we are considering piecewise polynomial approximation. Our function f is defined on interval a b closed and bounded interval a b taking real values. We divide this interval into n equal parts. So, each sub interval is going to have length h which is equal to b minus a divided by n. We fix k to be the degree of the polynomial and then on each interval we are going to fit a polynomial of degree less than or equal to k. We have already considered the case of piecewise linear interpolation. In that case we had considered the interval interpolation points to be the partition points. So, there are n plus 1 partition points we are dividing the interval into n equal parts. So, there are n plus 1 partition points. So, for each interval ti to ti plus 1 we look at the value of the function at ti and value at ti plus 1 and join it by straight line. So, that gives us a polynomial of degree less than or equal to 1 because we are considering the interpolation points to be the end points of the interval. Our function which is a piecewise linear function it is going to be a continuous function and last time we saw that the maximum error norm of f minus p n its maximum norm or infinity norm is less than or equal to constant times h square h is b minus a by n b minus a is fixed. So, as n tends to infinity h square will tend to 0 and error will tend to 0. So, thus the degree of the polynomial in each sub interval was fixed to be less than or equal to 1 and then we increase the number of intervals and then under the assumption of function to be 2 times continuously differentiable we have proved the convergence of piecewise linear interpolation. Today we are going to consider piecewise quadratic interpolation and piecewise cubic interpolation. We will show that in the case of piecewise quadratic interpolation the maximum error is going to be less than or equal to constant times h cube. So, instead of h square we get h cube. So, we have got faster convergence and in case of piecewise cubic we will get the maximum error to be less than or equal to constant times h raise to 4. So, still a faster convergence. Now, let me consider the notations which we are going to use. So, for k bigger than or equal to 1 c k a b is going to consist of functions which are k times continuously differentiable on interval a b. This is going to be a vector space. For a continuous function we have already defined the maximum norm that is maximum of mod f x x belonging to a b. Throughout this lecture we are going to consider the following uniform partition of interval a b which is a is equal to t 0 less than t 1 less than t n is equal to b. For i is equal to 0 1 up to n minus 1 t i plus 1 minus t i which is equal to h that is going to be equal to b minus a by n. Let us recall the error in the interpolating polynomial. I had told you that this is a very important relation. So, p n is a polynomial of degree less than or equal to n interpolating the given function at n plus 1 points x 0 x 1 x n. Then the we have got an expression for the error f x minus p n x and if the function f is n plus 1 times differentiable then we get the error in terms of the derivative of the function evaluated at some point. The error consists of two parts. It consists of divided difference f of x 0 x 1 x n x which if the function is n plus 1 times differentiable then it is going to be equal to n plus first derivative evaluated at some point c. That point c is going to depend on x. It also depends on x 0 x 1 x n, but x 0 x 1 x n are fixed and x varies over interval a b. So, this is one part and other part is the function w x that is x minus x 0 x minus x 1 x minus x n. Last time we have seen that maximum norm of w is equal to x 1 x n. So, this norm is minimized by choosing x 0 x 1 x n to be Chebyshev points. Now, today we are going to replace the interval a b by interval t i to t i plus 1 and in each polynomial in each sub interval we are going to have a polynomial of degree less than or equal to k. So, we are going to join two polynomials together. Let us look at first the case of linear polynomial. So, let p 0 x to be a 0 plus a 1 x, x belongs to 0 to 1. Then, let q 0 x to be equal to b 0 plus b 1 x, x belonging to 1 to 2. So, we have interval 0, 1 and 2. So, we have interval 0, 1 and 2. If we do not impose any continuity condition at 1, then we have got four constants which can vary independently. So, a 0, a 1, b 0, b 1. Suppose we impose continuity that means we want p 0 at 1 should be equal to q 0 at 1. The problem of continuity comes only at the partition points.  In the interval 1 to 2, the polynomial q 0 is going to be infinitely many times differentiable. So, the question is whether the values at 1, whether they match. So, that gives us a 0 plus a 1 is equal to b 0 plus b 1. So, that means now if I fix a 0, a 1 and b 0, then I have no choice for b 1.  So, earlier we could vary a 0, a 1, b 0, b 1 independently. Now, we can vary only a 1, b 1, b 1 independently. So, we want three constants independently a 0, a 1 and b 0 and then b 1 is determined in terms of a 0, a 1 and b 0. Then suppose we want the derivatives to be continuous. So, we want p 0 dash at 1 is equal to q 0 dash at 1, p 0 dash at x is equal to q 0 dash at x is equal to a 1, q 0 dash at x is equal to b 1 and hence we get a 1 to be equal to b 1. Now we have two relations. We have got this relation and we have got this relation. So, from these two relations, one concludes that a 0 has to be equal to b 0, a 1 has to be equal to b 1. That means it is a single polynomial. So, if we look at two polynomials of degree less than or equal to 1, then we can ask the resulting function to be continuous and still retain the piecewise nature. But if we ask the function to be continuous and differentiable, then it becomes the single polynomial. Next let us look at the case of quadratic polynomials. So, look at p 0 x to be equal to a 0 plus a 1 x plus a 2 x square x belonging to 0 to 1 and q 0 x to be equal to b 0 plus b 1 x plus b 2 x square x belonging to 1 to 2. So, we have got six constants which can vary independently. Continuity at point 1 implies that a 0 is plus a 1 plus a 2 has to be equal to b 0 plus b 1 plus b 2. So, that means we have got only five constants which are independent. Then suppose we want p 0 dash at 1 is equal to q 0 dash at 1. Then we have a 1 plus 2 a 2 should be equal to b 1 plus 2 b 2. So, one more degree of liberty is lost. If we say that p 0 double dash 1 is equal to q 0 is equal to q 0 is equal to double dash 1, then this will imply that 2 a 2 should be equal to 2 b 2. So, thus from here we have got b 2 is equal to a 2. From here we get b 1 to be equal to a 1 and from here we get b 0 to be equal to a 0. So, as in the case of quadratic polynomials, the piecewise linear polynomials, if you say that two quadratic polynomials p 2 and q 2, they should together form a two times continuously differentiable function. In that case it becomes a single polynomial defined on these two intervals. So, if we want to retain the piecewise structure, then for quadratic polynomial we can go up to the continuity of the derivative. If we increase the degree of the polynomial, that means if we consider two cubic polynomials, then we can demand that the resulting function should be two times differentiable. And still we can have two different cubic polynomials respectively on the interval 0 to 1 and interval 1 to 2. If for cubic polynomials we ask for the function to be continuous, first derivative, second derivative, third derivative to be continuous, then it will reduce to a single cubic polynomial. So, when we demand the continuity or the continuity of the derivatives, then for each of this demand the degree of liberty gets reduced by 1. So, now we are going to here we had considered only two polynomials. Now, we are going to consider n intervals and on each interval we consider a polynomial. So, first we are going to look at the dimension of such a space and then piecewise linear case we have already considered. So, we will go to piecewise quadratic and piecewise cubic. So, as I said our interval a b is going to be divided into n equal parts and then you consider x n to be set of all f or set of all g defined on interval a b such that restriction of g to interval t i to t i plus 1 is a polynomial of degree less than or equal to k. So, we have got n intervals in each interval it is a polynomial of degree less than or equal to k. That means we have got k plus 1 coefficients at our disposal. So, total there are going to be n into k plus 1 and that is going to be the dimension of the space x n. If we consider the space y n where once again g restricted to t i to t i plus 1 is a polynomial of degree less than or equal to k, but we impose the function to be continuous. So, we have got this is our interval a b then we are subdividing into n into k plus 1. So, this is t 0, t 1, t 2, t n minus 1, t n. Here it is a polynomial of degree less than or equal to k. So, we saw that the total degrees of liberty they are n into k plus 1. Now, we will want these two polynomials to be continuous. So, that will impose one constraint then continuity at t 2 and continuity at t n minus 1. So, we are imposing total n minus 1 constraints. So, it will be minus n minus 1 and thus we get the dimension to be equal to n k plus 1. So, this is if you are assuming continuity. If we want the function its derivative and f double dash to be continuous then the dimension will be equal to n into k plus 1 minus at each point t 1, t 2, t n minus 1. We are putting two constraints. So, that will be minus 2 into minus 1. So, that will be the dimension of the space. So, this was general case. Now, let me recall the piecewise linear continuous polynomials. We have got n intervals in each interval linear means we have got two coefficients. So, it is 2 into n at interior nodes and the continuity considerations. So, they are n minus 1, t 1, t 2, t n minus 1. So, the dimension of the space is 2 n minus n minus 1. So, that is going to be n plus 1. So, this is going to be dimension of our vector space which consists of piecewise linear continuous functions. If we look at the value of the function f at the partition points that means t 0, t 1 up to t n. So, these partition points they are n plus 1 in number. So, if we try to determine a piecewise linear continuous function which matches with the given function at t 0, t 1 up to t n then such a function is going to be unique. This part we had seen last time that for linear polynomial interpolation norm of f minus p 1 its infinity norm is less than or equal to norm f double dash infinity divided by 2 into b minus a by 2 square. Our function is going to be linear on each interval t i to t i plus 1. So, this bound we will write for the interval t i to t i plus 1 and the bound will be norm f double dash infinity norm divided by 2, b minus a will be replaced by h by 2 square. So, that will be for the maximum of modulus of f x minus g 1 x where g 1 is our piecewise linear continuous function for x belonging to t i to t i plus 1, but then the bound is going to be independent of i. So, the same bound works for each interval t i to t i plus 1 and we get norm of f minus g n its infinity norm to be less than or equal to norm f double dash infinity divided by 8 h square. So, it is the bound for the linear polynomial we had applied to piecewise linear polynomial on each interval t i to t i plus 1. So, thus we have got the error in the piecewise linear continuous functions to be less than or equal to constant times h square where constant is second derivative nor f double dash its infinity norm divided by h. What we are going to do is consider an example, look at the function f x is equal to root x, x belonging to closed interval 1 to 2. So, root x on interval 1 to 2 is going to be infinitely many times differentiable, what we want is it should be differentiable twice and then we will look at the second derivative. So, we can find a value of norm f double dash infinity in this special case f x is equal to root x, then we will look at this upper bound and suppose we want the error to be less than 10 raise to minus 6, then we will get our error which involves b minus a by n whole square multiplied by some constant to be less than 10 raise to minus 6 and from that we can get the value of n. So, by looking at the upper bound we can beforehand tell what n