In the last lecture, we have started discussing method of characteristics for solving Cauchy problem for Quasi-Linear equations. In this lecture, we complete the proof of existence of solutions to Cauchy problem for Quasi-Linear equations using method of characteristics. So, outline of today's lecture is first we recall the assumptions and notations which are made about Cauchy-Linear equations. And then we describe method of characteristics. A quick recap of lecture 2.6 where we have completed this step 1 in the method of characteristics which is passing characteristics through points of gamma. Today, we start elaborating on step 2 where we say what is the finding a candidate solution and then show that that is indeed a solution. And then a small remark on the proof of existence and uniqueness theorem that we are going to prove at the end of the step 3. So, assumptions and notations in the context of Quasi-Linear partial differential equations, first order equations. This has been revisited many times 2.1, 2.5 and as last time also in lecture 2.6. So, Quasi-Linear equation we denote by ql it stands for this equation ux plus buy equal to c, abc are functions c1 functions on omega 3, omega 3 is an open subset of r3 such that a square plus b square is not equal to 0 at every point of omega 3. And the projection of omega 3 to x-ray plane is denoted by omega 2. Quasi problem given f space curve that is a curve in r3 which is given parametrically in this form where the parameter is running in interval i and fgh are c1 functions such that notation gamma 2 important, gamma 2 is a projection of gamma 2 x-ray plane that satisfies this condition. Gamma 2 is actually x equal to fs, y equal to gs as s varies in i. We want that f prime and g prime do not vanish at the same time at every point on gamma 2. Then we want to find a solution that means a function defined on some domain such that u on gamma 2 takes the value h which is a prescribed datum curve. Of course, we do not expect the solution domain to contain entire gamma 2 but a small portion of gamma 2 is also allowed. Let us look at the method of characteristics from which was discussed in the last lecture. The inspiration for method of characteristics is the following theorem which said give me a surface of the form z equal to uxy then saying that this in this surface is an integral surface is same as saying that this surface is union of characteristic curves corresponding to quasi linear equations. Since we want to construct an integral surface in other words wants to find a solution or a surface corresponding to a solution we will start with this idea of constructing some quantity some object which is union of characteristic curves. That is the idea behind this method of characteristics. So, step 1 is through every point of gamma you pass a characteristic curve. So, gamma is here we want to find a surface which contains this gamma. So, what we do is we pass through this characteristic curves like that and you see this is going to give a surface. So, characteristic this given datum curve is described by S and the characteristic curves for each fixed this point this characteristic passing through that is defined by T in R. So, we have a description of some object geometrical object in R 3 which is parameterized by 2 parameters S, T. So, as S and T vary we hope you will get a surface that was the idea. So, that is why we start with this step 1 where we pass characteristic curves like this. Then we say that the surface that we are going to generate would be of the following type where the third coordinate is going to be a function of the first 2 coordinates. That is what we hope and that we will achieve in the step 2. We will define a function u, u is a function of 2 variables x and y using inverse function theorem. And then we show that u is indeed the solution. Step 1 what we did is finding integral surface containing the datum curve or a piece of datum curve means we need to weave a surface as I have just shown you in the picture and if you weave that surface through passing curves through points of gamma and curves are characteristic curves then that theorem gives us a hope that the surface we get is going to be an integral surface. Here this is the datum curve these are characteristic curves this is the same picture which I have just drawn. So, for this what we do take a point on gamma any point on gamma looks like FSGSHS and then solve this system of characteristic ODE with initial conditions FSGS and HS it means that when you look at this gamma you take at this point FSGSHS the point looks like this this is 3 tuple and then you pass a curve through that. That will be denoted by XTS, YTS, ZTS this is what we get at the end of this step at t equal to 0 it is going through this point that is how we are solving we are giving the initial conditions to this system of characteristic ODE. Now by Cauchy-Lipschitz Picard's theorem it has a unique solution let it be denoted by like this X equal to XTS, Y equal to YTS, Z equal to ZTS. Where is it defined? S is in I and t belongs to JS which is an interval of course the interval contains 0 note the dependence of S on JS that is why it is written JS but we have seen a lemma on reparameterization which says we may take JS equal to R. So, now let us proceed to the remaining steps in this method. Second step is finding a candidate solution how do we get that? Of course we have to take help of this what we have been achieved so far. So, this is this mathematical object is described by two parameters and this lives in R3 so we hope it is a surface that is the idea. As I pointed out earlier S represents a parameter running on the datum curve gamma and t in R it is parameterizing characteristic curve passing through this point on gamma which is visited when t equal to 0. Of course gamma already lies on 1 by design initial conditions. The parameterized surface given by 1 I have put it in inverted commas here because it is not clear it is a parameterized object is clear surface means we expect something that is why I have put here till the confirmation comes. It is expected to be an integral surface because we were motivated by the theorem and therefore we hope that we get an integral surface. But this is described using parameters t and S now we need to write it as z equal to u of xy. So, what does this mean? It means that we would like to express the parametric surface given by 1 in this form z equal to u xy for some function u we need to find some function that means we need to find a domain on which this function is defined on these holes. In other words write the third coordinate as function of the first two coordinates using one that is the only thing that we have in our hands namely the family of characteristic curves passing through points of gamma. So, the strategy for finding u is this express t and S as functions of x and y we have x and y given in terms of t and S. Now, suppose I write t and S in terms of x and y go ahead and imagine this is the formula we write t equal to capital T of xy and S as capital S of xy. Now, we go and substitute in this relation z equal to z T S and define a function this way z of t equal to capital T xy S equal to capital S xy. So, we got some function of xy call it u xy this should work because z itself was motivated like that. The equation for z was derived after using that it is going to be the value of solution along the character along the base characteristic of x T S by T S refer to lecture 2.6. So, therefore it should work but there can be very technical difficulties we will see what they are. So, express t and S as functions of x and y like that recall that x and y given by x T S and y T S that together represent base characteristic curves that is this. Now, this suggests applying inverse function theorem to this function then you will get t and S in terms of x and y. So, if you want to apply inverse function theorem you have to see what is the setup of the inverse function theorem it involves a function it involves a C 1 function and certain Jacobian condition. So, we need to check all those hypotheses then only we can apply a theorem. So, therefore answers to the following questions will be needed what is the domain of this function is a function continuously differentiable is a Jacobian condition satisfied these three conditions. Usually one ignores questions 1 and 2 and just concentrate on third one at length. In fact, the third is not a condition it is a Jacobian condition satisfied is never checked that is put as part of the assumption and then one proves the theorem but still 1 and 2 are still important. So, you ask this question why you will understand later on. So, answer I will not give you think about think about this question. Okay, what is the domain of this function as we observed for each fixed S T varies in J S. So, it is going to be subset of R cross I for each fixed S T varies in J S. Okay, that is all we can describe it is not even clear what the set is is an open set etc. But anyway we will do away with this kind of answering these kind of difficult questions. So, it complicates further analysis. However, we may assume without loss of generality that A is equal to J cross I where J is a sub interval