonna try my luck with filler spots, which is all good. It's just marinating more guys. That's all. All right, so here we go. No problem. Like I said, you can read more about it on the Instagram post. I think it explains it pretty good. And how the wait times are right now for certain submissions. Alright guys, so I think we open up a new case with number five and like Joe just kind of scribbled through these boxes. So letting you know that they're all from the same case to do like some scribbling. So this is a five box break guys. If I have at least at least enough for two more breaks number 78. And like I said, this was one of the bigger releases that I had missed when I was out. But I did a couple of these boxes recently on on Jaspies.com and this stuff is very beautiful. This stuff is really nice. Alrighty, good luck guys. David Bertens, Deondre Jordan, Danny Green, Jerry Smith, and we got a green Lonzo ball. Number 74 out of 99. We had Kendrick Nunn rookie. Joe Harris and his canter Simons. Carson Edwards rookie. We have been Dari Wetherspoon. We got a taco fall. Josh Richardson, Lansing Galloway, Brooke Lopez, little Alan Iverson, Tobias Harris, a pink Vince Carter. That's number 225, I believe 22 out of 25. Atlanta Hawks. Tim Hardaway Jr. Derek Favors, JJ Redick, Kyle Guy, we have Kevin Hooter, and Kelly Dobbery, Jr. Alright, I'm taking Ohio State, guys. Alright, next one, guys. Remember, guys, for that Mac, the first off the line basketball, all we need is that last pack filler. We can break that as well tonight. Kuzma, Siakam, Fournier, Jalen McDaniels, we got a Nicholas Claxton, red, and then this looks like one of those shimmers, which I believe are one of the most popular. PJ Washington, Jr. Robin Lopez. Frank Jalen theCUBE, we got a Tyler hero for the heat going to Young Min. We got JJ Redick. Dort. I don't know who's Ricky's in here. Nice. Kurox. Holiday Buddy Heald. Dirk. George Hill. Austin Rivers, number two 49. Steph Curry. Josh Jackson, Ilya Silva, Grant Williams, Robert Covington, and Julius Randall. Alright, next one, guys. What up, Jordan? How's it going, buddy? Nice little Zion. How about that? Zion Williamson for the Pelicans. That is going to James. You got a Damian Lillard, number two 35. Tribute. Like Ham Reddish. Yelosilva Vince Carter, Damian Williams, Mefiondu, and his cancer. Terrence Ross, Kyrie Irving, Fusovich, McKinney. You have Deandre Hunter, Miles Bridges, and Jonathan Isaac Green. Number two 99. No, what is going on over there? I have no idea, man. Bama DeBio, Blake Griffin, Admiral Skollfield, and, wow, an high voltage. Kevin Durant. Brooklyn Nets, that is going to Brian. We got Cody Martin. That's a rookie as well. And how about a LeBron James for the Lakers? When a Kevin R. Yeah, that's right. Is today the day that he's actually doing the giveaway today? I know you had like the last couple of weeks to submit and, you know, do the following and get answered to win. Oh, that's cool. Oh, did you really? Nice, Jordan. That's sweet, man. Craig Smart, Rudy Gay, Lou Dort. Ricky Rubio. Also, Rob, I see you have the thunder. So you got a couple of dorts. I might be interested in one of those if you're willing to sell one of those, man. Let me know. Brandon Clark, Jake Crowder, Seja McCollum, Lonzo Ball, Terrence Davis, Alex Caruso, Paul George, Daniel House Jr. Jermay Grant, Dylan Wendler, we got a Chuck, we got a Talon Horton Tucker. Whoo. Talon Horton Tucker. Oh, that man. What about him in preseason? Everybody was, it's stuff skyrocketing guys in preseason. Evan Fournier Green, Jared Allen, Kawhi Leonard, Malik Monk. Another high voltage Kevin Durant. Wow. I like Dora. I mean, I'm a big Thunder fan, but I'm more for just collecting him. That's all. Tobias Harris and Carrie Irving. Oh, no, Rob. I gotta send you something or do something. I appreciate that. Whoa, this was cut wrong or that's just the cut line maybe. I don't know. I didn't realize that. All righty, last one. Our last box is Gordon Lonnie Walker, Paul George, Mike Scott and a J.R. Smith. 299. You got a Deandre Hunter, Terry Rosier, Drew Holiday, Cam Reddish, Daniel Gafford, Steph Curry, Ignace Brazdaikis, Courtney Lee, Trey Young, Thomas Bryant, Shaq Diesel. How about a little bowl bowl for Denver? It's going to Robert. Very nice. Lonzo Ball. That's a purple. I think that's number two thirty-five. You got Zubac, Danny Green, Marky Morris and Darius Garland. Very nice. Terence Ferguson and a Thomas Bryant. I think he just got injured though out for the season in Torney, Seattle, I want to say. Well, there you go, guys. That was Hoops Premium Stock. We're going to top load all these penny sleep cards and then we're going to penny sleep every single card. Just have it protected as well. So, do not worry, guys. All right. So, there you go, guys. Lot of value in this. These cards are beautiful. All right, Robert. Thank you, brother. Lot of value in this stuff. Really, really nice stuff here. So, I got a lot of stuff there. Everything ships and then we got a couple Kevin Durant, LeBron and one Zion. So, there you go, guys. Appreciate it. Number seven should be posted up soon. You guys want to get another one? Go to jaspyscasebreak.com.