we should use to get the value of n we should use. So, as to obtain the desired accuracy the same example we are also going to consider for the piece wise quadratic polynomial. So, we have f x is equal to root x, x belonging to 1 to 2, we have norm of f minus g n infinity to be less than or equal to 0. So, we have to find the value of f dash x is equal to norm f double dash infinity divided by 8 into h square, where h is b minus a by n and in this case it is going to be 1 by n. f dash x is going to be 1 upon 2 root x, f double dash x will be equal to minus 1 by 4 x raise to 3 by 2. When you consider mod of f double dash x, this is going to be 1 upon 4 x raise to 3 by 2. So, this is a decreasing function on interval 1 to 2 and hence maximum of mod of f double dash x, x belonging to 1 to 2, this will be attained at the left end 0.1 and then you get it to be equal to 1 by 4. So, we have norm of f minus g n infinity norm to be less than or equal to 1 by 4. So, we have norm of f minus g n infinity norm 1 upon 32 and then 1 upon n square because h is equal to 1 by n. Now, this will be less than say 10 raise to minus 6 provided n square is bigger than 10 raise to 6 divided by 32. So, if you choose n is equal to 200, then that will work. So, in fact, any number in to be bigger than 200 will work and this is a rough estimate. So, this was about piecewise linear interpolation. Now, we are going to consider piecewise quadratic. So, first let us look at quadratic polynomials. So, one single quadratic polynomial defined on interval a b. For quadratic polynomial interpolation, we need three points. So, let us choose those three points to be two end points of the interval a b and the midpoint of the interval a b because we are choosing the end points and afterwards when we consider the interval a b sub divide into n equal parts and for each sub interval t i to t i plus 1, we will be choosing two end points and the midpoint. So, because the partition points are going to be the interpolation points for piecewise quadratic function, our resulting function is going to be continuous. So, let me first recall that if f is from a b to r, p to x is a polynomial of degree less than or equal to 2 such that p 2 at a is equal to f of a, p 2 at the midpoint a plus b by 2 is f of a plus b by 2.   So, let me call this as w x. So, we have f x minus p 2 x is equal to f triple dash at some point c divided by 3 factorial and then w x. So, when we look at the error norm of f minus p 2 infinity norm, this will be less than or equal to norm f triple dash divided by 3 factorial and into norm of w infinity. So, w x is our function x minus a x minus a plus b by 2 x minus b. We can find the infinity norm for this function. We have got a continuous function. When we want to find its maximum, what we have to do is look at the value at the end points. Look at the value at the end points, look at the value where the derivative vanishes or derivative does not exist. So, such points they are known as the critical points. So, we get hopefully finitely many points, two end points and the critical points. Compare the value of the function at this finitely many points, whichever is the maximum that is going to be absolute maximum of the function, whichever is the minimum it is absolute minimum. You should not go for second derivative test because the second derivative tells you only about the local maximum or local minimum. So, let us calculate maximum of mod of w x, x belonging to a b. So, for the sake of convenience, we will make a change of variable and then we will calculate. So, we have maximum of modulus of w x, x belonging to a b, maximum of modulus of w x, x belonging to a b, maximum of modulus of w is equal to maximum of mod of x minus a, x minus a plus b by 2, x minus b, x belonging to a b. So, let y be equal to x minus a plus b by 2 and k to be equal to b minus a by 2. Then, when we look at y plus k, that will be nothing but x minus a, y minus k will be x minus b and hence, this will be x minus b. This maximum, it reduces to maximum of mod of y minus a or it is going to be maximum of y plus k, y, y minus k, where y is going to vary over, x varies over interval a b. So, x minus a plus b by 2, that is going to vary over minus k to k, since k is b minus a by 2. So, we have this is maximum of mod of y cube minus y k square, y belonging to x minus k to k. This function vanishes at the two end points, minus k and k. Now, let us look at the critical points. So, critical point will be given by 3 y square minus k square is equal to 0, this is the derivative value. So, you get y is equal to plus or minus k by root 3. So, these are critical points. When you put y is equal to k by root 3, then modulus of y cube minus y k square is going to be k cube by 3 root 3 minus k cube by root 3. So, this will be 2 k cube divided by 3 root 3, it is modulus. And for y is equal to minus k by root 3, you are going to get the same value. So, the maximum of this k cube minus k by root is going to be equal to 2 upon 3 root 3 and k cube is b minus a by 2 whole cube. So, that is for the norm w infinity and then for the error in the quadratic polynomial, we have got norm w infinity and then norm f triple dash infinity divided by 3 factorial. So, this is our norm of f minus p 2 infinity which is less than or equal to norm f triple dash infinity upon 9 root 3 b minus a by 2 cube. So, this was for a single polynomial. Now, we consider piecewise quadratic. We look at uniform partition x n is the vectors space of functions which are continuous and which are such that on each sub interval, it is a polynomial of degree less than or equal to 2. So, the dimension of the space x n is going to be n intervals on each interval a polynomial of degree less than or equal to 2 that means 3 coefficients. So, it is 3 n and then interior partition points are n minus 1 in number t 1 t 2 t n minus 1 you want continuity. So, subtract n minus 1 and then you get 2 n plus 1. So, now, in order to have a unique g n belonging to x n, we need to provide 2 n plus 1 condition. So, consider g n which belongs to x n and which interpolates the given function at t i and also at the mid points of the sub interval. Such a polynomial such a piecewise polynomial is going to be unique and now, look at the error. So, the error for single polynomial was norm f triple dash infinity upon 9 root 3 b minus a by 2 cube. Look at the interval t i to t i plus 1. So, replace the interval a b by t i to t i plus 1 then upper bound is norm f triple dash infinity by 9 root 3 into h by 2 cube. We could have taken maximum for the third derivative only on the interval t i to t i plus 1. Still, it would have provided us an upper bound, but we wanted upper bound which is independent of i. So, for each interval t i to t i plus 1, we have got the same bound. So, if you take maximum over the whole interval a b, it is going to be the same bound and notice that here the error is less than or equal to constant times h cube. The constant will involve the third derivative of the function and then the constants 9 root 3 and then 8 in the denominator. We look at again the same function f x is equal to root x. So, the for this root x for piecewise linear continuous approximation, we saw that the in order to have the error to be less than 10 raise to minus 6, we have to choose n to be about 200. Now, here because our error is less than or equal to constant times h cube, definitely the constant here and the constant in piecewise linear, these two constants they are different. For piecewise linear approximation, the constant was depending on the second derivative. Now, here it is depending on third derivative. So, here for piecewise quadratic, the convergence is proved under the condition that f is 3 times differentiable. So, now, we have constant times h cube and then we hope that the accuracy 10 raise to minus 6 should be achieved for a smaller value of n because h cube converges faster to 0 than h square and this is the constant. So, this turns out to be the case and when you do a bit of computation, we had calculated up to f double dash. Now, we have to calculate f triple dash. So, that is going to be 3 by 8 x raise to 5 by 2. Once again, 1 upon x raise to 5 by 2 is a decreasing function on closed interval 1 to 2 and hence the maximum will be attained at the left end point which is 1 and hence norm of f triple dash infinity is equal to 3 by 8. Then, we get norm of f minus g n infinity which is less than or equal to a constant into 1 by n cube. So, this upper bound will be less than 10 raise to minus 6 provided n cube is bigger than 10 raise to 6 divided by 192 into root 3. So, in this case, n is equal to 20 will work. The values of n is equal to 200 and n is equal to 20, these are rough estimates. You can try to find the better estimate. So far, we had considered piecewise interpolation where you were interpolating the function. Now, we are going to consider a piecewise cubic polynomial which interpolates function values and the derivative values at the partition points. So, here the interpolation which we are going to explain, it can be done provided your derivative values of the function, they are also available. This may not be always the case. If only function values are available, then we cannot do this piecewise cubic hermite interpolation. Now, the method or the methodology is the same. Consider the derivative values of cubic hermite interpolation and then do that for each sub interval. We have already seen cubic polynomial interpolation which interpolates the given function at two end points a and b and its derivative values at a and b. In order to determine a cubic polynomial, since there are four coefficients to be determined, we need four interpolation conditions. These four interpolation conditions can be either four distinct interpolation points or four points which are which can be repeated. So, we are repeating the left end point a twice and right end point b twice. In this case, we had seen that the error involves the divided difference based on a repeated twice, b repeated twice x and multiplied by function w x which is x minus a into x minus b whole square. Already, we had seen that maximum of modulus of x minus a x minus b it is attained at the midpoint a plus b by 2. So, for the square x minus a square x minus b square, again the maximum is going to be attained at a plus b by 2 and that gives us a upper bound for the cubic hermite interpolation. This result for a single cubic hermite interpolation we will apply for each interval and then obtain a piecewise cubic function which is not only continuous, but it is differentiable. If we had decided to choose four distinct points in each interval, like for linear we had chosen two points to be end points, for quadratic we had chosen two points to be end points, three points which involve two end points and the midpoint. So, similarly for the piecewise cubic interpolation we could have chosen two end points and two other points in the interior of the interval. So, we get four distinct point. If you do that, then because we are choosing the interpolation points, our interpolation points include the partition points the function will be continuous, but then for the cubic hermite interpolation by very fact that you are interpolating the given function and its derivative. At the two end points our function is going to be continuously differentiable function. So, you get continuously differentiable function, you are going to get the error to be less than or equal to constant times h raise to 4, the drawback is you may not know the derivative values at the points T i. So, what one will like to have is have a piecewise cubic polynomial, then because it is cubic I can ask for overall continuity to be C 2 continuity. That means the function is continuous, the derivative is continuous and second derivative is continuous. So, still I retain the piecewise structure and then we will also like to have the order of convergence to be h raise to 4. So, that is the cubic spline interpolation point. So, we will be considering that and so far for the we could write down explicit formula for the polynomials in terms of the function values for cubic spline we will have to solve a system of equations. So, piecewise cubic hermite interpolation and cubic spline interpolation is going to be the topic of our next lecture. Thank you.


