So far, we have discussed about NP completeness by showing satisfiability problem is NP complete. Through a chain of NP complete problems, we have conducted reduction and then shown that satisfiability problem is NP complete. Of course, NP satisfiability problem was the first to show that it is NP complete by Stefan Koch and this is the Koch's theorem that we have established in our lectures. Of course, so far the examples that we have presented to show that these problems are NP complete, they are essentially related to touring missions and some tiling problem and then we have finally, reduced this problem and establish satisfiability problem. So, you know how to establish a problem is NP complete by now through these lectures. So, in this lecture I will present some more NP complete problems which are of some importance in this in practice and we will establish them there NP complete. These problems are among the sort of very first that are established to be NP complete. As I had mentioned in sometime around 1970-71, Stefan Koch has established that satisfiability problem is NP complete and then within one year time Koch established about 21 problems that mostly they are graph theoretic problems they are NP complete. You know graph models are essentially the models for very practical real world problems. So, they are essentially sort of transfer to graph theoretic questions and established those are NP complete. So, in this lecture I cover few more NP complete problems which are of which are mainly graph theoretic problems and having some importance with some real world problems. So, in this connection first I will prove that 3 set is NP complete. Of course, I had already mentioned what is k set is able to problem k set that is given a Boolean formula in its c n f in which each disjunction I mean the sum has at most k literals each sum has at most k literals. Now, given such an instance of the problem we have to cross check whether or not the formula is satisfiable that is what is k set problem. Now, I have already mentioned that if you take k equal to 2 this 2 set is can have a polynomial time algorithm. Now, if k greater than or equal to 3 of course, I will establish that 3 set is NP complete then you can understand for k greater than or equal to 3 these are hard problems. So, to show 3 set is NP complete you know by definition we have to establish the 3 set is in NP and 3 set is NP hard that means 3 set is as hard as any other NP complete problem that is what we have to look into. Observing that 3 set is in NP it is similar to what we have observed that satisfiability problem is in NP. So, you can mimic in the similar lines and write that 3 set is also in NP. Now, to establish that 3 set is NP hard what do we do since we have already established that satisfiability is NP complete we will give a reduction polynomial time reduction to 3 set from a set. That means what we have to do given a Boolean formula f in c n f we construct a Boolean formula f prime an instance of 3 set such that f is satisfiable if and only if f prime is satisfiable this was this what we have to do the meaning of this particular statement is essentially this is what we have to do. Now, you take what do we do if you look at this satisfiability problem instance satisfiability problem instance you will have you know a formula is essentially something like you know some certain sums will be there. So, say c 1 and c 2 and so on c k for example, where each c is a disjunction sum I means it is of the form you know some literals say l 1 or l 2 or l 3 at maximum you will have 3 literals because in case of 3 set now in case of satisfiability problem here you can have any number of literals that is there is no restriction for satisfiability problem.  So, you may have something like this each clause will be of the form. Now, what do we concentrate if we can write each clause any an equivalent formula an equivalent formula which fits in 3 set then we can do it for every c i and thus this f prime whatever that we are constructing that an equivalent formula which is having in each clause say for example, c 1 is now somehow divided into say c 1 1 and c 1 2 and so on say some c 1 k 1 c 1 k 1. So, this is corresponding to c 1 and similarly for c 2 and for c 3. So, what are we going to do each some c 1 now when we are writing this conjunction of disjunction of literals each c i j here will maintain that it will have at most 3 literals by maintaining this we will construct a formula f prime an equivalent formula equivalent formula we mean whenever this is satisfiable this is satisfiable if and only if this is satisfiable likewise we will construct the truth assignment you can have if it satisfies this and this satisfies here and vice versa that is what is essentially we are targeting to. So, that means it is sufficient to concentrate on one disjunction one sum if I do you can do it for the other sums and establish the