of R. I is where S is running the parameter S is running and I J is where the T is running. Okay, we take J equal to R this is what we said thanks to our lemma and reparameterization. So, therefore we may assume that without loss of generality that A is equal to J cross I where J is a sub interval of R. In fact, we can assume J is equal to R due to this lemma that I just quoted reparameterization of characteristic curves or by restricting to a small piece near gamma that is also okay. So, you have the point P at that point restrict to a small piece then it is true that one that J can be chosen independent of S. This idea comes when one discusses differentiable dependence of or continuous dependence of solutions on the initial data. So, see the ODE book by Wolfgang for this book is a good book we will be referring once more at least in this lecture okay, done. So, domain is R cross I fine or R cross some sub interval of I if you follow Wolfgang approach if you follow our approach it is R cross I. Now, next question is it C1 function if you look at this X and Y are solutions of certain ODE therefore with respect to T they will be C1 what about with respect to S? What about together? Answer is yes it is C1 function if partial derivative with respect to T and with respect to S both are continuous functions then it turns out that it is differentiable with respect to the two variables together as we know existence of partial derivatives of functions of several variables does not mean that the function is differentiable forget about that function need not even continuous at a point but all directional derivatives exists. But if partial derivatives are continuous then the function turns out to be differentiable. So, and it will be C1 therefore we are interested in XT, XS whether they are continuous functions XT as I said it is ABC they are nice functions. So, there is no problem what about with respect to S? Where is S? S is appearing in obtaining XTS and YTS through the initial conditions we solved X of 0S and Y of 0S as FS and GS and we assume F and GS C1 functions of the variable S that is the reason why this function is C1 of J cross I. It comes from differentiable dependence of solutions to initial value problems on the parameters in the theory of ODE's for instance see the book by Wolfgang on ordinary differential equations. Now let us implement step 2. Step 2 and question 3 we have to answer is the Jacobian condition satisfied by this function? We are interested in the invertibility of this function. So, we have to compute the Jacobian M-function theorem says compute the Jacobian that should be nonzero Jacobian is this. Now we have to compute XT at TS, YT at TS, XS and YS at the point TS. Now XT at TS is A it is known, YT at TS is also known, XS at TS is not known what we actually know is XS at 0 comma S that is the reason why we stick T equal to 0 now. Since we do not know the function explicitly we cannot compute JTS for an arbitrary point TS in J cross I which you can read it as R cross I. However using characteristic ODE and initial conditions because initial conditions for characteristic ODE are given a T equal to 0 therefore there something can be done. We can compute J 0S and that is given by this XT, YT are solving ODE's. So, from there you get AB X of 0 is 0S is FS therefore derivative of that with respect to S is F dash of S similarly YS is G dash of S. So, we want this to be nonzero. So, if you assume that J of 0 comma S0 is nonzero at a point S0 then we can apply the inverse function theorem I am written in the brackets local because there is something called something else called global inverse function theorem but the usual inverse function theorem is a local inverse function theorem thereby step 2 will be successfully implemented. This assumption on Jacobian is called the transversality condition. The quasi linear equation QL and the initial curve or the datum curve gamma are set to satisfy transversality condition. So, the transversality condition is two people together satisfy that that is quasi linear equation and initial curve. At a point S in I if the two curves which are listed here the first curve is a base characteristic curve. So, at point S right. So, look at the point FSGSHS that is going to be on the datum curve through that you find the characteristic curve project to R2 XY plane that will be the base characteristic curve. Thus base characteristic curve and gamma 2 which is the projection of the datum curve gamma these two curves intersect of course they intersect at the point FSGS non tangentially. That means the tangent lines do not coincide. If the transversality condition is satisfied at a point then it will continue to hold in an interval containing S0 because the transversality condition is non-zeroness of certain determinant. And whatever is appearing inside the determinant are continuous functions of S. Therefore, at some point non-zero it will continue to be non-zero near by that point that is the reason. In this case what would happen is imagine this is our gamma 2 that base characteristic curve will come like that. It will actually cut base characteristic curve will not be like this. For example, it will not be like that because here they share the same tangent. So, this is not allowed okay then transversality condition is not satisfied because if you see in the determinant what are the two things which are coming there the two columns. Xt yt is the tangent of the base characteristic curve and this is a tangent of the gamma 2 and asking that these non-zero means they are not parallel. And they are passing through this point FSGS 0, S sorry they are passing through the point FSGS yes and we are asking that the tangent line is not the same that is why it is the transversality condition. So, here it is transversal here it is not transversal.  So, analytically the transversality condition translates to this being non-zero this determinant. It rules out the possibility of base characteristic curves touching the projected datum curve namely gamma 2 that we have illustrated in the picture touching means shearing the tangent line and coinciding of course. Then also you have touching for a long time this is maybe at a point but this is if you if you intersect along a piece of it it means that they share tangent throughout that piece because curve is same. So, such things are ruled out. Please spend some time on understanding this geometrically because it will be useful when we are trying to give examples or solve or guess what would happen in certain situations for quasi linear equations some examples it will help. So, this justifies the use of the word transversal it means to cut it cuts. So, assume that Jacobian is not 0 at the point s 0 even if Jacobian is not 0 for all s in i j of 0 s is not 0 for all s in i it is not even useful because the only way you apply implicit inverse function theorem is at a point s equal to s 0. So, I am assuming at a point it is non-zero of course s equal to s 0 means the point on gamma will be f s 0 g s 0 h s 0. By inverse function theorem there exists an open subset of j cross i containing this point 0, s 0 and an open set on the other side omega 2 containing the point x of 0 s 0 y of 0 s 0 because our function is that which is actually f s 0 g s 0 and a continuously differential function from f to e because this should be the inverse of the function phi. So, psi circle phi will be identity on e and phi circle psi will be identity on f these are the identity functions okay i e means now we have a picture see this is a thing phi we have e to f. So, originally phi is defined on some other set which is j cross i or r cross i or j cross some i dash and then we assume that at 0 comma s not Jacobian condition is satisfied non-zeroness of the Jacobian. Therefore, you can find an open set e which contains the point 0 comma s not and an open set f which contains the image of phi under that whatever is the map phi we have it maps t s to x t s y t s. So, x of 0 s not y of 0 s not which is f s not s not and a mapping psi such that these compositions you go from e to f come back from f to e or go from f to e first and then come back to f both will be identity functions this where we are using t s coordinates we are using x y coordinates the variables names. So, therefore, so psi of x y you denote it as t x y comma s x y and this equality is hold because they are inverses of each other that will give you t equal to t x y s equal to s x y. As I told you earlier z t s was designed to be the value of the solution at this point x t s y t s refer to lecture 2.6. Therefore, this motivates us the definition of a candidate solution by this you defined on f to r given by u x y equal to z of t x y s x y. As the function is a composition of 2 c 1 functions this is c 1 inside and z itself is c 1 function therefore the composition will be c 1 function therefore u is a c 1 function. Step 2 is successfully completed we have defined what is a candidate solution. So, in lecture 2.8 that is the next lecture we will discuss the geometric meaning of transversality condition and what happens when this condition is not satisfied what will happen when it is satisfied we are going to prove a result today and what happens