folks. So, welcome to yet another session on our NPTEL on nonlinear and adaptive control. I am Srikanth Sukumar from Systems and Control IIT Bombay. As always, we look at our inspirational image to motivate us to develop algorithms that will allow us to drive systems such as these on Mars and Moon and other such explorations autonomously. So, without delaying any further, we move on to our lecture material. So, last time, until last time, we were looking at, I mean, we first looked at completing the proof of the norm requirements for the two norm, which is the Euclidean norm. In the process, we also saw a short little proof of Cauchy Schwarz inequality, which is rather specific to this Euclidean sort of space. So, today, of course, you will look at a little bit more general proof of this. Then, we looked at the notions of convergence, Cauchy sequence, what is this convergence? What is Cauchy sequence? And we also saw some examples of Cauchy sequence and convergence not being really an identical concept per se, but there being small differences between the two and the possibility of constructing a Cauchy sequence that is in fact not convergent by creating a sort of weird vector space. Now, we also spoke about complete norm linear space or Banach spaces until now. And the good news was that all the spaces that were really in consideration or that are going to be in consideration in the future in this course are all going to be Banach spaces. So, this is definitely something that's comforting for us that we don't have to actually verify every time whether the spaces we are working with are Banach spaces or not. So, we don't deal with such very specific, very special cases in this course and in most applications, you would not find such cases. Then, we looked at the notion of an inner product space, which is slightly more general than a norm linear space. We also saw the definitions of what is an inner product. And we in fact also saw that if you're given an inner product space, there is also a corresponding norm which can be defined as the norm of x with x in this capital X being defined as the inner product of x with itself. So, this is sort of where we were last time. So, I will actually just start here and call this lecture five. So, I think let's see. I think I did not label this completely because we started somewhere here, I would say. So, let's not worry so much about these numberings per se, but this is sort of where we started the lecture four and so we are at lecture five now. So, just like the notion of a complete non-linear space, there is of course also the equivalent notion of a complete inner product linear space. So, when do we say that this space is a Hilbert space? We call the space Hilbert space, if first of all, we have an inner product space that is x with the inner product operation, which is complete with the corresponding norm. So, not with any arbitrary norm. So, maybe the inner product space admits a different norm also, but we are not concerned with that. If the inner product space is such that the associated norm defined as this, in this associated norm defined in this way, the vector space is complete, that is all Cauchy sequences converge, then the space is called a Hilbert space. Then the space is called a Hilbert space. Again, this is not difficult, we will let's go back again. I mean, if we take our usual Rn, let's see, write an example, as our usual Rn with the inner product being defined by x transpose y, x comma y in Rn, if this is the inner product space we are considering, then the norm that we get, so the norm that we get, in fact, let me be very careful, I think there is a slight error in how we have done this. So, this norm, in fact, should have a one half, there should be a one half. In this definition of the norm, there should be a inner product of x with x to the power half. So, the example that we were giving is that, if I have Rn with x transpose y, that is the usual dot product, scalar dot product that we know being the corresponding inner product, then if you look at the norm, the associated norm, it is simply, let's see, let me write it in this top product notation, it's simply going to be x transpose x to the power half and this is nothing but the two norm of the vector as we have defined until now. This is exactly the two norm of the vector and we already know that Rn with the two norm, as you can see from this example, Rn with the two norm is in fact a complete linear space. And therefore, Rn with the two