things. Now, each disjunction in f with more than k literals with more than 3 literals say for example, if you take one sum disjunction which is having say for example, x 1 or x 2 and so on or x k if you have like that this will be replaced by the following sum what do we do we will take this formula c prime corresponding to that. So, corresponding to this c 1 whatever that we are writing here if it is having more than 3 literals then only I will do this if say for example, c 2 has only one literals. So, no problem if it has 2 literals no problem because it is anyway fitting to the instance of 3 set. So, if it is having more than 3 literals then only we do this process whatever I am now mentioning. So, that is what is essentially c 1 prime for example. So, what is the formula here I am writing just carefully see this. So, this disjunction c will be replaced by this c prime in the formula f what is this if we have this literals say x 1 or x 2 and so on x k do not just think that these are variables this I could have written may be some l 1 or l 2 likewise some l 1 or l 2 likewise because this can be you know complement of the Boolean variable whatever that under consideration. So, consider a sum x 1 or x 2 and so on x k what we are doing this x 1 or x 2 or y 1 now what is y 1 we are going to choose now some k minus 3 new Boolean variables and construct a sum in the following. So, x 1 or x 2 or y 1 and y 1 bar because these are Boolean variables this is a negation of this y 1 or x 3 whatever is the third component here in this in c what you have that you put it here or y 2. So, what I am doing here y 1 bar or y 3 or y 2 I will explain you. So, and then this is y 2 bar or y 4 y 3 then if you look at here I have 4 then I have 2 here. So, if when I go to this x k minus 2 then I will have the here this if you continue this way y k minus 4 bar and here y k minus 3. So, these are the new variables that we have chosen and this y k minus 3 bar I mean complement of that negation or x k minus 1 or x k. So, this is what is the formula constructed. Now, let me explain you the theme like if you look at any of this sum here say if you look at y 1 bar or x 3 or y 2 y 1 complement or x 3 or y 2. Now, we look at this this is equivalent to you know this is same thing as y 1 implies x 3 or y 2. So, what is the meaning how do we look at the truth assignments for this. If y 1 is true then x 3 is true or y 2 should be true you know. So, that is what is the formula equivalence here. So, the theme here is if y 1 is true then one of these should be true right. So, with that we have constructed this formula and now we can see that this c is satisfiable if and only if c prime is satisfiable that is what we will look at now. Now, the point is one side you can see that we are going to observe that c is satisfiable if and only if c prime is satisfiable. That means, if you have a truth assignment which satisfies c I will construct a truth assignment that satisfies c prime and vice versa. That is what we have to do now assume you have a truth assignment t that satisfies c. That means, you look at this sum a truth assignment satisfying this sum we mean in at least one of this x i's will be satisfied by the truth assignment right. So, that is t x i equal to 1 for some i. Now, what I will ask you to do here let j be the smallest index such that t x j is equal to 1. So, that means, here in this x 1 or x 2 and so on x k what we have looked into here. So, at x j we have truth value 1 and before that all are receiving truth value 0 that is how we have. Now, what do we do we will extend this truth assignment t to all these variables this all this Boolean variables x 1 x 2 x x k and the new variables y 1 y 2 and so on y k minus 3. And we give truth assignment this way because this is an extension. So, that therefore, what are the truth assignment t that gives for all x i's that will be the same. And now we have to tell for other variables this y 1 y 2 y k minus 3. So, what are we going to assign here is this t prime at y l if you take any variable here in among y i's for all l with less than equal to j minus 2 we give truth value 1 otherwise we give 0. So, now you can cross check that this t prime satisfies that c prime. How it is look at the formula because what I have to say that this t prime satisfies each of this sums this sum this sum and so on all these things. Now, we know that till x j all the way all these literals x i's they are receiving truth value 0. But, now from the definition of t prime you see till l less than equal to j minus 2 we have we have assigned truth value 1 for this y l. So, that means in the if x j is somewhere here in between now y 1 has received truth value 1. Therefore, this sum will be satisfied by t prime similarly, y 2 is receiving truth value 1. So, this will also be satisfied and so on. Now, wherever we have say for example, x j here. So, that has been satisfied that has been satisfied