if it is not satisfied will be analyzed in the next lecture. Can we still hope to solve Cauchy problem we have to see answers will be in the next lecture. Now, let us proceed to step 3. What is step 3? Candidate solution which we got in step 2 is actually solution by definition of u we have this formula and z t s equal to u x t s y t s we also have this. Now, what is that we want to show u solves PDE right. So, we have to compute u x u y etc. But that may not be useful let us see differentiating the last equation that is this equation with respect to t we get of course we use chain rule and we get this. Now, why did we not choose the first equation you could have chosen that differentiate with respect to x with respect to y go back and substitute in the equation and check whether it is satisfied or not this is usually the situation all the time whenever we have change of variables and PDEs are there we need to differentiate some equation but there are always two equations for that only one of them would be useful in the sense it may be a quicker way of getting the solution not wrong other one is not wrong but you may not be able to achieve anything with that but this one will definitely cure you will learn this as many times as you make mistake of differentiating this u that is true. Now, it is going to work differentiate we got this formula now x t s y t s and z t s are solution to Kara ODE. Therefore, here points will come out to be too long just to make the notation short I am using x t s y t s z t s as p t s. Now, z t is c at x t s y t s z t s but now it is p t s u x is as it is x t is a at x t s y t s z t s now it is a of p t s similarly the other term we have this. So, in terms of x y enough the last equation reads as c of x y u x y equal to u x of x y and a of x y u x y u y of b x y x y this means u is a solution to the Quasarine equation Cauchy data is already taken care we already observed that Cauchy data will be satisfied the datum curve will be on the integral surface. So, step 3 is also successfully completed after the three steps we have obtained an integral surface defined by a function u from f to r of course here we have used inverse function theorem that means existential to start with and s contains this point p 0 s contains a piece of the datum curve near p 0 in fact that much part of gamma for which f s g s belongs to this open set f if you could apply if there is a global inverse theorem if you could apply that then we would have got a solution defined still on a subset of omega 2 but that would have contained the entire datum curve gamma that integral surface would contain entire datum curve gamma the solution may still not be defined on whole of omega 2 this local global we had be very very careful what we have been dealing with is we are looking at a point on the datum curve and showing solution a surface exist nearby that which is integral surface that means we are discussing something like local with respect to the datum curve and if you say global inverse function theorem it must be giving global with respect to the datum curve but not with respect to the domain more on this we will see in a future lecture local there are two different concepts local with respect to datum curve local with respect to the domain omega 2. So, theorem is here hypothesis standard hypothesis on QL I am not going to recall here and the Cauchy data assume that the transversality condition holds at a point s 0 and denote p 0 f s 0 g s 0 h s 0 same thing let us call x 0 y 0 z 0 for convenience conclusions Cauchy problem has a solution defined on a neighborhood of the point x 0 y 0 and let d 1 d 2 be open neighborhoods of the point x 0 y 0 let u 1 u 2 be defined on d 1 d 2 respectively be solutions to Cauchy problem and then u 1 will be identically equal to u 2 on some subset of d 1 intersection d 2 of course it contains the point x 0 y 0 that is solution to Cauchy problem is locally unique near the datum curve gamma. So, let us prove conclusion one that is precisely steps 1 2 3 that gives the proof of conclusion one that is about existence now let us do the uniqueness let s 1 and s 2 be a as given one is defined on d 1 other one is defined on d 2 and take this point p 0 which is in gamma now p 0 is there and gamma it is there in s 1 as well as s 2 and as a consequence p 0 is there and s 1 p 0 is there and s 1 therefore a part of what is s 1 it is an integral surface therefore a part of gamma will be there on s 1 intersection s 2 it will be there on s 1 another part will be in s 2 there will be a common portion which will be there on both s 1 s 2 that is there. Then we have seen a corollary in lecture 2.6 in the last lecture it said some part of the each of the characteristic curves of q l passing through points of gamma dash will also be there on s 1 and also s 2 of course the corollary was with respect to one surface but here with respect to two or two surfaces yes you can see the corollary yeah this characteristic curve will be there on both s 1 and s 2. Due to transversality condition that is why step 2 is very very important step 2 the T s and x y the maps phi and psi are inverses of each other that is very very important. Due to the transversality condition any point on a characteristic curve has the form x y u 1 of x y and also x y u 2 of x y whenever you are in this common area f intersection d 1 intersection d 2 f you remember is the domain of definition for psi this proves that s 1 coincides with s 2 in a neighborhood of p 0. Hence u 1 is identically equal to u 2 on some neighborhood of x 0 y 0 this proves uniqueness now quick remark on the proof of existence and uniqueness theorem in fact it is going to be another proof of step 3 step 3 was very simple we had to simply compute the derivative substitute we see that the candidate solution is indeed a solution we can give a geometric proof of that surface s was described using this parametric equations right x t s y t s z t s fine the curves s equal to constant what are they they are characteristic curves this is gamma fix s on this this is x t s so if s is fixed it is a characteristic curve characteristic curve means what is the tangential direction first of all it is a curve therefore x t y t z t is a trans is the tangential direction so x t y t z t this is tangent but it is characteristic curve therefore the direction is actually a a b c remember this okay and characteristic curves lies on the surface s for s we know something which is normal okay characteristic curves lie on s x t y t is also a tangential direction to s and x t y t z t is nothing but characteristic direction we have observed what is the normal normal is a b c right no u x u y minus 1 yeah u x u y minus 1 that is a normal z equal to u x y if it is your surface normal is along this direction u x u y and minus 1 okay normal and what is the tangential direction one of them is a b c that has to be 0 and this is nothing but the equation therefore the surface s is an integral surface so any proof of the uniqueness assumption in the theorem is incomplete if the transfer surrogate assumption is not used because uniqueness fails with the failure of transversality condition when transversality condition fails there are examples where uniqueness fails there are examples where existence fails anything can happen so we will discuss that later so if you are proving uniqueness you must be using transversality condition is satisfied so people may not write this much explicitly in books but they will say that step two which is what we have said right they make use of step two without explicitly mentioning to you so you should not get confused the summary is that we consider the Cauchy problem for QL we proved the existence of a local solution assuming transversality condition holds nothing no more assumptions via method of characteristics we proved local uniqueness of solutions and in step two we noted a few possible reasons for existence of only a local solution which will be seen in next lectures and why you may not expect global global I already mentioned there are at least two notions of local solutions and hence global solutions so in the next lecture we will take up as promised what is transversality condition geometrically and what happens when it fails what are the possibilities thank you