norm is a complete linear space and hence, this inner product space satisfies all the requirements for being a Hilbert space. So, again, the good news is that most of the space times we work with Euclidean spaces, that is Rn, RP, Rk and so on. And in all these cases, the vector space, the inner product that we consider is the dot product. If we take the dot product as the corresponding inner product, then Rn with this dot product is a Hilbert space. And therefore, we are able to handle most of the cases that we are going to cover in this course. So, most of the examples that we covered in this course are in fact going to be Hilbert spaces. So, moving forward, until now we did look at the matrix norm. So, let me again write this, we already introduced this, but not completely. So, we are going to try to do a better job of defining and explaining what the terms mean. So, we defined the induced matrix norm earlier as somehow representing the maximum magnification that a matrix provides to any vector in the vector space. So, the induced matrix norm, so I mean I will actually want to write this definition a little bit more generally as this is the soup over all x in a vector space. So, we essentially have to have not just a vector space, but a non-linear vector space. It is not just enough to have a vector space, but we also need the norm without which we cannot actually write these quantities. So, the important thing to note is that the induced matrix norm is defined as this that is it is the somehow the maximum magnification provided by the matrix to any vector in the vector space. So, although I have written it in this way, it is for the purpose of whatever we are going to do, it is sufficient for us to have x as some Rn. So, that is what we are going to assume that x is actually some Euclidean space some Rn. So, one of the things to remember is of course, we are we have to assume that it is not the zero vector because otherwise I have a division by zero and it does not make sense anymore. And other thing is I am somehow introducing this notation of a supremum a soup. So, what is the soup? The soup or the supremum is simply the least upper bound. So, the supremum is the purpose of the supremum is to sort of generalize the notion of a maximum. So, the purpose of the supremum is to generalize the notion of a maximum. So, what is the supremum? The supremum as we say here is the smallest value y such that for all x in s, x is less than equal to 1. Now, suppose I did not write and remove this from the definition. Suppose I remove this particular qualifier from the definition that is the smallest value, then this is nothing but the definition of a upper bound of any set. This should be obvious to you. However, the fact that we have added this qualifier makes it the supremum. That is, it is the least upper bound. It is not any arbitrary upper bound. For example, if I took any set, let us say I took a set of the form, I took an open set 0 comma 1. So, what is an upper bound? How many upper bounds? 1 is an upper bound, 2 is an upper bound, 3 is an upper bound. So, y equal to 1, 2, 3. Anything works. However, if you are looking at the least upper bound, this is the only answer that you can actually say is correct is 1. Nothing else works. You can sort of think about this very carefully. So, you can think about just make a thought experiment. Suppose I take, let me call my soup. So, soup, if soup of 0, 1 is equal to say 1 minus epsilon. Some epsilon positive. Suppose I took some, suppose I claim that my supremum is, in fact, 1 minus epsilon for some positive epsilon. Now, the question is, is this even positive? So, think about it. This quantity 1 minus epsilon should understand that because this set is an open set 0, 1 is an open set. So, what does it mean to be an open set? It means that any term arbitrarily close to the boundary is also in the set. So, it should be obvious to you that, it should be obvious to you that you have 1 minus epsilon plus delta belongs to 0, 1. So, where some delta positive, and 1 minus, as long as 1 minus epsilon plus delta is less than 1. As long as these two conditions are satisfied, as long as these two conditions are satisfied, such 1 minus epsilon plus delta is contained in the set 0, 1. This is by virtue of the fact that this is an open set, which means that if any number which is arbitrarily close to 1, but less than 1 is contained in the set. So, you can imagine this is always possible to choose. I can always choose a delta positive such that 1 minus epsilon plus delta is less than 1, because all you need is that, what do I need? I just need that. This essentially implies that delta has to be less than epsilon. As long as I choose a delta which is less than epsilon, I am good. Now, the amazing thing is this 1 minus epsilon plus delta.  