What's up everybody it's Jason here for jazzy's case breaks comm 2018-19 pinning in a war basketball just sold out It is a first off the line edition one box break random teams number one So let's chase luga donchage guys and Again random teams everybody gets a random team if you bought a spot straight up you are in Mike So much galah spa mojo and if you want spots for mosaic sell a 401 to 402 you're also in So congratulations and good luck to everybody everybody gets a random team in the NBA. Let's go. Let's roll it And we got ourselves a four and a two There you go six times one two three four five Six at the final time Ryan Shackleton down to Mike six times here now one two three four five Six, I know do you remember Christian bill when he came out and What was it called again? It was the fighter right one of those ones with With Mark Wahlberg, I think he like lost like so much weight Mami heat down to Detroit Pistons Ryan you have the heat Derek with the blazers Parker you have the 76ers Kevin you have the jazz part of the rafters robber with the bulls Paul with the Kings Ryan with the Grizzlies Mike with the Bucks Zack with the Celtics Parker with the wars rich You have the clippers Mike with the Pacers Derek with the pelicans Parker with the wizards Jeremy you have the Lakers Logan with the Nets Kevin you have the Knicks Dylan with the Phoenix Suns and Ryan from mosaic sell a pack 402 lands the Mavericks Parker with the Timberwolves Ryan with the Hornets Gary with the Rockets Mike with the magic Matthew the cows Paul the Thunder Jeffrey with the Hawks Derek with the nuggets Parker with the spurs and Mike You have the Detroit Pistons, so again anybody wants to trade It is officially open. That's for jazz Kevin's proposing. I mean our Logan's proposing to Kevin. Sure. I've got a trade anybody else so jazz for the Nets right, and we'll just have that one thing us That's for jazz. I'm both of you guys want this in sell a pack of 402 That's nice. I've got one trade guys, and that's for jazz. And here you go. Just a one box break a first off the line war And if you guys want to see more 1819 guys Even up next would be a pack break to help fill up opulence That is a full case Gold plate or something. All right. First one we got is Andrew Wiggins to 85 his T. Will's gonna Parker We got a Jason Tatum. I've been second year and two out of 25 Zack with the Boston Celtics. We got a Darren Fox 6265 dish night Sacramento Kings It's going to Paul Got a little Aaron holiday new wave jerseys number to 99 Indiana Pacers going to Mike's homage lost by mojo Lonzo ball Number to 65 Lakers What a Jeremy We got a Zire Smith Rookie patch autograph to 99 1799 for the Philadelphia 76ers going to Parker We got a Miles Turner For the Indiana Pacers We got a rainy nights of Glen Rice 299 Charlotte Hornets Going to Ryan We've got a Lulu autograph 29 out of 99 Luka Donchich raining nights There you go Not an RPA, but Still Luka Donchich. There you go That's going to Ryan who got them average from mosaics L pack 402. There you go. Very nice We've got a two out of nine Thunder edition of Paul George Okay, see thunder When a Paul and the last one here is just a little Chandler Hutchinson To 85 little rookie for the Bulls That is going to Robert uncle And there you go guys mission complete not an RPA, but a little Luka Donchich auto 29 out of 99 Which is a rookie from Noir Basketball There you go. Very nice Ryan. Congratulations, man So if you guys want to stick around and watch some more 1819 coming up next is opulence That's a full case break as well. I appreciate you guys. Jasper's case breaks calm


Now that conservatives have a comfortable 63 majority on the Supreme Court, take a guess as to what is one of the first things that's on the chopping block. Of course, it's Roe v. Wade. And the Supreme Court will indeed be hearing a case about abortion. So as Politico reports, the Supreme Court announced on Monday that it will reconsider the right to an abortion it established almost 50 years ago, agreeing to review Mississippi's ban on the procedure after 15 weeks of pregnancy. The court's decision to take a case directly challenging Roe v. Wade, the landmark 1973 decision that legalized abortion nationwide, suggests that the court's new 63 conservative majority is ready to eliminate or, more likely, curtail the right to terminate a pregnancy. In a one-line order, the court said it will review just one question that cuts to the heart of Roe, whether all bans on abortion before a fetus can survive outside the womb are unconstitutional. The Supreme Court will likely hear arguments on the case in the fall, meaning a ruling could come down in summer of 2022, just a few months ahead of midterm elections that will decide party control of the narrowly divided House and Senate. The decision to take the case also sets up a clash between a new presidential administration supportive of abortion rights and red states intent on limiting or banning the procedure outright. Since the Mississippi ban on virtually all abortions after 15 weeks of pregnancy was enacted in 2018, it has been blocked by lower courts that have cited Roe's viability standard. Most medical experts believe viability occurs around 24 weeks of pregnancy. Mississippi was part of a wave of Republican-leaning states that passed bans on abortion early in pregnancy in recent years, knowing they would be rejected by lower courts, but hoping they would provide the Supreme Court with the opportunity to revisit the Roe decision. The addition of Justice Amy Coney Barrett to the Supreme Court last fall expanded the bench's conservative majority, raising expectations that the nation's highest court would curtail abortion access. But what states like Mississippi were doing in passing these laws that they knew were unconstitutional was trying to instigate a challenge to Roe v. Wade because, I mean, the court tipped in their favor. And solely but surely it was tipping further and further in their favor. Back in 2018, you know, Ruth Bader Ginsburg, they probably knew that she wasn't necessarily healthy and that Amy Coney Barrett could soon replace her or some conservative justice could replace her. So they knew exactly what they were doing. And now predictably the court is going to be revisiting Roe v. Wade. So the question is, will the Supreme Court actually further curtail the rights granted to women under Roe? And the answer is most likely. If they don't straight up override the precedent of Roe v. Wade, that will actually be surprising. But at a minimum, I expect them to side with Mississippi here. This is what conservatives have been fighting for for a very long time. It's been a rallying point in elections. So we'll see what happens, but this is a very bad sign of what's to come. And for those of you who claim to be pro-life but are also conspicuously silent when it comes to wars, I just want to let you know that this isn't actually going to curtail the number of abortions taking place in the United States. What will happen is abortions will continue, but they will continue illegally and unsafely for women. And as a result, we may see more unsafe abortions kill women as a result. Now sure, this doesn't mean that abortion will automatically be banned in all 50 states. This just means that states can restrict abortion if it is the case that the Supreme Court does indeed overrule the precedent set by Roe v. Wade. But I mean, let's say, hypothetically speaking, they overrule Roe v. Wade. How long until we see like a dozen, two dozen states possibly just outright ban abortion? Probably immediately. Probably immediately. So this is a very, very bad sign. And if Democrats aren't serious about packing the Supreme Court or introducing a constitutional amendment to codify the right to an abortion into law, which isn't going to pass either, then we may just watch disaster unfold. And that's a really scary thought, but this is not necessarily surprising given the make up of the Supreme Court.