Personal finance practice problem using Excel. Estimated home price from monthly income, part number two. Get ready to get financially fit by practicing personal finance. Here we are in our Excel worksheet. If you don't have access to the Excel worksheet, that's okay because in a prior presentation, we basically put this together from a blank sheet. So you can go back there and start there from the blank sheet. If you so choose, we're gonna be continuing on here with the practice problem. If you do have access to the sheets, there's three tabs down below. An example tab, a practice tab and a blank tab. The example tab in essence being an answer key. Let's take a look at it now. In prior presentations, we got the information on the left-hand side. We used it then to populate the data on the right-hand side. We first thought from the perspective of a financial institution using certain heuristics of the financial students' institution such as a bank to determine based on our gross income how much they think we might be able to afford for the home purchase process and then distilling that down to how much we can possibly afford just simply for the financing of the loan. We then use that to determine how much loan we might be able to purchase based on that information. And then we use that and the estimated down payment to get to the amount that we might be able to afford to purchase on the home and then we double checked our number. Now we wanna think about the financial statements. In other words, you might look at this first number from the bank and say, well, the bank is using certain types of heuristics kind of shortcuts, you might think, instead of using what you would think that they would use, which would just be the financial statements. And notice again, from the banking side of things, there's a couple of different reasons or there's multiple reasons they might use a different kind of shortcut type of system. One is that they might not be able to trust the financial statements from a bunch of different kind of individuals. They'd have to kind of verify them and there's costs related to verifying and so on, whereas they can fairly easily verify the income number with W-2 statements and stuff and like paycheck stubs and so on and so forth. So if they can use the paycheck stubs, then they have more verification. There's less kind of variables if they can use that as a heuristic from there. And then you also have the situation where there's gonna be regulations in terms of government regulations that they're complying with, banking regulations and so on. So you could be in a situation where you're saying, hey, look, the number that the bank is coming to may not be something that you think is appropriate based on your personal financial statements as you do your financial statements. Now the personal financial statements may be something that the bank requires or asks for and may not depends on kind of the circumstances, but we want to do our own income statement as well in any case. And remember when we're dealing with the financial institution, we have kind of two separate goals. The institution on their end, they're trying to determine if we can pay back the loan, of course, and they wanna be feeling quite secure that we can in order to give the loan. On our perspective with the financial institution, we're not going there for budgetary advice. I'm not going to the financial institution to give me budgetary advice. I'm going there to see how much to get the maximum amount of capacity for loan capacity that I think I might need in the event that I might need it.  So those are two kind of objectives. When you're talking to the bank, you typically want to look as good as possible, of course, and so that you get the highest loan possible that you want to be honest and upfront, but you want to give them the numbers that would provide you with the best loan possibilities. On our budgeting side for the personal use, that's when we actually want to budget and see what we can actually afford and how much loan we would want to be taking. Those are two different things. We're not just going to take how much loan the bank will offer. We're going to take the loans up to the amount that we could afford and we would like the bank to be willing to give us a more loan in the event that we needed it, right? So we want that number to be as high as possible, typically. So that's going to be the example, the practice tab. We're going to have the information that already has some cells populated over here so you can do less Excel formatting in it. And then the blank tab is where we built it basically from scratch and this is where we're going to be continuing on here. So now we're just going to construct in essence a financial statement and compare and contrast that to kind of the number we got down here so that we can practice our own budgeting. So let's hide some cells. I'm going to go back on over here. I'm going to make a skinny K first. I'm going to go to column G and I'm going to go to the home tab and we're going to go to the paint for us. I'm going to make a skinny K and then I'm going to hide the cells from C to J from C on over to J. I'm going to right click and hide those cells, not delete them, just hide them from CJ. I knew a CJ one time, someone that probably stood for something but CJ. Any case, here's the financial statement. So we got the balance sheet which shows where we stand as of a point in time but we're more focused oftentimes kind of like on the income statement. Possibly we want to break down our income statement on a year by year basis. Now note, when you're looking at your income statement, there's a couple of things you might, the ways that you might put this together. You might actually try to compile this information just from your paycheck stubs and your W-2s and then your bills and so on to get a good idea of it. That's one way, that's fine. Or you might be using software or something like that and basically going from the bank statement in essence, oftentimes being more on a cash basis method. Just realize when you're looking at the income side of things, it gets a little tricky because the income often has these withholdings involved in it. So if you're on a cash basis system and you have like a QuickBooks or other kind of software that's basically constructing your financial statements and pulling this stuff in from the bank feeds, from the bank, from the financial institution or if you are building your financial statements from the bank statement and your W-2 employee, the amount that hits your bank is actually the net check and not the gross check. Now remember when you talk to the financial institution, they often want the gross amount and that's often what you would like to give the financial institution because you want to give them the highest amount because that will give them the most likely result in the highest loan kind of value on their end. On our end, when we get down to our net income, we want to take into consideration whether or not we're calculating the income statement with the gross or net and then just be aware of that, be conscious of it as we do the financial statements. So that's one thing we'll take into consideration. So we've got our income, we're gonna assume this is like two W-2 incomes possibly by a husband and wife, married or possibly just someone working like two jobs kind of thing on the gross pay and then the net pay, we're gonna assume the difference between the two being the taxes that were withheld but there also could be benefits that were withheld too. And then we've got the expenses and we just listed out a couple expenses so we can practice putting together our income statement here. So let's just construct an income statement. We're gonna do it kind of more in a cash basis, cash flow kind of income statement. So this is gonna be a monthly income statement.  Let's make this home tab.  We'll make it black and white. And then we're gonna go with our income. Income is the top of the income statement. And we've got the two incomes. I'm gonna use two columns here because I'm gonna kind of do the income on a gross basis and then on a net basis. Those are the two ways you might see the income statement. They're both okay as long as you know that how you're reporting the income. So I'm gonna use two different columns and I'm gonna assume the first one, we've got the gross amount. It's not really gross. It's just, you know, that's the top line after that it's not like it's icky. That's not like a slimy one or anything. And then this one, and let's put the amounts here. This is gonna be equal to the gross income, which we're gonna say the 47,000. That's for a year. I'm gonna divide it by 12 to get the monthly amount. Now you might get the gross income from your paycheck stub, for example. You might get it from your W-2. But if you look at your W-2, note that there's three income lines on it. There's box one, there's box three and box five. Box one really only has the income that is related or subject to federal income taxes. So that might even be reduced by something like a 401k plan or something like that. Box, the Medicare one, box three is for Social Security, box five is the Medicare. That's probably actually the highest one, which is closest to your gross income. And then your net income is gonna be over here, which if you're getting the information from like, I'm gonna take this divided by 12, if you're getting the information from like the bank, that's what actually hit the bank. So if you're constructing your financial statements from the bank, you're more likely to be using the net income over here. And you just wanna be aware of that because it's already net of at least taxes. And those taxes are your taxes, right? They're actually expenses to you. They're taxes that you paid that were taken from you by your employer. So we're gonna say down here, and then we're gonna say that we have our other income, the gross side of things, the gross income, so gross. Income is so gross. Why is income, income is good. Income is, that's the grossest income.  