So, 1 minus epsilon plus delta is greater than 1 minus epsilon.  The supremum is unique, is the supremum. But we just show that it is not possible, because there exists a term in the set which is larger than this supposed to be claimed supremum. Now, how can something be a supremum when it is not even an upper bound? So, arguing like this, you can very easily conclude that 1 is in fact the supremum of the set. So, you can argue that hoop of 0, 1 is exactly equal to 1. So, this is something that I hope is clear to you. So, this should sort of illustrate to you what is a supremum. It is not essential that the supremum of a set line the set itself. Usually, when this happens, we replace the term supremum by the term maximum. We no longer use supremum, but we use maximum. So, I hope this much is sort of clear to you. Another example to illustrate the idea of a supremum for continuous functions is to look at say a function of the kind fx equals 1 minus e to the power minus x, where x is a non-negative real number. Now, how do we compute supremum? Now, we have been defining supremum of sets only, but it is not very difficult to generalize to functions, because what will I do? Supremum is simply the, a supremum of the function sup of f is simply sup of image of f. When I am looking at supremum of f, I am just looking at supremum of the image of f. That is, I take all the points that are of the form fx. This is what is called image of f. And I take the supremum of this set, so, supremum of image of f. Now, the important thing to see is that the set E, so, let me be careful. The set E, that the way I have defined, and the way I have defined the set E, that is, image of f is actually equal to 0, 1, which is close dot 0 and open at 1. What does it mean? That is, the set contains 0. Why does it contain 0? If I plug in for x equal to 0, then f of x is 0. And it goes almost to 1, but never goes to 1, because when does the value of the function become 1, you get the function value at 1 when this is 0, which means x has to go to infinity. And infinity is not contained in real numbers. This is something that all of you should know, if you do not should remember. So, that infinity not part of reals and therefore, you have this implication that 1 is not part of the set E. So, what is then the soup? So, what is then the soup of this? The soup of image of f is actually equal to 1, just like this previous example. Here it was open on both ends. Here it is open on one end and closed on the other, but it does not matter. We are looking at the supremum. So, only this end matters. So, this function actually has a supremum at 1 and this 1 is not in the set E. So, this is important to remember. So, there are two conditions under which soup becomes max. Well, the most general condition is when the supremum is contained in the set itself. That is i E if a soup of E belongs to E or if you have a set of finite number of elements. In fact, if the supremum of E is contained in E and the finite number of if a set as finite number of elements then the soup of E is contained in E. Therefore, this is in fact the most general condition under which the soup is written as a max. When the supremum is contained within the set itself that is soup E belongs to E, then and only then you write the supremum as a maximum. Otherwise, the notation supremum is used. So, I really hope you understand what is the meaning of supremum. In our context, what we now need to realize is that finding the supremum here is not too easy. It is not very obvious how I would find the supremum. In fact, if you just ask me an ad hoc question as to find a supremum of this kind of a function, what I would do is the brute force way would be to sweep over all sorts of vectors x and then compute this and keep trying to find the maximum. This is a rather painstaking process. However, if you have some knowledge of eigenvalues and some smart tricks, you can of course get simple formulae in some cases and we will discuss what those cases are very soon. We will discuss those cases soon. So, we are also interested in a few important matrix properties. Most of the times we will deal with symmetric matrices, symmetric square matrices especially I mean when I say most of the times, most of the times in analysis in Lyapunov analysis, we are dealing with symmetric square matrices and so we are interested in properties of these symmetric square matrices. So, what are these properties? The first is that all eigenvalues are real. I hope you know this. So, most of these things you should already have known or seen. A matrix A is called positive definite that is a symmetric square matrix A is called positive definite if and only if the corresponding quadratic form is strictly positive for all non-zero vectors alpha. So, if I take any non-zero vector alpha and Rn, I am already assuming A is in N by N matrix, then