Hi everybody, I'm Jen with opensource.com and I have a brand new top five articles of the week for you today, February 24th at number five six open source tools I use on my Windows machine When Jimmy gets a new work computer, it's always a Windows machine So he makes quick work of installing these six favorite open source applications At number four using Python to find corrupted images in this tutorial Jason van Gompster teaches readers how to root out corruption in their image folders with a Python script at number three ten command line tools for data analysis in Linux step one You may find some data you want to analyze Step two might not be so sure So Jason Baker shows us how the command line can be faster and more powerful than using a spreadsheet At number two internet enable your microcontroller projects for under six dollars with ESP a 266 After many hours spent on home open hardware projects David X shares a primer with readers of the ESP a 266 board And finally at number one an introduction to the Linux boot and startup processes David both writes that understanding the Linux boot startup processes is important to being able to both configure Linux and To resolving startup issues so find out what it takes to get your system initialized and ready to run applications That's it for the top five this week. Thanks for joining me. See you next time


Good morning, everyone. Thank you for getting up so early to see me. I am Neon Rain. This is actually not Julian. This is Joe Grant, the amazing one and only Joe Grant. You see, as usual with DEF CON, there always comes some problems. My biofeedback device is actually being held at customs because heart rate monitors equal terrorism. So I actually do not have the device today to be able to show you biotetrus. And unfortunately, without the heart rate monitor, biotetrus is kind of just tetris. So if you wanted to get up and walk out the door, I entirely understand because I do not have my demo. I can tell you how super cool it is and it's so cool. And if I can get the device, they say it's supposed to be coming at some point before the end of CON. I will try to put up flyers or something and say, please come play biotetrus with me. But I don't have the device, so there will not be a demo. So don't throw stuff at me. Please don't hate me. It's not my fault. But there is a link to the source code, right? We do have the source code. We definitely, we do have that. Unfortunately, we don't have the device to show you, but I do have the source code. And I can explain the device and stuff that we were using. So is it OK if I go on? Are you guys OK with that? There's still really cool information here. Yeah, I wasn't going to speak, but Joe was like, and I'm sure you guys all know Joe Grant. He designed those amazing badges you're wearing, everybody. I'm lucky enough to be able to call him my friend. So when I told him that I wasn't going to speak, because I didn't have the device, he was like, you're speaking. If I have to be there. So you guys get to see a special surprise performance from Joe. Nobody really knows. He's here right now. Except for you guys. So you guys be like, yeah, I totally got to see Joe Grant. Suckers.  Yeah, basically we're going to do a little dance. And hopefully you guys will like that. So basically that's what we're going to do. Hopefully for next year I will be able to have it or something. So I'm so sorry. I'm so sorry I'm Canadian. I'm so, so sorry. Because if we didn't have it, like if I would have been from this country, it probably wouldn't have been half the amount of problems. I'm Canadian. I have to apologize for being Canadian. We love you anyway. Thank you. Thank you. So I guess I will at least tell you about the human brain. So we can totally start with that. So I will tell you about how I got into, everyone always says, how on earth did you get into neuroscience? And it's a very, very good question. Because I don't know if you notice, I don't look like the average scientist. What happened is I actually have an anxiety disorder that is not treated by chemicals. There's no pharmaceuticals that can help me. So you can only imagine how scared to death I am of you guys right now. So what happened is being born and bred a hacker and a do-it-yourselfer, what I said was, well, I don't need you. I will come up with a way to figure out how to fix myself. And that's basically how I started out doing this. And as far as I know, the only person at DEF CON that's silly enough to actually go this way. I have no education because I have an anxiety disorder. I can't go to school. And I have not hold down a job for a long period of time. But I have come up with a lot of super cool things because I have a lot of time on my hands. So hopefully you'll still consider it cool, even though we're missing the most essential part of my talk. So basically what we're going to go through is we're going to go through the Brain 101, the video game myths, the history of game peripherals, which I'm really excited about that Joe is doing it. Because if I was doing it, I'd be totally screwed. But Joe remembers all this as well. Old guard, that's the thing about being old guard, you're old. I'm not that old. Sometimes I feel that way, though. So then we'll quickly go through industry current trends and the proposal, and I don't have the demonstration. So this is your Brain 101, or all the stuff you forgot from Cyclase.  So what I decided to do this year is I'm just going to simplify everything. We're going to throw away all the jargon. We're going to throw away all the diagrams. We're going to forget all those kooky Latin names because people have explained to me they just don't find that fun. So we're going to cover all the questions that a lot of people come up and they ask me about the brain, and there are quite a few. So this will be your FAQ on the human brain, that wild and crazy organ, your own personal three pound universe. So what is the brain?   Neurons pass information to each other using electrical and chemical signals. The amount of cells in your brain can be as sprinkling like you'll find in simple vertebrae or found in the billions as in the human brain. How are memories stored in the human brain? Understanding how memories are stored is an essential step in understanding ourselves. What we call memory is simply the way we store and recall things we've experienced with our bodies.  Our brain holds everything we've learned, but because experiences can be so varied from one person to another, it's our memories that give our life cohesion, and it's what truly makes us unique. How does the human brain think and solve problems? Think of neurons as multi-armed creatures, like starfish, but they don't touch one another and they come in many different shapes and sizes. Now imagine this multi-armed starfish has a little tiny battery in it, and when the tiny battery is activated, an electrical impulse passes through the multi-armed creature until it reaches the gap between it and its nearest neighbor. Chemicals then cause a bridge in order to allow the electrical impulse to cross and trigger the battery in its neighbor. And does this domino effect is that how neurons allow us to think? Now solving problems can happen in two different ways. It can be a type of thought or calculation, pulling information from memory, filtering new information, and then combining it all and weighing up our alternatives. But you can also make decisions with a primitive emotional part of your brain, and that's what we call a gut instinct. We usually use both of these ways in tandem to come up with our best problem solving. Am I so nervous? Why do we dream? I always find this one so interesting. You may be surprised to hear this, but there is no proven reason why we dream, and if dreaming at all serves any importance at all, there's many modern theories about why we dream, but the understanding about anything to do with dreams is still completely incomplete. We know for sure that we do dream, and we know that some animals that have the same sleep cycles as us dream as well. Dreams once were only thought to occur during a stage called REM sleep, I'm sure you've all heard of it, but we now know that they also occur at a lesser extent in non-REM sleep as well. REM sleep dreams are the ones you usually have that are bizarre and like you're being chased by zombies and tutus, and when you try to explain them in the morning, everybody just thinks you're kind of crazy, so that's generally your REM dreams.  But it could be completely possible that there's not a time that we are asleep that you're not having some sort of dream of some sort. What's really intriguing about REM sleep is the fact that the brain chemistry in your brain is completely different than it is when you're in a waking state, but when you're sleeping, the electrical energy is exactly the same. Your chemistry changes because it's what helps you stay in one place and it's what triggers things in your mind, so if you're dreaming about fighting things in your sleep, you don't fight. It's why drugs like Ambien have a problem because they do cause you to sleep, but they don't turn off that little switch that causes you not to move. So what's also interesting in non-REM sleep is where all the snoring and the sleepwalking happens. You won't necessarily get that in REM sleep unless it's unusual or chemically produced. It's easy to fall into the belief that since we dream and since we have REM sleep, it's very, very important, but a lot of people don't realize that there's a lot of people who don't get REM sleep. A lot of major depression drugs take away REM sleep, so these people will not have REM sleep for over a period of years. And they don't suffer any damaging effects. So at present, we know that we don't know the exact purpose of REM sleep or even why we dream.  We have one of the largest ratios of brain weight to body weight of any species on this planet, but it's been a very, very long time in the making. Long ago, when brains first started out, they were pretty basic. There was just a group of specialized cells with the ability to transmit electrical signals to move other cells. And in an eat or be eaten world, mobility was a really good idea. And then what happened was the ability to actually sense the environment, so you were able to sense predators. And then after that, another group of cells developed into specialized structures that give you your emotions, and other cells relayed messages to the spinal column because your limbs were now further away. So a brain was pretty amazing when you consider all these factors together. A stimulus from the environment would cause an emotion that would lead to a physical action, fight, flight, or pursuit. So after a long while, a change started happening that led us towards your modern day brain. The brain began to sprout an area called the neocortex. Please forgive me for using this, but I couldn't make it any simpler. So it is the neocortex. And eventually, as time went by, the neocortex actually tripled in size to give us a brain more complex than any other animal. The neocortex actually engulfed the entire part of the brain stem so that's why you kind of got this bumpy thing that seems to look on the end. It's because the neocortex came out and that's what gives us our huge brains. And the coolest thing that we got from the neocortex was consciousness. And consciousness is what enables us to learn from our experiences. It helps us plan ahead and solve problems and make decisions. We are the only living creatures that use language like words or symbols that represent objects, actions, qualities, feelings or ideas. Other animals do communicate, but they do not communicate in as complex ways as we do. So of course the next question is if the neocortex gives us consciousness, what the heck is consciousness? We think we know who we are because we have the ability to be conscious of our own consciousness. But what the heck does that mean scientifically? Trying to understand the intimate mysteries of conscious experience has become increasingly popular these days. Every field of academia has seemed to be taking a renewed interest in what David Chalmers has called the hard problem of consciousness. Why should a physical system give rise to conscious experience at all? Once upon a time, the study of consciousness was just left to philosophers and religion because science had no reliable techniques that they could use to solve this hard problem. But now in recent years, we've developed startling new ways to study the brain and sciences back in the game with the hopes of finding a more convincing explanation of what exactly it is to be a conscious entity. At this point, scientists are arguing passionately who's over in the right playing field and the rifts between them run really deep. Some people even argue that the problem is too difficult, that no good explanation can ever be given. No one knows if consciousness will ever be explained in present day science. What is the difference between the mind and the brain? The American Heritage-Deadman Medical Dictionary defines the brain as the portion of the central nervous system that is enclosed within the cranium, continuous with the spinal cord, composed of gray and white matter. It is the primary center of the regulation and control of bodily activities, receiving and interpreting sensory impulses, transmitting information to the muscles and body and organs. It is also the seat of consciousness, thought, memory, and emotion. The same dictionary defines the mind as the human consciousness that originates in the brain and is manifested especially in thought, perception, emotion, will, memory, and imagination. So if you remember, I just told you that the biological basis of consciousness is still unanswered by modern day science. You will probably now find yourself completely and totally confused, which means you're on the right track. I'd like to welcome you to the mind-brain problem. Is the mind part of the brain? Is the brain part of the mind? If they're distinct, how do they interact with each other? And who exactly is in charge? Many modern day scientists have proposed that the mind is a product of brain activity, but whether it is or not, we don't actually know. Despite that there