And then this is gonna be equal to the net income on this one divided by 12. So there's our grossed net income on our income statement. I'm gonna go ahead and sum those up. Let's sum it up. And we're gonna call this total income. So we'll do with the gross side first, total gross income. Total gross. And then we're gonna say this is the total, the sum of the net income. We'll put some underlines here. Why not? Some home tabs, some font groups, some underline. Let's do some indentation. Selecting these items, we're gonna go to the home tab alignment and indent it. And then we'll indent this one again, double indentation. And then we'll put our expenses in place. Expenses, brackets. And I'm just gonna list them out here.  If you have software like a QuickBooks or other types of softwares that can compile this, they might be drawing this from basically what clears your bank and or credit card funds, for example. And so we can go down here. And software's getting better and better at being able to construct this kind of stuff from banking and finance stuff. So it might be worth doing. I'm just gonna copy this down. And so I went too far. So you've gone too far, but that's okay. I did it on purpose because I can just delete those. Calm down, it's okay. You've gone too far. And then we're gonna say the utilities is gonna be over here is 120. I'm just gonna copy that down. So we're gonna pull this utilities. And again, you could get this from your bank statement, from the actual statements themselves, compiling them, but you might want software to kind of construct this stuff. It's useful. And so we'll copy that down. I'm gonna put some underlying down here. Notice I put the car loan payment as kind of an expense. So that's a little bit tricky because really, if you're on an accrual basis, it would be a decrease to the principal rather than an expense and so on. But we're gonna assume it's kind of like a cash flow item that we're spending each time. And we're trying to think about this on a kind of cash flow basis to think about how much added cash flow we would have if we were to make a home purchase and so on. So let's go over here and say we got the font group and underline. And then this is gonna be the total expenses. Total expenses equals the S to the U to the M otherwise known as the sum. And then we're gonna say this is gonna be, this will be the net income, income or cash flow. I'm gonna say before taxes, even though the taxes are included in one of them, but not the other and the net, but not the other. But I'm gonna say before taxes and this is gonna be equal to the income minus the expenses. That's how these things work. This will be the same income minus the expenses. And then the key point, I just wanna point out down here is you've got your taxes, which I'm assuming is the difference between the gross and net income. You might have benefits as well, but I'm assuming it's just the taxes. And so I'm gonna say the difference between these two is the taxes. So I'm gonna come out to the same number except that the taxes are kind of included up here, which would be if you're using it on the cash basis, this is what actually hit the bank because they took it out on the employer side of things, your taxes. Okay, let's put an underline here. Underline here. We'll put an underline there and an underline there and I'll put a zero there so it has something to underline. And that's gonna give, let's do some indentations. You're getting ahead of yourself. Stop getting ahead of yourself. But I'm so slow.  This, I'm gonna indent this one. I'm gonna indent home tab, indent this one. And then this is gonna be the net income or cash flow after taxes. And this is gonna be equal to the total income minus this number here. Well, hold this. It's gonna be after tax, no, hold on a second. It's gonna be this number minus the taxes. That's what it's gonna be. That's what it's gonna be. And then this one is gonna be equal to this minus that. So we get to the same bottom line number. Let's put the double underline to indicate the bottom line. That's the bottom line which has two underlines. That's how you know. Okay, so then let's put some borders around it, blue and border it, blue and border.  Border blue. Let's do a spell checking. Did you mess up any spelling? No, cause I just copied it over. I didn't even type any spelling. I can't mess it up cause unless the data over here was messed up. So that is it. So then down, so notice again, we get to the bottom line. The main thing I just wanna point out what one is that when you put together the income statements, you might be putting it together gross or net. And then you wanna think about your own kind of net income. We wanna think about how much we can afford as a separate kind of thing to what the bank is doing. We might use this, the bank might ask for something like this in which case we're gonna give them the information. But the bank might be doing something different. And again, we're not really talking to the bank in order to do our personal budgeting. We're not, they're not a financial planner for us. They're trying to determine whether or not they're gonna give us a loan. And we're trying to get as much access to as much loan as we can. Not that we're gonna take as much loan as they'll give us, but because we want the ability to take as much loan as we need. So, and so if I unhide my cells over here from B to L, bull and unhide, then you could see that their number here on the affordable amount might be substantially different. And I'm just, obviously, we just kind of made these numbers up. They might be whatever heuristic that they're using is based on whatever standards, complete averages, you know, it's just kind of, they're trying to use just a number that covers everyone. It might be completely different to what we actually come out with because our actual finances, our actual circumstances might be quite different. We might have a completely different lifestyle than other people on their normal, that their normal heuristic, their average numbers that they're using are using. So that, and that's fine. So we wanna basically, again, talk to the bank and use whatever format that they're gonna use so that we can make the estimate of how much loan could be available to us. Again, not so that we could actually take out that much loan possibly, but so that we can try to get as much access to financing as we can have. And then we're gonna do our own financing, our own budgeting to determine how much loan we actually want to take out, how much loan we can actually afford. And for that, we would like to have as accurate as possible actual financial statements, balance sheet and income statement. Next time we'll take this information and we'll build our tables from it, our amortization tables and our grouping of that table by year. And that, of course, can help us with the budgeting from this point. So once we have the income statement before the loan kind of process takes into consideration, then, of course, we wanna think about, well, what would happen after or if there was a change, what's gonna be the difference from the standard now to the change. And those things will include things like, what's gonna be the cost of the home and so on, it taking into the interest, tax implications and that kind of stuff.


Hi, I am Shashi and I am going to help you with the following questions. Question is, using elementary transformations, find the inverse of each of the matrices if it exists. The given matrix is 4534. Let us now start with the solution. First upon, let us assume A is equal to matrix 4534. Find the inverse of A by elementary row transformation method. We will write A is equal to IA. We can write matrix 4534 is equal to matrix 1001, multiplied by A. We will apply sequence of row operations simultaneously on the matrix A on left hand side and the matrix I that is identity matrix on the right hand side. Till we obtain identity matrix on the left hand side. Now, first of all, we will make this element equal to 0. So, we will apply on R1 row operation 1 upon 4 R1. So, we can write, applying R1 row operation 1 upon 4 R1. So, we get 1, 5 upon 4, 3, 4 is equal to 1 upon 4, 0, 0, 1, multiplied by A. Now, to make this element equal to 0, we will apply on R2 row operation R2 minus 3 R1. So, we can write, applying on R2 row operation R2 minus 3 R1. We get matrix 1, 5 upon 4, 0, 1 upon 4 is equal to matrix 1 upon 4, 0, minus 3 upon 4, 1, multiplied by A. Now, to make this element equal to 1, we will apply on R2 row operation 4 R2. So, we can write, applying on R2 row operation 4 R2. So, we get 1, 5 upon 4, 0, 1 is equal to matrix 1 upon 4, 0, minus 3, 4 multiplied by A. Now, to make this element equal to 0, we will apply on R1 row operation R1 minus 5 upon 4 R2. So, we can write applying on R1 row operation R1 minus 5 upon 4 R2. We get 1, 0, 0, 1 is equal to matrix 4 minus 5 minus 3, 4 multiplied by A. Also, we know I is equal to A inverse multiplied by A. Now, this matrix is an identity matrix of 2 into 2 order. Now, comparing these two expressions, clearly we can see A inverse is represented by the matrix 4 minus 5 minus 3, 4. So, our required inverse is represented by the matrix 4 minus 5 minus 3, 4. This completes the session. Hope you understood the session. Take care and goodbye.