by the truth assignment t and therefore, t prime. Now, there after now you look at there after now you look at what is here. There after we are assigning this y i's truth value 0 for this j minus 2. So, whatever here if I have truth value 1 that is the least here of course, till this point we have assigned truth value 1 for all these y i's. So, you can concentrate on these things these variables and see these sums are satisfied. Now, there after for all the variables y we are assigning truth value 0 for y i's. Now, you concentrate on the first one now you concentrate on the first one. Now, this candidate we have assigned for y k minus y k minus 4 for example, we have truth value assigned truth value 0. So, it is complement will receive truth value 1 similarly, here. So, the truth assignment t prime. We have assigned this way for all y l truth value 1 if this l is less than equal to j minus 2 and elsewhere 0 I mean there after that means, when you are looking at the first one you have y 1 here this will assign truth value 1 and when I have y 2 here this will assign truth value 1 and so on. When I have got this x j which is having truth value 1. So, this will be satisfied by that t prime in which and there after what are the classes what are the sums that we have this clauses you just concentrate on the first one. Because we have assigned truth value 0 for all other cases here I have the candidate that you if you look at this y i prime. So, there after this first component now you concentrate on and this y i since we are giving truth assignment 0 with respect to t prime. So, this will receive truth value 1 and therefore, all this sums in this formula will be receiving truth value 1 with respect to this t prime the truth assignment and hence the c prime is satisfiable. Now, converse assume you have a truth assignment that satisfies this formula let me write this formula once again. So, the formula is this x 1 or x 2 y 1 y 1 bar x 3 y 2 y 2 bar x 4 y 3 and so on. And of course, at the end we have this y k minus 3 prime y k minus 3 complement or x k minus 1 or x k this is what we have. Now, if you have a truth assignment that satisfies this that means every one of this is getting satisfied by the truth assignment. Let me probably write that truth assignment say t 1 what is the meaning of this this t 1 satisfies this this this and all this sums will be satisfied by this. Now, what I have to say this c is satisfied by some truth assignment that is what we have to look into. So, here it is very easy now whatever this t 2 let me call which satisfies that c how do we look at now look at this t 1 which is this is satisfying this. For example, if this t 1 is giving truth value to y 1 1 only, but not for x 1 and x 2 if any one of this x 1 x 2 is getting truth value 1 by this t 1 we are through because the same t 1 will work to satisfy c because in c we have x 1 or x 2 and so on x k if x 1 or x 2 if any one of them is getting truth value 1 with respect to this t 1 then we are through if not this y 1 should get truth value 1. Now, when y 1 is getting truth value 1 then y 1 bar will receive truth value 0, but if you look at this sum if when y 1 is true this 1 is true or this 1 is true. Now, since y 1 is getting 1 this will receive truth value 0 when it is receiving truth value 0 since t 1 satisfies this sum also one of this y x 3 or y 2 one of them should get truth value 1. If x 3 is getting truth value 1 then we are through if we are not getting truth value 1 for this that means if we are getting 0 here this should get truth value 1 then only this will be satisfied by t 1 correct. Now, let us continue this way suppose if none of this x i's are receiving truth value 1 till this point that means till x k minus 2. Now, let us look at that means this y k minus 3 must be receiving truth value 1 through this t 1 correct. Therefore, this can date should receive truth value 0 because y k minus 3 is receiving truth value 1 and therefore, its complement must be 0 and now what are the variables left over here laterals this x k minus 1 x k. So, one of them should receive truth value 1 then only this clause will be satisfied is a very simple argument. Therefore, what is our t 2 this same t 1 will work for us. So, if you have a truth assignment t 1 which satisfies the formula c prime that is the that is this formula every one of its sum should be satisfied by the t 1. Now, the same t 1 will serve our purpose because if any one of this x i's is satisfied before you know in any one of this clauses then we are through if not what is going to happen if not the first one then second one in x 3 we have x 3 if it is satisfied then fine. If x 3 is not satisfied by that t 1 I mean is not receiving truth value 1 then it has to receive 0 which forces us to get truth value 1 for this y 2. So, you can argue this and continue till you know x k minus 2 if you are not getting truth value 1 with respect to the assignment