are a number of theories and of course the obvious interest in solving this problem, it has been hampered by a lack of experimental models to test such theories. So neuroscience is actually still holding out in hope of a solution. The next one's everybody's favorite question to ask me. Why do drugs and alcohol feel so good and why does addiction happen? The answer to this question starts so, so simply. All animals have a reward pathway in their brain. The reward pathway exists to help secure the survival of the species by making us feel good as positive reinforcements so the behavior is repeated. Pretty simple to start out with. It's what gives us our love for food and sex and companionship. Scientists think that all addictive drugs activate the brain's reward system. At first these act in the same way as natural rewards producing pleasure, but with increased use the drugs needed to stop, I totally have to, there we go. The drugs that are used to stop unpleasant symptoms that appear when it's not available. So it sees the facts that contribute to drug addiction. Addiction is actually different than physical dependence. The Merriam-Webster dictionary defines addiction, I love the dictionary, as compulsive psychological need for the use of habit forming substance characterized by a tolerance of well-defined psychological symptoms upon withdrawal. The same dictionary also uses the broad entry that you've probably heard of that defines addiction as persistent compulsive use of the substance known by the user to be physically, psychologically, and sociologically harmful. Physical dependence on a drug is neither necessary nor sufficient to define addiction. Where it gets tricky is that addiction and physical dependence often occur together. However, addiction can be present without physical dependence and vice versa. To make things even more complicated, not all drugs of abuse create dependency and people can experience withdrawal symptoms from drugs that don't addict them or make them physically or psychologically dependent. Some drugs of abuse produce tolerance and some don't and not everyone that uses a drug of abuse will become addicted. So I'm sure you wonder, how does this all relate to video games? Repetition and learning is how we learn. I'm sure you guys had to write things out over and over again when you were in school to learn how to spell. And video games shine when it makes it fun to repeat a task. Practice may not make perfect, but it's quite possible that it will make permanent. If you repeat a task often enough, it will connect in your brain and it will become easier. Things that you expect that you'll never do, you can do if you just keep repeating it. But repeating things have a tendency to be really boring and video games don't. So combining them both is the best way in order to come up with a learning device that we're not going to like want to run away from. My computer, there we go. And now I'm gonna move on to some video game myths because a lot of people do ask me stuff about video games and scientific research on video games. So we're gonna now move away from the human brain and we're gonna look back on a little bit of history. Rock and roll, comic books, films, and even novels have been blamed for corrupting the young and have been the subject of frequent controversy and censorship. With the majority of North America residents having at least one video game system in their home these days, the video game industry is now the latest phenomenon accused of sending society to hell in a handbasket. So I'm gonna move on to cleaning up some of the misconceptions behind five of the most common myths about video games. Myth number one, video games are mainly marketed towards children. And just like books, movies, and TV shows, video games are now out created for a diverse audience of all ages. And the fact is that some are simply not intended for children. It's true that most American kids do play video games, but there's been a shift in the video game market as first generation of video game players have continued into adulthood and they have now put aside their favorite pastime. Gaming technologies and graphics have become more realistic during the last years and they make video games completely and continually appealing to the youth that grew up on them. Game designers and corporations are now redirecting their marketing strategy to an older and more mature audience with plush paychecks who are nostalgically looking to video games for relief from stress and boredom from day-to-day life. To date, 62% of the console market and 66% of the PC market is aged 18 or older. Myth number two, children have become more violent since the introduction of video games. According to the US Department of Justice, Bureau of Justice Statistics report in 2005. Well, there is no real differences between 2004 and 2005 in the percentages of household experiencing any crime, both violent and property crime declined between 1994 and 2005. US households experiencing one or more crimes dropped from 25% in 1994 to 14% in 2005. The percentages declined each year between 1994 and 2001 for leveling off in 2002 through 2005. There are no year-to-year changes after 2001. I'm sure you're wondering what does this mean in plain English? Well, during that 40-year low in violent crime, Wolfenstein was introduced in 1992. Quake was introduced in 1996, and even the bad boy himself, Grand Theft Auto, made its debut during that time in 1998. The children who would have been playing video games during that period of time are now of legal drinking age, and by statistic standards, have grown into productive members of society. Myth number three, violent video games make children more aggressive. I'm sure you guys have all heard this in the media news.  So there actually is very little evidence from our current body of literature that violent video games that casually or correlationally are associated with increased aggressive behavior. It is true that violent video games appear to increase aggressive thoughts, but that does not mean violent video games will increase aggressive behavior. Wouldn't these same studies logically show that an increase in aggression when playing football in the school playground as well? There are several major gaps when it comes to the research in violent video game literature. One especially large gap is that all research takes place in a temporary period, and there are no long-term studies. I would like to say that again. There's no long-term studies on the ability of violent video games to make people aggressive. Also, other gaps include the sample size tends to be too small and too many of the studies. Some studies, when analyzing aggressive behavior, they've used dependent variables that are not true aggressive behavior, and some studies do not have violent and nonviolent games and are sufficiently different in actual violent content. You have to understand defining violent content is very, very hard. Do we include cartoon violence, or is that not realistic? Do we include non-cartoon violence? And even as it's hard for us to define what we consider violent, scientists don't use the same variables across the board. So if you read that there's this study, you don't know what they're considering violence or what they're considering aggressive. So you really, when you see these things, always, and the news is going crazy about them, always look at the studies because you will just shake your head and be like, well, I could totally, with bubblegum and a paperclip, have done a better study than what they've done here. This is my personal favorite. Girls just don't like video games. Historically, people working inside the video game industry and the video game market itself, they've been predominantly male. Also, many earlier adverts for video games depicted gamers as male. I can personally remember being with my brother as a young child, and they would always depict these boys playing the game and the girls were like, yay, watching. So I totally can remember that. So in the mid-1990s, real efforts were actually made to build games that appeal to girls. I'm sure you guys know of the runaway hit The Sims. The majority of people who played The Sims are actually female. This was the first game that really started bringing women in to play video games. The percentage of women playing games was steadily increased over the past decade. Women gamers are not bound to console gaming. The majority of casual gamers, 60% are women, and women age 40 and older, when they go online, spend a longer time playing games than men or teenagers do. There's now a generation of players who are comfortable with computers, and computer are literally enough to be with online socializing, and they all remember the traditional games of their childhood. So we now have people who are crossing over and getting older and are using video games in a way we've never had before. This one I love as well because I have so many friends that play World of Warcraft. So, so many friends that play World of Warcraft. Playing video games is socially isolating. Not only do video games have socially redeeming qualities, but they can actually help promote social interaction. 60% of frequent game players play with friends, 33% play with siblings, and 25% play with spells and parents. And these are older statistics. I'm sure with now with the Wii coming out, we're gonna see those statistics. Like I've never seen a system before bring so many people together, and families playing video games like never before. So I'm sure that those statistics will end up going up. When it comes to online gaming, the internet has created numerous gaming possibilities that were not possible with single-player games by giving the opportunity to choose to work as a team or to compete against human impotence. It's a place for social interaction and relationships that expands beyond the workplace and the home. And it has the ability to expose gamers to ideas, worldview, and cultures that they may not otherwise get a chance to experience. There is no research to date that actually shows that online video games are more addictive than single-player video games. That doesn't exist. It just seems a lot of people like World of Warcraft, so it has a tendency to cross over. And now we get to move on to the fantastic Joe Grant, and I am so looking forward to him doing the history of peripherals. The interesting question you need to ask yourself is those of you who are old like me will remember a lot of new ideas that have come out that have already previously existed. For 13 years, there was no new peripherals. Everything seems to be recycled, and you have to really wonder what happened. So Joe is gonna bring us to the most interesting history of peripherals, and these are not his slides, so please, please have sympathy as he kind of looks as you do to figure out what he's talking about. So everyone, Joe Grant. Cool, wow. That stuff was really cool. I like that you actually described it in a way that I could understand it, because normally I don't read books. All right. So I guess the history of peripherals, and this is tying together all of the brain stuff that Desiree talked about with trying to create some sort of game that's controlled by the human brain. So we're gonna go over some of the original peripherals that came out for some of the video game systems and kind of see where they started, see where they've been for a while, and then see where the industry's going, and why this type of thing that she's been working on is really pretty cool. In the early, in the 1980s, when the Atari came out, Colecovision and television. I like all the old stuff, so I was actually thrilled to be able to do these slides. Basically, anything that came out was pretty innovative we had. Well, I'll show you in a few slides, but there's different types of joysticks, different types of controllers and interfaces, and because there wasn't really a video game industry before that, anything that came out was pretty cool. So we had all sorts of different games, all sorts of different gameplay. But then what happened is in the mid-90s, and I'll show you the systems, but it was like PlayStation 2, Nintendo 64, all the controllers ended up being the same. There wasn't any new innovation, and it almost seemed like the audience, like the players, never really adopted anything new, or maybe it was just that the publishers didn't want to spend the money to develop new stuff. And then we see with the Wii and the popularity of DDR, there's some areas now that are pretty popular again, like Samba de Amigo is something really cool. I think it was on the PS2, but I mean, where else would you be able to dance around with the Maracas and stuff, and that was fun. But so there's a lot of new stuff that ended up coming out, and that turns out that now there's the resurgence in some of the older stuff, so you see a lot of things that were once available in the early, or late 80s, and early 80s I guess, now coming out again for these newer systems, and it's a whole new generation of people. So everybody, well maybe some of you guys remember the Atari? Cool. All right, so I'm not the only old one here. Oh, someone else remembers the Atari. Oh, the power button never works. All right, it's my wife. She always calls at the perfect time. She's being mobbed in the vendor area and has no idea what to do. So, that's all right. If she calls back, we'll answer it on speakerphone. All right, so 1972, Pong came out, and that was actually prior to the Atari. So this was a single-game system, Pong. You had your two paddles. You could actually, this version was the, I think the arcade version, and then a few years later was the home video game version. So you had two players, and all there were were two things. And a ball, and two players would play against each other, and you'd try to score more points. And there wasn't even any artificial intelligence at that point. You had to play against a friend. So there's part of social interaction right there. And actually, an interesting story about when Atari first released Pong, they beta-tested it at a place called Andy Caps, I think, somewhere in Sunnyvale. And it was in a bar, and they put this thing in there. And a few hours later, one of the engineers at Atari got a phone call. They're like, the machine's broken. You've got to come down and fix it. So he rushed down to try to fix the thing and open the coin box, and the entire coin box had overflowed.  