 So, we are in this specific talk concentrating on teaching of mathematics using Sylab. We have tried a number of softwares for teaching of mathematics and we found that this software Sylab is most suitable for teaching at the undergraduate level especially and to some extent at the postgraduate level. The difficulties for other softwares are certainly one has to purchase these softwares and if a curriculum is based on a specific software one has to purchase that many copies by all the institutions who are using that software and that becomes extremely difficult for the colleges because of the cost involved. So, moreover mathematics is considered as a difficult subject by the students and to make the concepts clear it is very useful if you use a specific software such as this Sylab. Here I would like to mention one point is packages presentation is made up of PDF screen with Latik. PDF screen is a package contribution from Indian person Siviradakrishnan from Trivendrum. We all use here free softwares and part of that exploration of that we got Sylab also is very useful. I would like to tell why we choose Sylab for mathematics. So, as Sir said it is free software. So, students also can try these things at home. If we teach them some proprietary software they are not able to legally use that software at home. That way we are asking students to do some illegal work. So, Sylab we found most useful in that way. Also Windows and Linux it is like a platform independent. So, there is no issues for which platform we are using. Then Sylab understands mathematics is polynomial as a data type. So, that is also something additional. So, we do not need to declare these things and then ask students to start. So, they are these things are available. Also Sylab has good inbuilt functions in mathematical point of view like rank and inverse all these things. We can directly start with these things because we asked students already what is the concept of this and then if these functions are readily available then we can do some further things with the help of Sylab. As was mentioned we can use operations like rank and inverse of a matrix to solve equations especially the students solve equations three equations in three unknowns. But if there are more equations and more unknowns then it is a difficult job. Even I find that students are afraid of looking at a determinant of a matrix which is of size four or five. So, these things become very easy if you use a software like this and therefore this software is very useful. Also if we have a singular matrix and then if you have a matrix equation like a x equal to b where a and b a b that its rank is the same as the rank of a and the rank is less than the number of variables then the number of solutions is infinite and the students are not very compatible with this situation. On the contrary we can do this very easily by using Sylab for example one example we have given here that a is a matrix three by three matrix b is a column vector then the augmented matrix is a b its ranks are same rank a is two rank of a b is also two then the system is consistent and we use row reduced echelon form this command r r e f a b and then we get the nice form of the augmented matrix a b and then we are able to give the solution of this system as x equal to 0.25 t plus 0.0625 and so on. So, t is a parameter and x y z are given in terms of this parameter. So, this thing if you show a number of examples of these this thing to the students then that they do not think it is very difficult. So, that is a very important use of a software like this. So, polynomials and their roots the these things one can verify easily with the help of Sylab and then student will also enjoy the result and the meaning of the theorem. So, here we have demonstrated one theorem if we have coefficients of the polynomial real coefficients polynomial odd degree then what is the result. So, if we take polynomial in with complex coefficients and that way it is showing you real part and imaginary part separately and if then we ask for root then we get three complex roots and usually we have in mind whenever complex root occurs they come in conjugate pair. So, this is something different what we are experimenting here and this is because we have complex conjugates are not real conjugates and in visual practice we always take generally real coefficients. So, we never try to see this what happen if this is like that. So, here you can see if we have real coefficients the vector v is having real coefficients and the polynomial form by that of degree 5 and here we have complex coefficients complex roots in occurring conjugate pair and one real root. So, existence of one real root is we can check like this. So, as was mentioned if you have an equation of odd degree at least one real root is there this can be checked very easily by this software and if the equation is complex then complex conjugates occurring conjugate pairs that can also be seen and the idea gets fixed in the mind of the students if you work out a number of examples of this type. People are afraid of even finding out a square root of i i is a square root of minus 1 what is the square root of i. So, for them or such things you can actually show that this is the square root of i and so on. So, that teaching of it is become somewhat easy if you use a software like this then things like Newton-Raphson method actually we know that Newton-Raphson method is a very fast method and has quadratic convergence. We can compare this method with other methods very easily by means of a software like this and even Newton-Raphson method how fast it is. So, if the error at a specific stage is epsilon next step error is epsilon square. So, you can see that if at a specific stage the answer is correct to say 2 places of decimal the next step the answer is correct to let us say 4 places of decimal approximately and you can the students will actually see this that the consecutive answers are correct are same up to 4 places and so on. So, that is why this is very useful for teaching of mathematics. Here another thing is when you take the initial approximation in Newton-Raphson methods like this you get a solution which is near to that initial approximation. So, for example, cos x has infinitely many zeros like pi by 2, 3 pi by 2 and so on. So, if you take initial value as let us say 10 then you will get a 0 of cos near that. So, that can also be checked by means of this software which is otherwise very difficult. We cannot just do computations and actually show this to the students by hand. So, here 10 is the initial value and we see that the iteration gives you 11.54 and so on. Next iteration is 10.93, next thing. So, g of answer we use previous answer as the value command answer a, s. So, then 10.99. So, you see that correct to 1 place. Immediately you see 10.995574 and next again is 10.995574. So, you see that quickly we can demonstrate this thing and the students see that it is really a fast procedure. Symbolic computation actually is not supported by Salab in general, but you can do certain computations like this. So, you can have a matrix in which the entries are rational functions in one variable x and we can treat this matrix find this determinant and so on. All these things are possible. So, this is one illustration. Determinant of this matrix is given here. Inverse of a symbolic matrix. So, it is too wide to fit in one screen. Therefore, each column we have separated in another slide. Just want to show you that it