t 1 then what will happen as x k minus 1 or x k should receive truth value 1 with respect to this truth assignment t 1 and hence what are the truth assignment which satisfies c prime will also satisfy c. Hence I conclude that c is satisfiable if and only if c prime is satisfiable. Now, as I had explained in the beginning you take any formula now each of this sum will replace it by you know if it is having more than 3 literals will replace it by this kind of constructive sum and construct this f prime which is an instance of 3 set and we can observe that f prime is satisfiable if and only if f is satisfiable. Thus the conclusion is we have reduced this 3 set to sorry satisfiable to problem to 3 set. Now, the question is whether this reduction is possible in polynomial time because you look at this less than equal to p what it indicates less than equal to is a reduction and p indicates that should work in polynomial time. Now, you tell me whether this works in polynomial time what is the construction that we are doing what are the input that you have if you have this you know each sum you consider because total input if you look at the formula the entire formula you have c 1 and c 2 and so on c k these are the disjunctions right these are the sums that we have. Now, with respect to the length of the literals which are appearing in c 1 now you look at if I have this x 1 or x 2 and so on x k is some k say number given to you. Now, what we are constructing here we have included with respect to k few more new variables which are not appearing anywhere in f. So, this you have to consider and construct this fixed formula because this formula is a fixed one with respect to that k and hence this construction with respect to the input parameter k we do not have you know anything bigger than you know just a multiple of k. And therefore, you can easily perform this task within polynomial time the constructing that formula in polynomial time and hence this reduction is a polynomial time reduction. Therefore, 3 set is NP hard to conclude that 3 set is NP complete. Now, the what I will do I will present few more graph theoretic problems as I had mentioned which are NP complete further purpose I might use this 3 set is a ability also because just we have established that 3 set is NP complete. I can use 3 set I mean I can use 3 set to reduce to the problems that we are targeting to other than satisfiability or any other problems which we have established so far. So, let me present this problem coloring problem in graphs. Let G equal to V e a pair V a graph I hope you are aware of this definition because V set of vertices this first component of this pair E a set of edges I am writing graph this is undirected graph that means you know I am not going to if you are considering an example I am not going to put any directions of this this undirected I mean something like this. Here each edge V is vertices let me give label say V 1, V 2, V 3, V 4 for example. Now, E you know edges we have put here between V 1 and V 2, V 2, V 3, V 1, V 3, V 2, V 4. So, since this is an undirected graph I may I may write this V 1, V 2 edge by unordered pair V 1, V 2, V 1, V 2. So, this is an element of E alright similarly V 2, V 3 unordered pair it is an element of E V 1, V 3 it is an element of E. So, here essentially E the cardinality of E in this example I have 1, 2, 3, 4 edges. So, unordered pairs here and vertex set of course it has 4 vertices. So, V cardinality is also 4 here anyway. So, let me that is what is I mean graph here. Let us consider a graph G now in a graph a k coloring of G this is a mapping is a function f from the vertex set to you know some k element set I might choose 0, 1 and so on k minus 1. Essentially coloring I mean here what do we do for each vertex we associate a number that means we label we give color may be you can take this set of colors may be red, blue, green that way. So, just to say that these colors are now named as 1, 2, 3 and so on k minus 1 some k minus 1 colors. So, some k colors 0, 1, 2 and so on k minus 1. So, essentially coloring a graph we mean you just give some color to each of its vertex. For example, this I give red for example, this I give green this may be some blue some yellow whatever and you can give may be yellow to this as well whatever it is merely just a function from vertex set to a set of colors. Here we are choosing colors 0 to k minus 1 for some k colors instead of naming red, blue, green whatever now we say a k coloring is a proper coloring we said is a proper coloring when whenever you know you see taken edge the colors which are given to you know this vertices the adjacent vertices should be different such a coloring we call it as a proper coloring. So, a k coloring of g is said to be proper any take any 2 vertices whenever there is an edge which we require the colors given to the vertices should be different whatever the in this example I had