So they empty the coin box and put it back in a few hours later, it overflowed again. Because people started realizing, wow, it's an awesome video game. It was the first arcade game that was actually a huge, huge success to the masses. So that was 1972, and they had the paddle controllers at that point, which was just a potentiometer. And for those of you that are intimately familiar with the Atari, you'll see that is not a paddle controller. That's a driving controller. And that was actually a rotary encoder instead. So there was a lot of really cool stuff going on in the late 70s and early 80s. So 77 is when Atari came out with the VCS, which is also known as the 2600, my favorite system. Even though I guess I was two. But it's one of those systems that everybody seems to recognize. And even if you're a teenager now, you probably have still seen pictures of this thing or played games on some of the newer systems that are based on the old system. So that really set the stage for home consoles. It also brought the first arcade joystick to the home environment. So we see the joystick with the one red button, which everyone identifies as old school. And that was basically just four digital switches left right up down and then another button. No analog stuff yet in there. Here's something that was a foot pedal that I don't think was very successful when it came out. And basically it would let you control stuff with your feet. And I think you had to actually plug a joystick in too. So you could do certain things with the joystick and certain things with your feet. And I think it was intended for flight games and really rudimentary flight simulator, stuff like that. I'd never seen one of these things until about 2002. And there's a whole homebrew community based around writing games for these old systems that I'm heavily involved in. And somebody had actually written a game called Thrust Plus, which is a takeoff of the old Commodore 64 game Thrust. And he ended up finding a huge stash of these foot controllers in some warehouse in Germany or something and designed his game to work with the foot controller. So it's actually the only game that I know of that's really designed to use that. And you can fly around and use the foot pedals to control your gas and brake and whatever. So that was something that was sort of interesting is people were trying a lot of new things back then. And some of them weren't successful and some of them weren't. 1982, the Vectrex came out, which is a really awesome system if you ever get ahold of one. It was a vector-based system. So as opposed to now, we have raster systems and we're drawing across the screen and whatever. This is totally vector-based. So you look and you'll see just discrete lines. It's almost like this thing's drawing in logo. And the images are really sharp. It's a really interesting type of system, totally standalone. These controllers were neat because they actually had little springs inside, so they'd always return to center, which was fun until they broke. And this was also, yeah, they had some other cool stuff. They had something called a light pen, so you could draw on the screen and do stuff with the light pen, which there's some actually pretty cool homebrew hacks out there about how to make your own. They also had the first 3D game, which was basically these goggles that are now really hard to find. Each game came with its own wheel. And inside the goggles, this wheel would spin and somehow synchronized to what was drawn on the screen, so you actually see these vector graphics in 3D. Really, really cool. And that system actually surprisingly didn't do that well compared to the Atari and Collegial Vision, but it's pretty innovative. 83. Everyone knows the toaster. These guys, I actually didn't even know it was 83. I think it was 83 in Japan and maybe 86 in the US or something, but in, I guess, 83 when Nintendo released the Famicom in Japan that became known as the Nintendo Entertainment System in the US, they had this D-pad controller that sort of became synonymous with games from this point on. The D-pad left, right, up, down, you could do diagonals. That was all digital, hence the D for digital. A, B, start, select. Everybody's seen that. And let's see, Joy-Pads use 8-bit CMOS shift register to tell console what film is being pressed without requiring a deck to wire. Yeah, okay. This is actually pretty cool, too, because before this, with the Atari joysticks, if you looked at the pinout, you'd have a connection for each thing. You'd have up, down, left, right button, and then you'd have ground. So when you hit one of the directions, you'd ground that connection just like a physical switch. But this was the first thing that actually had some intelligence inside the controller, so you'd have a serial interface, basically transmitting some stream of data with a less number of wires to the Nintendo, so the system could say, okay, I'm hitting A, B, A, B, left, right, up, down, and let me in. So that was sort of interesting, because now you don't need an individual wire for every single button you have on your controller. If you did, like with the PS2 or whatever, you'd have a huge, gigantic connector. So now the controllers are somewhat intelligent. 84 to 89, Nintendo just was huge. They took over the video game industry that had crashed, at least in the US, in like 83, 84, and just demolished. Nintendo's Apper was one of the first light guns. There was a light gun that came out, that was designed by my friend Ralph Beyer, who's very old now, that also designed the Odyssey, the original Odyssey system, who also arguably designed Pong, and his light gun was actually like a real rifle with a little light sensor on the end, and he was a sniper or like a rifleman in the army a long time ago. But now it's an actual product at this point, the Zapper. They were originally gray, but then they had to turn orange due to some legal issues, I guess, because kids were running around with them, and cops thought they were real. It doesn't look anything like a real gun. Oh, okay, 10 minutes, I better hurry up. Nintendo satellite. Nintendo had this thing, you could plug in and then wirelessly control a bunch of controllers, so you could sit on your bed and play, and don't have to sit right in front of the TV. The Rob, the ROB, was cool, that actually interfaced with the Nintendo, and was also controlled by signal sent from the TV, so it had an infrared sensor, and there was some sort of photo detector. So really, some pretty crazy stuff. The NES advantage was a huge controller. Powerpad, first kind of dance mat. The games pretty much sucked, except for world-class track meet, which was fun. They had even like a dance aerobics game, but that was pretty successful. And then the power glove was just really cyber, everyone knows that one.  Sega, the same type of stuff, they were competing, the Genesis was competing with Nintendo with the NES. They had some pretty uber stuff too, like that huge arm-mounted gun thing on the right. Their new controller had even more buttons, which you could somehow use if you could figure out what all of them meant. Super Nintendo, there's the original D-pad, but they've added a few more buttons, and now they have the shoulder buttons, so they've done some research and realized people like to use their fingers like triggers. And I guess this was the first one that had the diamond configuration of the buttons on the right, those four buttons. Sega Saturn, didn't do too well at all, but this gigantic controller is, I don't even know how much this thing weighed, but it's huge. It's like, I think they skipped the ergonomic user interface test. But anyway, it's gigantic, and I guess this was the first thing that used an analog stick. So in the upper left, you can see it's two potentiometers that determine X and Y movement.  And you'll see this ends up being the standard for a little while. Here's the cool Nintendo 64 one. They had a microphone in their games. Okay, now we're up to 2006, the PS3, the same looking thing that's been done since 1996. Oh my God, is this there? All right, I'm sorry, I have to do this. Hi, you're on speakerphone, and I'm giving a talk. I'm giving a talk, everyone say hi. All right, tell everyone to go see some talks and come back later, and buy some shirts. Oops. That was Romer, the guy that set up the vendor area. Tell him I'm sorry. Yeah, all right, bye. Oh, I'm in trouble now. So basically it's the same type of layout since 1994, 1996, nothing changed for 10 years, and that's sort of boring. Then finally, the Wii comes out. I should say the PS3 does have accelerometers in there. The Wii comes out, this thing, nobody thought it would be successful at all. It's a totally unique side of controller, the nunchuck thing, and accelerometers inside, some infrared detectors, and a huge amount of games being developed for the system, and all of a sudden people are like, wow, this thing's cool. We don't have to just sit on our couch and just move the joystick around. So it's like this whole new generation of marketeers come up and they're like, let's take advantage of these kids and make them exercise. So I think this is your slide. So I'm gonna have to tell you this really quickly because we're almost out of time. In 2007, there will be an amazing new thing being introduced. They will be combining, and I've been waiting for this for five years, but they will be combining video games with actual EEG headsets. As you can see, she has on. There will be, we will have mine to video games. We will never, now if they can just bring toilets to the couch, will be perfect. But so this will change video games forever. You guys next year, nothing will be the same ever again. So be aware that this is coming. I wanna mention one thing with that is Atari actually tried to do a mine control system back in 1983 or something. Right, that I was supposed to do. And basically it didn't work very well and you sort of just had to wiggle your face to make it move. And they were trying to say it was this type of stuff, but it wasn't, and it didn't work. So maybe they'll do it right this time. Right, and Nintendo also came out in Japan with something called Bio Tetris, which is what I was going to show you. I actually thought it was an original idea, but Nintendo had to run me again. So there actually is a thing called Tetris 64. So, and it had a little earpiece. So back in the 80s, they were already doing this. So you really have to wonder what took them so long to like get this all done. So basically I'm gonna have to skip really, really quick.  That's basically what that is. And we'll have to skip all this, because we're almost out. So basically with the right video games, you can do amazing things. We will be able to treat ADD, we'll be able to help people with anxiety disorders. There's already people who have video games instead of biofeedback, it's called neurofeedback, so you totally are able to connect this all together. It will basically help us build ourselves into doing things we never thought before in becoming better humans and having a great time while we're doing it. So I'm gonna just skip through everything. But basically what you're missing, what I was gonna be doing is basically how the game worked is it's hooked up to your heart rate. And I created it because I'm a really anxious, high-strung person. So how it works is when you would play it, if we could be playing it, is that as you get more stressed out and you play Tetris, the more stressed out you get, the faster the pieces would fall. And the less stressed out you get, the slower the pieces would fall. So it was actually pretty awesome. And hopefully you guys will come back next year and I will hopefully have it if America lets me.  You could emotionally control the game as well. If you got more stressed out and more angry, you would also be able to make the pieces go faster and slower. So the whole point would be in order to train yourself. A lot of people are surprised to find out that relaxation is actually something, this is why most people in North America have trouble with meditation. It's actually something that is learned. And if you are unable to recognize it, it can take a long time. Like meditation, it takes people a long time to be able to communicate with their bodies. That's why biofeedback exists in the first place. So you're able to have feedback to go, am I doing this right? Am I doing this right now? So they've actually done a lot of things with biofeedback and meditation. I actually in my quest to find out about how to deal with my anxiety actually stayed with Buddhist monks and they were great. And they totally helped me with that as well to understand the connection between the mind and the brain as you guys learned as well and how to work biofeedback as well. There's more slides that you won't have a chance to read. So basically we did do a biofeedback game. So we didn't get sued, it's actually called Brain Blocks. Brain Blocks, not Tetris, please don't confuse the two. And there is a galvanic one as well. So just like what the Scientologists use, but less creepy. And much more fun. So as you approach an intense emotional state, things would get better. So please, I don't know if you guys can see that, that code. There is, that frozennorth.net is where Julian works. He actually works in the video game industry and does educational software. So I don't know if you guys can read that. It's www.frozenorth.net. Defcon 15, Bio Blocks with an X, B-I-O-B-L-O-X.zip. Eventually I'll also have all the links that I use to write my speech because people always ask me for that and they're a lot. Just be aware they're like five or six pages of links. So they will eventually be up there. My co-speaker actually was unable to come because of a family emergency. So I'm not sure when it'll be up, but it will, I promise, someday. So hopefully you guys will be able to see that too. And we are so, one minute, we are so out of time. So I am Neon Rain and this is the amazing Joe Gran who just saved my butt. So give it up for Joe Gran.