is possible for big matrix. Here we are taking 4 by 4 matrix of polynomial in x. So, it is also possible to compute that much large things with the help of Salab. So, things like eigenvalues and eigenvectors. One can do this. One can actually numerically solve this equation determinant of a minus lambda i equal to 0. Find the equation, find its roots and get the eigenvalues. This we can do for 4 by 4 matrices and higher order matrices also. And there is also spec, which gives you the eigenvalues directly. So, one can compare these things. That is one thing. One can find out eigenvectors also by solving the corresponding linear equations. Of course, the calculation of eigenvectors cannot be treated because of the difficulty of computation. But these things can be done by means of a software. So, linear algebra concepts can be demonstrated rather easily by using a free software like Salab. Now, exponential of a matrix. I was happy to know that this exponential of a matrix is incorporated in this. Element-wise exponential is also there. And exponential of a matrix. This is a concept, which is not discussed at the under the level, but at the post-legal level it is discussed because of its applications and differential equations. So, exponential of a matrix is i plus a plus square of 1, 2 factorial plus and so on, the usual series. And then if a is a matrix of size, let us say for 4 by 4, it will be difficult to calculate this infinite series. But one can have this function in Salab and one can try these commands. Exponential of a and logarithm of a and so on. The command for that is expm. So, you see here that this matrix a is 12, 3, 7, 5, its exponential is 5, found out this is actually element-wise exponential. And then log of that is also element-wise. And then you go to this thing expm of a. This is the matrix exponential. So, e raise to the matrix. So, here that is infinite series is calculated. And then its logarithm is the original matrix. So, something is like exponential of a plus b is equal to exponential of a into exponential of b. It is not in general true, but it is true when a and b commute. So, results of this type can be verified easily by using Salab. So, plotting 2D and 3D graphs is in general very hectic. And it is difficult to visualize graphs of the function for two-dimensional as well as three-dimensional. So, with the help of Salab with very simple command it is possible. And here we can define inline function like this and we can just use plot command to obtain the result. And students and as well as teachers research students found it very interesting to have such good graphics with simple commands. Also we can have three-dimensional graphs with the help of Salab very for that we should be aware of vectors and for range we have to aware of vector and dot product dot point element wise product it is operations it does. So, likewise we can very easily demonstrate surfaces and it found that in vector graphics also differential geometry to obtain vector fields and all these things all these things are possible with Salab. Also I would like to mention one point here Salab draws these graphs and functions correctly or automatically correctly. So, many times it so happen that teachers they want to write some book or research papers we have to students we have to include some good graphics or correct accurate graphics and this Salab with the help of Salab we can include these graphics very in one or two steps. So, let take we can include this Salab graphics. Okay, so the students find this drawing graphs very difficult and this Salab has very high probability of drawing graphs for example, using x and y and r and theta also polar form is also possible and the students can draw the graphs and then they can see using the Salab. Some other applications I have given here for banking like if a person deposits a sum of rupees 10,000 at the beginning of every year for 10 years and then the r is the rate of interest and then if the final sum he gets is at the end of 10 years what is the rate of interest. So, usually from small r we find out capital R, capital R is 1 plus r upon 100 and then we get an equation in terms of capital R and this equation is to be solved to get the value of small r. This equation is like this and that is why because solving of such an equation is involved a question of this type is not asked in any business mathematics book and now we can treat a question of this type that find the rate of interest provided the final amount is given to you. So, this is how this is useful even at the level of business mathematics. For teaching of mathematics limit of a sequence one can teach very effectively using this Salab for example, he can actually find out the values of 1 plus 1 upon n raise to n for various values of n verify that this sequence is an increasing sequence whereas if I take 1 minus 1 upon n raise to n that is decreasing and so on. So, though these things can be verified and the students get convinced and remember these things if you actually show these things using a using Salab. Solutions of differential equations that can be done if you take a differential equation let us say ordinary differential equation d y by d x equal to some function even that thing the students can see the solution curve by using Salab by using ODE command and that becomes very useful for teaching of differential equations. There was a discussion about F50 also pass Fourier transform which has a number of applications in signal processing and other areas and this can also be computed using Salab. So, one example we have given that this is 1 2 3 etcetera up to 12 this is a vector if you apply FFT you get a vector of length 12 transform vector and if you use I FFT of this answer then you get the original vector actually you don't get it in the expected form because of the competitions involved. So, if you clear the answer clean the answer then you get the original vector 1 2 3 power up to 12 this is a thing. So, these are various observations useful for teaching of mathematics. One or two things I want to mention here that a number of softwares are useful for research in mathematics and a number of problems like pore color problem or wearings problem for force powers. For example, wearings problem for force powers was a joint Indo-French collaborative effort and the long-standing wearings problem was solved by professor Bharat Sogramandam and Desheva and Resh. So, the theoretical part was done by both the mathematicians whereas the competition part was done by phase mathematicians using certain softwares. So, that's important. However, unfortunately Salab does not support the exact competitions at present. So, I hope that in due course of time when all the basic aims and objectives are done the Salab team can also look at at least elementary things about exact competitions because these are also useful for teaching of mathematics. To give an example things like 100 factor very simple things exact competition of 100 factor can be done factorization of polynomials with integer coefficients which is of interest at the high school level also can be done provided we add these exact competitions and this is like cryptography. RSA cryptosystem is a very simple application of what students learn at the first year basic level elementary number theory. This can be demonstrated very easily by other softwares like cache but this can also be incorporated and then the software will be still more effective for teaching of mathematics. Thank you.