So that's obviously a very common problem for patients that I see in clinic, but I think for all of us I don't think there's anyone who hasn't had problems sleeping at some stage Sleeps are very natural very healthy process. It is something that We shouldn't be chasing after or obsessing over So that's the first thing to say Don't worry if you have a bad night's sleep nothing bad is going to happen But there are a few things that we can do to try and improve our sleep overall We all are governed by light and dark cycles what we call the circadian rhythm and you everyone's circadian rhythm is slightly different But never quite 24 hours So we rely on certain cues throughout the day to make sure that our circadian rhythm stays in check And waking up at the same time is probably probably the most important if you take away one tip So having a time that you wake up and it's got to be the same weekdays and weekends I'm afraid people who sleep in at the weekend is something we call social jet lag And that means you haven't slept properly during the week so prioritizing Sleep and wake times is important Getting lots of light in the morning is very important because that tells your brain that it's morning And during the day as well getting exposure ideally just sunlight Ditto at night time so I tend to tell people they should have a sort of 60 to 90 minute buffer zone before bedtime where you're preparing for sleep Some of that is making sure that your bedroom is a sanctuary a haven a nest Which promotes good sleep so dark cool no flashing lights No electronic devices and really you just go there to sleep and nothing else And preparing for sleep means winding down from the day I tell people to write down their problems there to do lists so that it's out of their brain and Can be dealt with and not ruminated on before they go to sleep Caffeine's a problem. So no caffeine after lunch time It takes six hours for half of the amount of caffeine to leave your body. So it hangs around for a long time So definitely no caffeine after lunch time if you're having trouble sleeping Try to avoid heavy meals just before bed. So ideally a couple of hours of not eating before you go to bed And then having some relaxing wind down time.  So if you can manage 20 to 30 minutes of moderate intensity exercise a day Then that helps enormously with sleep quality Ideally not just well definitely not just before bedtime. It has the opposite effect. So at least a couple of hours before bedtime and if you can do most of those and Stick to a regular sleep schedule sleep shouldn't be a problem But it's important to not obsess over it too much Those would be my top tips


The 7-star Typhlosion Terrorite Event is now back in Pokemon Scarlet and Violet for its second phase through. We're gonna go through all the details in today's video as well as the best builds for you to beat this Pokemon in your game. So as of recording this video on the 21st of April and running through until the 23rd of April, Typhlosion will be returning in Pokemon Scarlet and Violet. The details of this raid, if we hop over to our trusty source at Sarabee, we can see that it will be set to level 100, it will be the Ghost Terror type and it will have the mightiest mark. Only one of these is catchable per save file, so if you caught this Typhlosion the first time round, when it appeared from the 14th to the 16th, you're not gonna be able to catch it this time round, but you can go in, battle it, beat it and primarily, you're gonna do that for the amazing item drops that you're gonna see here that you can get picked up from it. Ability patches included with this, Ghost Terror Shards and level up candies as well as some other high cost items. Including Ability Capsules, Bottle Caps and PP Ups. The move set that Typhlosion is going to run is primarily gonna be Eruption, Shadow Ball, Player of Earthquake and then only one additional move and that will be Sunny Day. And although Typhlosion is a pretty tough Pokemon, if you go and go into it with the correct Pokemon, it is one of the easiest 7 star raid events to beat. So in our preview article that we put up just before the event went live with our best builds, one of the Pokemon that I featured in that was Meow's Garada. Now when the event went live, I normally generally do a bunch of testing on which Pokemon are the best to go into this raid with. Furigraph worked out well and Annihilate seemed to be the other one on top, but Meow's Garada was something I did think could work, but just couldn't figure out how. Until over the weekend and I tested a new idea and I believe Meow's Garada to be one of the best Pokemon to solo this raid with. So the build for the Meow's Garada that we're gonna feature in today's video is this one indeed. It is very similar to the one we covered in our preview article. It is Grass and Dark with a Dark Teratiping. The item is a Shellbell. It is essential for this for recovery once you start doing some big damage and it all comes down to the move set which is Skill Swap, Substitute, Nasty Pullout and Dark Pulse. Has an EV spread of 252 HP, 252 Special Attack and a Modest Nature. And that is everything that you're gonna need to know on this particular Meow's Garada. Just make sure it's got that Dark Teratiping, Shellbell item, this move set and then the EV spread that we've just mentioned. All the details will be down in the description if you want to take a closer look at it. Now the basic premise with this Meow's Garada is to turn one Skill Swap. You will steal the Flash Fire from the Typhlosion giving you complete immunity to the big threatening super effective fire type attacks for the rest of the game. Now the issue was before with the Typhlosion.  Then hit the field again, you can Skill Swap but you're only gonna be Skill Swapping overgrow off the Typhlosion making the rest of the raid completely impossible. So to get around this, what you need to do is turn one Skill Swap with the Meow's Garada, then turn two, you go for the Substitute. That will then give you the immunity, the turn that you need where it nullifies abilities and then goes for the eruption in the sun. It will destroy the Substitute but you will still be active on the field, you will still have that Flash Fire and then after that you'll be able to go for those nasty plots, boost yourself up and then start firing dog pulses off until you can terastalize. You don't need to worry about your abilities being nullified again for the rest of the battle and you will make easy quick work of Typhlosion with this Meow's Garada and it is a great build and I'm pleased we finally got it to work. But it is one of those Pokemon I feel can do a really good job and it is a little bit different from just using an Islayip every single week when these 7 Star Raids come around. Obviously the other build that we did feature in our last week episode that is one of the best ones going into this and it's a one that a lot of you probably already have to beat 6 Star Terrorids but an Islayip is a great Pokemon. You want that Ghost Terror typing on there, Shellbell as the item, EV spread of 252 Attack, 252 Special Defense just to take those big special attacking moves from Typhlosion a little bit easier, Define Ability and then you want just basically the moveset is only Rage Fist but you can throw a Screech in there. It will speed things up if you can get a couple of Screeches off. Turn 1, Turn 2, Turn 3 before you faint then come back just spam Rage Fist after this and you'll make easy work of the Typhlosion. Like I say a lot of you probably already have this build in your game so it's not going to be too hard to grab and then just go in and farm for these items. And the final Pokemon we will mention is for a Rigoraf. It was the main Pokemon that I featured last time and it was the main Pokemon that we use Skill Swap with because it has the natural bulk to be able to take the big attacks from Typhlosion.  You don't need to worry about the ability for a Rigoraf because we will be Skill Swapping it away and obtaining Flash Fire. EV spread though is 252 HP and then 252 Special Defense on the Furigraf just to make sure you can take again big special attacks a lot easier from the Typhlosion. Again with a Shell Bell item for recovery but that is the Furigraf and as always these Pokemon will all be down in the description below. But the Meow Squarda like I say is the one that we're featuring today and I'm super pleased that we were able to get it to work just working around when the abilities were nullified and Typhlosion has that free turn because we weren't sure about that in the preview article but because of the substitute we're now able to utilize this Pokemon and make good use of it and it does a really good job and if this Terrorite event is something you want to farm for those high cost items after you've beat the Typhlosion with whatever Pokemon you go into you can then reset your dens and spawn a new 7 star raid and just keep going. You need to do this by hitting your home menu coming down into System then into System Settings down to date and time make sure your clock is off via the internet and then just press through the date and time click OK, come back into home and then back into game and you'll see all of your dens will repopulate in the area and you'll be able to relocate the 7 star Terrorite den to go again and you can just repeat this process over and over again until you're satisfied with the amount of items you've got. Remember you can keep this event in your game until the next time you go online the event itself as I say will be ending on the 23rd of April but if you don't go online after this date the event will stay active in your game until the next time you do go online after the 23rd. So that is everything for today's video. I hope you have fun with this second phase of the Typhlosion 7 star Terrorite event. If you found today's video useful please drop a like it is massively helpful and let me know what your thoughts are on the Meow Scarada set and if you do use it in your games to take on this Typhlosion I guarantee it will work and you'll have a lot of fun with it in the process. Do subscribe to the channel as well so you don't miss any future content covering Pokemon Scarlet and Violet. Thank you again for tuning in. Have a great rest of your day and I'll catch you all in another video very soon. So until then friends, take care of yourselves and bye bye.