I think a lot of the everyday ways we interact with maps, right, we think of them as this like neutral objective tool. But in a lot of ways, maps have tremendous representational power, right? They've been used to extract resources, to erase communities, to invisibleize histories, right? And I think a lot of our work in the project is really thinking about how to activate the representational and political potential of map making. The Anti-Eviction Mapping Project is a volunteer collective. We have chapters in New York City, Los Angeles, and San Francisco Bay Area. And kind of at the core of what we do, we produce maps and narrative work about gentrification, displacement, and resistance. The project emerged in 2013 during a moment in the San Francisco Bay Area of heightened evictions, and largely in response to what's collectively known as the Tech Boom 2.0. The project has produced dozens of maps that are really targeted at using data to support anti-identification and tenant struggles. And we produce our maps in partnership with grassroots organizers really in an effort to think through ways technology can be used. And leverage in anti-identification fights. We often use the term counter mapping to describe the work that we do. So really thinking through ways the maps that we make can be a tool to curb predatory real estate practices and also to expose the material consequences of gentrification that on a more objective or neutral map wouldn't be made visible. We're really invested in as a collective not producing just maps for map's sake or just releasing data just to have it be kind of floating out in the world, but really thinking through ways we can produce maps that are accompanied by action and support building momentum behind local tenant fights and struggles. My staying power project, which is similarly intended to center the first person narratives of NYC residents and was really informed by a map that we made in 2018 called the worst of Victor's NYC map. And one of the things that that map surfaced is that NYC was one of the top of Victor's in 2018. And although it's a lot more difficult right to put a face to nitro evictions in the way that you can with a private landlord in thinking through a project to implement with my laundry map project residency, I was really interested in thinking through ways I could sort of use those same narrative based tools to think through the current state of NYC and create space for residents to really drive the narratives around their lived experiences in public housing.


So, welcome to lecture L24. In this lecture what we will do is we will see 2 case studies one on inductance calculation for an induction motor and second would be for inductance calculation for gap coefficient reactor. So, in the first case study let us understand how to calculate no load inductance of a 3 phase induction motor. So, what we will assume here is we will assume that rotor speed is synchronous. Effectively what we are assuming is slip is 0 and hence there are no induced currents in the rotor structure and flux in the motor structure is completely due to the stator winding currents. So, in such case wherein we do these approximations then magnetizing inductance can be calculated by using a simple magnetostatic simulation. So, what is the governing partial differential equation? It will be say del square a equal to minus mu j and I have as I have already told you in some other lecture that 1 over mu this mu is taken as 1 over mu on this left hand side to make it more general. So, that changes in material properties can be easily accounted for in the FEM formulation. So, remember j is in z direction because this is a 2 dimensional approximation and hence a also will be in z direction. So, and of course, then that is why it becomes a very simple formulation in which unknowns are only magnetic vector potential values at various nodes in the domain direction of a is already fixed which is in z direction. Now, let us go further. Now, this slide shows mesh of the chosen induction motor geometry. Geometry details are taken from this reference and we are modeled only one quarter of the full geometry. So, now let us revisit calculation of flux from magnetic vector potential which we had seen in basics of electromagnetic. Now, this we will study with reference to a coil in a rotating machine. Now, this is a coil in a rotating machine. So, this is one coil side, this is the other coil side and these two are under two opposite poles. So, now we know psi flux is close counter integral a dot dl and in this case it will be ac1 minus ac2 multiplied by l. Why? Because these two horizontal segments of this contour will not contribute to this line integral because the length vector and a vectors they are orthogonal to each other. So, dot product will be 0. So, this we have seen in earlier. So, now here as I mentioned to you these two coil sides of a coil are under two opposite poles in a two pole machine and then the flux is like this for current direction shown. So, now this let us go further. So, now remember this is the axis of symmetry and this is coil side 1, this is coil side 2. So, this coil side 1 corresponds to this coil side here and this coil side 2 corresponds to this. So, now consider that each coil side in the figure is as discretized into two finite elements as shown here. So, coil side 1 is divided into two finite elements similarly coil side 2. So, the return path for this the current in this element would be this and for current in this finite element the return path would be to this element of coil side 2. Now, let ac1 and ac2 be the average values of magnetic vector potentials in this coil side 1 and coil side 2 respectively. So, now ac1 then can be given as integral acd as integrated over the coil area and they divided by ac because we are calculating the average value. So, this is the average value of magnetic vector potential in coil side 1.  So, now instead of integrating writing it like this we can write summation over two elements and then instead of the integral over the whole coil area it will be integration over element area and e goes from 1 to 2. So, now we can further simplify this as we can split this summation and then we can write summation over individual elements as shown here. So, remember this a bracket 1 and a bracket 2 these are the magnetic vector potentials over element 1 and element 2 respectively whereas ac1 is the overall average magnetic vector potential over this coil side 1. So, further you know expanding it what we have done this a1, a bracket 1 and a bracket 2 they are written in terms of shape functions like this n1 a1 plus n2 a2 plus n3 a3 for element 1. Similarly, the other magnetic vector potential is n1 a1 plus n2 a2 plus n3 a3 for element 2. And now we know that due to symmetry ac1 will be equal to minus ac2 because by symmetry the magnetic vector potential here and the corresponding point mirror point here they will be exactly opposite in science. So, then the flux is counter integral close counter integral a dot dl. So, that will be ac1 minus ac2 into L and putting this relation you will have twice ac1 into L. Now in place of ac1 this now substitute this expression. So, this twice L into ac1 so ac1 is 1 by ac into this whole bracket further simplifying it now what we are doing here we are bringing out magnetic vector potentials out of the integrals because only n i's are the functions of x and y. So, similarly for the element 2 here. So, now and we know you can refer lecture 18 slide number 4 we have seen this formula there integral over the element area n i ds is simply delta by 3 area of the element by 3. So, then substituting that then you simply get this a1 of element 1 into area of element 1 divided by 3 plus this plus this and similarly for the element 2. So, delta 1 and delta 2 are the areas of the triangular elements 1 and 2. So, now same expression which was derived on the last slide is written here and now what we do this term this term we can write as a product of 2 matrices and multiplied by delta by 3. So, a1 a2 a3 rho vector into column vector 1 1 1 into delta by 3 is nothing but this. Similarly, this expression we can write the way it is written here. So, now this whole thing can be written elegantly using 2 summation signs as here and remember what we are calculating this is psi. So, psi is twice L by SC summation over 2 elements and then for individual element summation over for its 3 nodes delta e by 3 a i e. So, a1 e a2 e a3 e you will get when i runs from 1 to 3 and then you will get this entire this expression when you actually expand this expression. So, now this is for you know 1 ton we considered only 1 ton coin if there are n tons and that too if there were you know more than 1 pole pair then that has to be you know taken into account. So, here for simplicity to start with we are taking number of pole pairs as 1. So, then total flux linkages which is nothing but number of turns into flux will be given by n capital N Np psi and Np here is 1. So, that can be replaced by just 1. So, it will not appear in the expression further. So, n and this psi this expression we are just putting it here. Now, this expression can be further written elegantly as done here where now this g e element level matrix g e is given by this twice n l by sc delta e by 3 column vector 1 1 1. So, now this you know size of a e t you know it is 1 by 3 because a e is a column vector a e is a column vector and when it is transpose will be 1 by 3. So, a e is 3 by 1 a transpose will be 1 by 3 and g e will be 3 by 1. So, product of these 2 will give you 1 by 1 matrix. So, now this is over this product is over 1 element and then we are summing it over both the elements. So, now if we actually go to the global level then we can actually from this local element level matrices we can get global matrices corresponding to both elements together and then this size of global matrices will be 1 by 4 will be for a transpose and g will be 4 by 1. Now, why 4 by 1 and 1 by 4 because here there are 2 elements but the total number of nodes are not 6 total number of nodes are 4 only. So, 1 2 3 and 4 because 2 nodes are common. So, the size of a matrix is 1 by 4 a e transpose is 1 by 4 and similarly g matrix would be 4 by 1. So, for these 2 element example we have quickly understood how we can go from element level to global level. Now, we actually you know go to the induction motor example that we saw in the first 2 slides. So, we are talking of jth phase coil. So, there are 3 phases. So, we are talking of jth phase. So, j can be either 1 2 or 3. So, now just reproducing the expression that we have already derived earlier with now NP taken explicitly here because NP now we are not assuming as 1 there can be more than 1 pole pair. So, NP is number of pole pairs and now this capital N which is number of turns is given by for an induction motor NCS by NPL. NCS is total number of conductors in a slot and divided by NPL is the number of parallel parts. So, NCS upon NPL will give you the electrical number of turns and here SC we should remember that it is coil sides total cross sectional area and now this we have already seen here you know from local. Now, what is the difference here as compared to the last slide earlier it was e was going from 1 to 2. Now, here e is going from 1 to N e, N e is the total number of elements in the domain. Now, here you see this we are doing it for jth phase coil.  