I compute alpha transpose A alpha, it comes out to be positive and this holds for every possible alpha in Rn. Now, again this is another condition which is not easy to verify because then I have to actually sweep over all possible values of alpha, seems ridiculous. So, there are simpler and equivalent conditions. First one of them is that all eigenvalues of A are strictly positive. This is where the fact that eigenvalues are real plays a role. If the eigenvalues are not real, I cannot talk about positivity of the eigenvalues. The second is there exists a non-singular Q such that A can be decomposed as QQ transpose and the third equivalent condition is that every principal minor of A is positive. So, these are the three equivalent conditions for this original condition. Finally, there is a very nice inequality that we use quite often in all our results, in all our derivations and that is that the quadratic form alpha transpose A alpha for any symmetric square matrix is lower and upper bounded by, is lower bounded by lambda mean A alpha transpose alpha and upper bounded by lambda max A alpha transpose alpha. So, what is this lambda mean and lambda max? These are the smallest and the largest eigenvalues of A. These are the smallest and the largest eigenvalues of A. I hope these properties are very clear to you. So, these are very, very critical properties and we will regularly invoke these especially this one, especially this one. All right. Then we want to look at some of the simpler to compute induced norms. I will come to the first one in the end. The first one is the infinity norm and the infinity norm in. So, notice that if I want to compute any induced norm, any particular induced norm say P, then I just have to take the P vector norm in all of this. That is it. So, if I want to, if I want to compute the infinity norm, then it is simply the maximum absolute row sum. So, I take the absolute row sum and whichever is the maximum value that is the infinity norm of the matrix. The second is the A1 norm and that is the maximum absolute column sum. So, I take the column sum and then whichever is the largest column sum that is the one norm. The two norm, the two induced matrix norm is the largest singular value. That is it is the largest eigenvalue of A transpose A square root it. So, these are the three norms. So, let us look at a quick example. All right. Let us look at a quick example of a matrix and what will be the norm. Suppose I take A as, again in this case, the matrix does not have to be a square matrix. I will take a rectangular matrix 2, 3, 1, 1, 0, 0. Let me make my life a little bit simpler. All right. So, what will be the infinity norm? In fact, let me put some different sign also. So, that I do not write. So, what is the infinity norm? So, I just compute the, let me compute the row sums and column sums. So, this is the row sum. The absolute row sum is 5. Absolute row sum is 2. Absolute row sum is 0. Absolute column sum is 3 and absolute column sum is 4. So, as per our formula, the infinity norm of A is what? Is the largest absolute row sum. So, that is 5. What is the one norm? It is the largest absolute column sum and that is 4. Now, what is the two norm? Now, here I have to do some work. I have to compute A transpose A. So, let us see. I have to compute A transpose A which is 2 minus 3 minus 1, 1, 0, 0 and whatever, 2 minus 3 minus 1, 1, 0, 0. Next. So, let me actually get this guy here. So, what is this? So, this is actually equal to, this will first of all be a 2 by 2 vector. So, we all have to remember because it is a 2 by 3 times 3 by 2 matrix. It will be 2 by 2 matrix. I am sorry. So, this is 4 plus 1, 5. Then you have minus 6, minus 1, minus 7. Then you have minus 6, minus 1, minus 7 because of symmetry 9 and 1, 10. So, this is A transpose A. So, now I am going to do the rest. I am sorry. In the interest of time, all I have to do is compute the largest eigenvalue of 5 minus 7, minus 7 and 10. All right. So, that is what it is. All right. So, what did we talk about today? We spoke about the notion of the Hilbert spaces. Then we discussed in a little bit of detail the definition of the induced matrix norm. What is the meaning of the supremum? And how to compute the supremum? When does the supremum become a maximum? So on. Then we looked a little bit at the relevant matrix properties for symmetric square matrices that interest us. And finally, we also saw how to compute the induced norm for some special norm, for some special cases like the 1 norm, 2 norm and the infinity norm, which is mostly what we will end up using. And finally also, we did see how to, well, I mean, we have not seen it yet, but we do plan to look at the Cauchy Schwartz inequality and a more general proof of the Cauchy Schwartz inequality. All right. All right, folks. Thank you. That will be all for today.