Hi everyone. Today I'm going to talk to you about your first 90 days as a new PM. I'm excited to share with you some tips that have helped me transition into new product roles over my career. A little bit about myself. I'm Swaroop Desai and I'm a product manager at Meta. I work on Facebook Watch, which is a video and entertainment platform for Facebook app. A little background about myself. I started my journey as an intern product manager at Thomas and Reuters and have navigated my way through various different industries and companies over the course of my career, from media to consulting to finance and finance. There were some key learnings along the way that I picked up over the course and I would like to share those with you and hopefully help you transition easier into your new role. Regardless of the industry or the level of role, these guidelines should help you reduce the ambiguity and give you some structure as you embark on a new journey. Here because you either got a new role as a PM or you're on the path of taking the next steps towards your new role. Either way, this is a great first step. What I'm going to discuss today applies to a broad spectrum of PM roles, whether you're a fresh graduate or taking on a senior leadership role. The key ingredients to starting off right as a successful PM stay more or less the same. And the first impressions that you make are here to stay. So it is worthwhile and it pays off to put in that effort in your first few days. A senior leader actually said this to me once when I started on your role and this has stuck with me. He said, Swaroop, the amount of time and effort you put into your first 90 days at work has the ability to generate disproportionate returns over the time period that you hold the job. And this is so true. So I think this is something that you should keep in mind as you start your onboarding journey. So what should your first 90 days as a new PM look like? I'm going to take the next 20 minutes to share top 10 tips to transition into your new role. And I'm going to wrap this up with five key takeaways that you can walk away with. So let's get started. First, I want to say congratulations. Congratulations on the new role. Going through interviews, negotiating salaries and landing the job of your dreams needs a lot of grit and hard work. But thankfully, the hard part is behind you and the good part is coming up. As exciting as it is to start a new role at a new company or even at your existing company, it comes with its set of challenges and anxiety. It's almost like the first day of school, that kind of feeling. So as a product manager, this adds another level of complexity. As a role itself needs you to have a good breadth and depth of knowledge in the industry space. So till the time you feel onboarded ramped up and relevant to the team, this can induce a lot of stress and anxiety. Add COVID to the mix and remote onboarding that throws its own set of challenges. So I'm going to try and simplify this for you by sharing my experiences and tips so that you can make your transition fun, memorable and hopefully a time of no stress or may I say no stress and anxiety. So let's begin with the top 10 tips of a PM in a new role. Number one, learn, learn, learn. Two, roadmap for product you. Three, build relationships. Number four, establish credibility and secure early wins. Number five, align with your team. Number six, focus on execution. Number seven, strategy. Number eight, take charge, take control. Number nine, launch your feature or product. And number 10, land and celebrate. So let's break this down. Breaking this up by the timeline to give you some sense of milestones. Your first 30 days are all about learning and preparing yourself to take on the role. Your next 30 days are going to be all about understanding the fine print and starting with baby wins.  Breaking down your transition into key milestones as Michael Watkins breaks it down into his book, The First 90 Days. Your phase one is when you're just being considered for the job or you're considering yourself for a new role. Your phase two is your pre-entry. That's when you're learning behind the scenes. You know that you have the job and you're preparing yourself to take on the new role. So this includes a whole set of challenges including your mental preparation, understanding what the role expects from you and all those kind of things. And finally the phase three, which is the most important one, which we are going to focus most of our conversation around is your first 90 days. So what does your first 30, 60 and 90 day look like? So let's begin with the first one. Learn, learn, learn. Where should I start? Peak of Product Manager is fundamentally about how technology can support and drive the business forward by building amazing products for customers. To effectively contribute, you need to understand where the organization is and where it wants to go. Understand the destination relative to where you are today. This will help you get a sense of how you can contribute to this destination. Regardless of the size of the company, you must understand this delta. So how can you do this? One, learn about your role and responsibilities. Understand what the leadership expects from you and also what the team expects from you. Two, ground yourself by getting a good industry context. Read about the industry, research, important posts about prior milestones. Three, understand your team and responsibilities. It is extremely helpful to have a good understanding of what the org structure is. One, upward, what does your leadership look like? Two, lateral, your day-to-day. Three, your cost-cross-functional peers. What's the privacy, legal, any other partner teams that you need to work with on your day-to-day? Four, your direct reports. If you are a manager, understand what your direct reports are and what their expectations are from a manager. Five, your peers. It's extremely important to have a good relationship with your peers because they're going to help you get through your role. One tool that I find useful is called the career cold start algorithm. The concept is simple. When you start a new role, pick five people and ask them to tell you everything you should know to perform your role efficiently. Ask them a prepared set of standard questions, such as the challenges they face, the opportunities that exist, etc. And ask each of them the same set of questions. Then at the end of the conversation, ask them to refer you to five new people that you should talk to and continue doing this till you reach a point of diminishing returns. What this technique does is it allows you to ramp up very quickly in your understanding of the space. It gives you a quick sense of what the challenges are and understand those challenges and opportunities through the lenses of people in different roles that you have spoken to. And all of this while talking and covering a lot of ground in terms of the team that you should know. So as I say, get ready to drink from the fire hose. One book that I highly recommend that has been particularly helpful for me is The First 90 Days by Michael Watkins. It has some good suggestions to help you hit the ground running in your first 90 days. Learn the three P's, your product, your process and your people. In terms of your product, dog food product, read previous product dogs. Play with the features till you get a good hang of the workflows. A great way is also by starting to do like a quick audit review of how you find the product to be. Understand who your customers are. Extremely, extremely important and something that will come in handy for you throughout your product journey. Understand the metrics. How do you measure success? In terms of the process, you need to understand the process that the team has in place today. This will help you understand if there are any gaps and if you should be fixing anything. Take copious notes. Hold back from trying to point out the obvious flaws. This is your time to understand and identify. Be patient on sharing your thoughts with the team. In terms of the people, understand your people. Understand what motivates them. What is that each engineer's unique skill set so that you can help them reach their best by understanding how you can best support them. Accelerate hitting the break even point. What we want to do here is to accelerate hitting that breaking point. What does that mean? You want to reach the point where the value you consume is equal to the value you produced for your team as quickly as possible in your 90 day onboard vehicle. At this point, your