So, a is at here this a will be N into 1 and this gj so this you know a transpose will be 1 by N and gj will be N by 1. So, again this will give you 1 into 1. So, lambda flux linkages would be just a scalar number 1 into 1. So, now this lambda j is a transpose gj. So, the size of a transpose is 1 by N we just I explained and gj is N by 1 where N is the number of nodes in the whole domain and gj matrix will have non-zero entries only for nodes within the jth phase coil. So, corresponding to the nodes of the jth phase coil gj entries will be non-zero rest entries will be 0. Similarly, for some other coil some other nodes the entries corresponding to some other nodes corresponding to that coil will be non-zero. So, 3 global matrices can be computed for 3 phases as I just explained briefly and then the magnetizing inductance of jth phase coil can be simply calculated by flux linkages upon current. Now, you can see here using the formulation just described we have you know got this flux plot and this the flux count tools are you know as expected we are getting. So, then the inductance calculated would also be then you know as per expectation. Now, we will see the second case study of this lecture which is inductance calculation for a gap core shunt reactor. Now, what is this shunt reactor? This shunt reactor has a central core with air gaps. So, these are the air gaps. So, these are although they are called as air gaps there is some you know non-magnetic and non-metallic material in all these gaps. So, that this whole thing is mechanically stable and these you know gaps basically are adjusted the length of or the height of the gaps is adjusted to get a particular value of inductance. Now, what is the inductance? Inductance is mu 0 n square s by L because we have seen in basics of electromagnetics that inductance is given by n square upon the reluctance and here L upon mu 0 s is the reluctance is it not and reluctance in this whole magnetic circuit is mainly and predominantly calculate contributed by these air gaps which also have explained during the basics that reluctance of air gap is very high as compared to the magnetic material because all this rectangular bridge that you see in cross section that is the magnetic material high permeability magnetic material. So, if the relative permeability of this magnetic material is 1000 in a very simplified calculation one can say that 1 mm of air gap is equal to 1000 mm of this high permeable ion material. So, most of the reluctance is offered to the flux by these air gaps and hence most of the energy also is stored in all these air gaps. Now, remember this each of these packets rectangular packets that you see in this cross section is actually a circular packet right like this. So, this is having actually a circular cross section when you take a cross section and look on this from the top. So, you will get a cross section which is circular when you take a cross section here and look from the top you will get a circular cross section right. So, now the field solution can be calculated by using Poisson's equation in magnetostatics. Now, let us you know see the understand the dimensions. So, first of all this you know core packet the diameter is this is D and diameter is phi 80 mm coil depth is 175. So, this distance is 175, core to coil distance is 135, core to coil this distance is 135, coil to yoke this distance is 150 and coil to end limit. So, this distance also is 150 right. So, now ampere turns for this shunt reactor are number of turns is 2800 and current is 68.73 and area of cross section is pi D square by 4 and if you substitute 0.58 meters here then you will get 0.6242 meter square as area and total length of the air all 8 air gaps together because there are 8 air gaps here. So, 1, 2 and this is 8 air gap. So, total length of all air gaps together is 320 mm. So, if you just put it in this formula mu 0 mu 0 n square S by L and put in all the values then you will get inductance as 8.13. But we will see later that this is far from the actual value because of you know these air gaps flux does not remain entirely within this you know central this you know magnetic path flux actually comes out of this and it fringes between these you know air gaps as well as there is some flux in the coil and that flux outside the core also contributes to the inductance which is not accounted here in this formula. I also again mentioned during the basics of electromagnetics that inductance wherever there is a flux at the corresponding flux tube you can associate some inductance to that. So, this inductance is only controlled is actually this inductance calculated this way is representing inductance of flux only contained within this central this magnetic path and not you know the flux which is outside. So, finite element method will help you to you know account all the flux and get accurate value of inductance as we will see now. So, this is what the flux solution obtained by the flux plot is a flux plot obtained using finite element you know core. Now, here we have modelled only half the geometry right. So, here there is another half identical but we have because there is a symmetry we have taken only half the you know geometry. So, remember we are actually solving this in x y coordinate system. So, what we will be getting energy is per meter depth. So, we have to multiply by pi into mean diameter which is a circumferential length to get the entire energy you know stored in respective you know parts of this geometry. So, now the inductance calculation we will see how to do. So, now we have divided the whole geometry into 5 parts core air gaps in the core, core winding gap winding and rest window portion. So, I will explain you these 5 parts again. So, when I say core it is only this core magnetic part high permeable magnetic part here. Second is air gaps. So, basically it is all these air gaps. Third is core to coil. So, this region here and this region here. Next is the actual coil and then rest of the portion means this portion between coil and end limit. So, these are the 5 parts and remember we have modelled only one half of this whole geometry and then the other half we will take care by multiplying by you know pi into mean diameter. So, that will give us the total circumferential length because we have to remember that this core entire core with gaps as well as this coil they are circular. So, they are symmetrical about this central axis. Now, you can see here maximum energy is you know stored in the air gaps 1.92 into 10 raised to 4. Then it is you know 6.15 into 10 raised to 3 between core to winding because the flux is fringing between the gaps. So, there are also considerable energies there. But in the winding it is further less whereas in the core and in the rest of the winding portion it is very very less. In core obviously it is very less because the permeability is very high and the reluctance is very low. So, the energy stored is very low. For the same flux density the energy density will be given by half mu h square is it not. So, since mu is high for the same b h value will be very small and if h value is small half mu h square will be very small which is the energy density that energy density multiplied by volume will give the energy. So, now total stored energy is addition of all these pi numbers and now remember this is the total energy j because why we have already this the values given by the FEM code where joules per meter in each individual area to that to those you know values we have multiplied by corresponding pi into mean diameter. So, you can easily verify these are the corresponding mean diameters of those you know phi you know sections here, phi parts of the geometry. So, total stored energy is addition of all these phi. So, the i is 68.73 and now if you equate half L as square as this then L you will get as 11.64 and now actually by FEM if we just calculate if the inductance by considering only energy in air gaps which is 1.92 into 10 raise to 4 then we get you know inductance as 8.13 which matches with the one calculated by simple analytical formula. But as I said earlier other 4 energy values are not accounted here that is why we get this value much smaller as compared to the actual value which is 11.64 the same actually you know solution we can get by further reducing the model. So, earlier we made only you know we took the we basically obtained a solution by considering only half the model. Now in this case we will consider only quarter model because then we can because we can exploit a symmetry around this horizontal axis in the middle of the coil. Earlier we exploited symmetry along this axis now we are exploiting symmetry along this horizontal axis. So, number of ampere turns then we will have to define just 50 percent here in this coil block right and the total energy will be the energy obtained from this one fourth model you have to multiplied by 2 to basically get the energy for the both the parts this part as well as the part down right and then the total energy that was obtained was 2.78 into 10s to 4 which is quite close to that obtained considering the half model. And then the you know inductance we can calculate as usual as done in the previous slide and we get it as 11.77 Henry. So, the last point that I want to discuss here is you know why to have so many air gaps in the middle core structure why not have one single air gap that question may arise in minds of those who are actually not knowing the design aspects of shunt reactor typical shunt gap core shunt reactor. What happens is there are two problems associated with this if you have a single air gap of course it is much easier from the point of your design as well as from the point of your manufacturing. But then there are two issues one is now you can see the fringing is very high and the effective you know length of this flux counters also is more right. So, the inductance that you would get by only by having only this one air gap would be quite different than that obtained by using multiple gaps. So, that is one thing that one has to remember. Secondly, because of this heavy fringing flux that flux hits this winding winding area particularly the middle of the winding can you see here there is a heavy flux which is incident on the this winding area that can cause excessive eddy current losses and hot spot and that can even damage this winding within no time. So, that is why we should have multiple air gas which are of smaller heights so that you can avoid all these all such quality problems and you can estimate the inductance also quite accurately you know and get the value within the expected range. Thank you.