was being just tying his shoes and now he's getting smacked in the arse and still don't pay sugar what's with BTS with their asses how's it going everyone it's Logan here we're back with another reaction video and today we're gonna be reacting to some BTS funny moments because I feel like it okay when we say BTS what does BTS stand for? being the sexiest but anyways bantang saw you around we're gonna be reacting to them today why did I say it like that I don't know I don't know it's just like recently I've just been wanting to like do some funny moment reactions to like BTS I want to do a blackpink one a twice one Stray Kids one I really just want to do some funny moments on these guys within the next few days so hopefully you guys are fine with that because I'm gonna be fine with that but yeah if there's any other BTS videos you want me to react to I'll be freaking happy to do that seriously happy to do that because I really do love these guys I say this every time I make a BTS video because no one's subscribed let's check the analytics quickly wow won't you look at that oh those numbers hurt don't they only six point three percent of my watch time has been from my actual subscribers so that's why whenever I like make a video on BTS I always have to say this I've been a fan of BTS and dope and I just love their music their wings album got me to really love them and you know I just I can't wait to just get to do more videos on them and and all so I feel like cuz like they're one of the groups that go into kpop so I feel like I do need to do a lot more videos on them you know those numbers actually really really hurt me like really really hurt me we're here to feel better by reacting to the BTS funny moments 2018 try not to laugh challenge M I don't know the M stands for like medium maybe I don't know but anyways let's just get straight into it wait no try not to laugh challenge not laugh challenge oh my god I love this song ah let's go Jen I love them I've seen that clip a lot no dude what is this I gotta I gotta I gotta not laugh this is hard there's a reason why these guys I love by everyone they just talks they just talks from now on I will not laugh I'm gonna fight oh my god I'm smiling not laughing man when I don't came out oh nothing myself roast no don't laugh okay it's gonna be so hard oh bro sugar is my bias but I swear man all of them are bias rick is respect Jen I love Jen man like with him of animals so funny oh is it like whenever you laugh you get hit I feel like I see Jen and I man so simple just both idiots and we're scared of animals just this is hard try not to laugh with BTS impossible I still already want to react to bomb voyage season 4 because they're in New Zealand I live in New Zealand but I'm not sure if I could post damn I don't know if I can actually put it on to YouTube that's the thing I'd like oh junkers gonna laugh watch it 3 2 4 night 4 night 4 night 4 night I haven't played that game in months I'm kidding this is trying not to laugh help help okay let's get it when you're a carrot yeah it's impressive oh when you floss bro Jen's a monster just tying his shoes and now he's getting smacked in the arse and stood on by sugar what's with BTS with their asses Jen's a crackhead I'm loving this too much I need to do more BTS videos actually when did I start reacting to BTS not today with not today in spring day that's when I started reacting to them so maybe I should react to all their music videos bro this was insane when it came I think there should be a BTS try to laugh challenge because I think all you need to do is just play any clip from BTS and you would laugh but what's with them smacking asses it's like it's like probably afraid of the clips of them smacking asses and I'm like why it's funny but why I need to know and this is why I like I don't know much about BTS behind the scenes but like just for when it comes to music I know them a lot but when is them behind the scenes I don't really know them after their music you know I'm a massive fan of their music but I don't really know their personalities that well I know like I've seen clips of them and all that stuff but I don't really know them actually why why do they make each other's asses I need to know but anyways thank you guys for watching this video if you guys enjoyed show some support by leaving a like comment down below how much you love BTS subscribe for more BTS content and yeah other than that hopefully you guys have a great day and keep on smiling and yeah