net contribution is still zero. But it will start picking up and you will eventually reach the point where your output significantly outpaces the value that you consume. Next, putting together a roadmap for product you. Roadmap for product you. Your first roadmap should be for product you. Define what success looks like. What does 2.0 version of you look like? The ramp up version of you. Set milestones and goals. Share this plan with your manager and team. This will help you hold yourself accountable and also build trust. Here's a quick roadmap template that I like. Think about the learning objectives that you want to accomplish. Set milestones for yourself against those learning objectives. Try to put together an estimated timeline to diagnose yourself. And have a POC that you know you can rely on to get some support on. Having such a roadmap template definitely helps you guide and structure your onboarding much more. Something that's also helpful is having a 30-60 90 day plan. In my previous roles, I have tried to put together a 30-60 90 day plan and I typically like to share it with my manager and get their inputs. This helps you quickly put forth your ideas on how you would like to accomplish your onboarding. While also giving your manager a chance to provide more inputs and help you course correctly. So some elements of a good 30-60 90 day plan is what are your priorities? How are you planning to achieve those priorities? What are your learning goals? And how will you have some milestones to measure success? So next, build relationships. Building relationships is usually the easiest, takes little time and goes a long way. So make sure you do that and invest time in doing this. Let your team know that you genuinely care about them. Know them as people. Be a good listener. Understand everyone's point of view. Each person on the team brings in a different point of view. Show them that you can be trusted and that you will represent them accurately and that you are a good sound leader to guide the team towards success. Next, establish credibility. Secure early wins. Secure early wins. This is something that can be any small successes that you can help the team accomplish. Demonstrate that you are invested in the team's success. Help them make small decisions. At this point, you should have enough context which will help you make some small, tiny decisions for the team by helping them evaluate the trade-offs. Use suitable decision-making frameworks and bring in the fresh perspective and share those with everyone. Next, align with your team. This is extremely important. Aligning with your team. Develop a shared vision and get a buy-in from the team. Deep alignment comes from strong processes where people have had a chance to express their perspectives and understand others well. Build inclusive processes. Allow people to share their thoughts in multiple modes. Async via docs or in-person meetings. Different people on the team communicate differently and it's your role to understand them. Ensure that everyone who wanted to had a chance to speak in the meetings. Sometimes, this means a longer or follow-up meetings as well. Go out of your way to make sure that everybody feels accommodated. Be mindful of developing biases. This is something that happens so subconsciously but if you're aware of this, you might be mindful and prevent that from happening. Provide that psychological safety for everyone to be able to share their ideas and give them the credit for that. This will help you bring out the best ideas and build truly inclusive and successful products. Next, focus on execution. This one's my favorite. This should be easy, right? It's simple. Keep things simple. Have a good process. Make sure you have a good understanding of workflows and have an understanding of the design. Improve processes. Identify any gaps that are missing in the current process and modify and make changes to suit the team. Run efficient and effective team meetings and escalate whenever necessary because this helps the team speed up things and continue the pace and momentum going. Strategy. Understand the current mission. What is the goal? The vision. What does your product look like five years from now? The strategy. How is the play of team planning to meet the goal? Identify the gaps in the current strategy. It's important for you to understand certain things that did or did not work well with the strategy that was in place. Bring some problem space with the team. Make sure that you stick to the problem space and not let yourself or the team jump into a solution. Build a solution and a point of view together as a team. And then this will help you tweak the whole strategy so that you can come up with a new strategy which will actually work. Next, now that you have the learning under your belt and you also have a small few wins and you have a handle on the execution, it's time for you to take charge. Take charge. You got this. Start driving meetings with minimum support. You should have a good layer of land by now. Practice written collaboration on critical documents. This is a good practice. It will not just help you keep your documents organized, but it will also help new hires joining the team ramp up easily if you have all your documents ordered. Share your ideas verbally written and visually through designs and examples. This helps reach your message to everyone in the way they like to communicate best. Drive consensus on ideas within the team and externally. This is extremely important. As a product manager, you're not there to make the decisions for the team. You're there to make sure that the team is able to come to those decisions together and also make sure that you're taking your stakeholders along with you. And finally, take charge of all the PM duties. You need to help the team and so that the team can focus on the important deliverables that they need to focus on. Make sure that you are removing all the blockers that you possibly can in your limits for your team. Next, launch your new feature or product, but it'll be too soon for a product in your first 90 days. So let's say it's a new feature that you already have a good understanding of that you're ready to launch. Work on your PRD. Put together your product spec. Share this with external stakeholders and get alignment, especially in the world of remote. Make sure that you over communicate. Always err on the side of over communicating. Roll out your first new feature or idea. Follow a pre-agreed process of deployment so that everybody is aware of what the next steps should be. Communicate the launch internally as well as externally. And this could depend on the type of team or product that you're building. But make sure that you are, again, trying to err on the side of over communicating. Make sure you plan for your public tests. Have a test plan. Do all the user testing wherever applicable. Monitor the dashboards for critical metric movement. This is extremely important to know how the new product or feature idea that you are launching is impacting. And lastly, don't forget to monitor the guardrails so that you're not negatively impacting anything important. And last but not the least, land and celebrate. Land and celebrate. Share results for the launch with the team and the leadership and do this on a cadence. Help the team see how their efforts laddered up to the broader goals and always iterate as needed. Finally, and most importantly, take the time to celebrate your wins as a team and reflect on the learnings. There's so much that you can gain from just looking at what went well and what didn't go well to help you improve and build even better processes and products. So wrapping up, five key takeaways. The sooner that you hit the break even point, the better it is for yourself as well as the team. This will make you and the team perceives yourself as someone who doesn't need too much handholding and you will feel welcome on the team. Make sure you build your relationships because that's going to take you a long, long way. Ramp up your learning and make sure you focus on the product, the industry and the process. Establish credibility with the team through early wins. Take ownership once you feel ready. And finally, be your team's advocate and support them in every way possible. Thank you everyone for your time today. I hope these tips come in handy and make your onboarding experience simple and fun. Enjoy your onboarding. Feel free to reach out to me if you have any questions or if you simply want to connect, you can find me on Facebook or LinkedIn. That's Varuk Desai. Thanks.