written this is anyway proper coloring. For example, if I choose here also yellow now you see this is not a proper coloring. Now, a graph is said to be a graph is k colorable it is k colorable if it has a proper k coloring. So, using some k colors you will be able to give a proper coloring to that graph we say graph is k colorable. Now, regarding this problem so what is the problem here you will be given a graph and given some k color colors and you will be asked whether this is k colorable. So, I give you a graph of 10 vertices for example, and I give you some 3 colors and ask you whether there is possibility of having some coloring using this 3 colors and the coloring should be a proper coloring. You can always have a coloring because there is simply assignment of colors you just keep on, but what we are looking at what we are looking at you can have a proper coloring we mean no 2 adjacent vertices should have the same color. So, that is what is the problem. Now, of course the problem we look at is a decision problem here because this I had already mentioned when I am talking about satisfiability. We are not worried whether you give me what is called a solution for this. That means, if you have in this context if you have such a k coloring what is that coloring I am not worried about we are worried about the counter decision problem. We mean if you are given a graph and some k colors whether it is k colorable or not you have to say s or no this is the problem that we are looking into for NP complete problems the decision problems. In this connection this 3 colorability is the problem given a graph G whether or not G is 3 colorable. And we will establish that this 3 colorability problem is NP complete. Once again to establish this is NP complete we have to observe that this is NP problem as well as it is NP hard to observe that this is NP problem what do we require you are given a coloring you have to only cross check whether it is proper coloring or not. So, that is very easy you just take each edge and cross check the vertices end of that edge whether they are having the same color or not. If they are having same color somewhere for some edge you simply say that this is not a proper coloring otherwise you report yes this is a proper coloring. So, what is the time you require you have to just traverse through all the edges and see what is the what is the color given to the end of those edges. So, that is also it is very quick that you can give an algorithm that works in a polynomial time with respect to the input here the input is graph you have vertices edges. So, you look at you have to simply just with respect to the input edges that you it will work in a polynomial time. So, it is not a problem for you to check whether it is in NP. Now, what do we do we reduce this 3 set problem which we have established it is NP complete to this 3 colorability in polynomial time of course. So, what I have to do I given an instance f of 3 set I will construct an instance of 3 colorability that means I graph. So, given a Boolean formula in 3 set that means each of which has utmost 3 literals what do we do we will construct a graph and such that whenever this formula is satisfiable we have the corresponding graph is 3 colorable and vice versa this is what we have to establish. Now, what I will do instead of considering I mean you take an arbitrary instance of 3 set we can always now say in 3 set I will always consider the formula in which each sum has exactly 3 literals because you know if I have only 1 literals I can always extend say for example, I have only x 1 what I will do I will just adjoin few more few more literals it is easy for you because this are you can put with 0. For example, this is equivalent to x 1 or say another variable you bring into picture x 2 and x 2 bar this is 0. Now, if you expand this x 1 or x 2 and x 1 or x 2 bar x 2 and x 1 or x 2 bar. So, when I have a clause with only 1 literal I can now write I can make it equivalent formula in which I can bring 2 literals each and similarly if you want one more literal to bring into the picture what you do here x 1 or x 2 and now for example, are x 3 and x 3 complement this is same thing with this and this can it also. So, what do I get here similarly this x 1 or x 2 or x 3 and let me put a bracket here x 1 or x 2 or x 3 complement you see you can easily extend a given formula in which you can have you know exactly 3 literals. So, what I will now assume without loss of generality that means, what are the formula that I am considering an instance of 3 set I assume that each of its sum has exactly 3 literals.  How do we do that let f be this formula say u 1 1 u 1 2 and so on u 1 3 you know these are the 3 literals as I had just mentioned in this sum I assume that I have that k sums in the formula f where each u i here what I mentioned these are literals or the variables say x 1 x 2 x n these are the variables that I have x 1 x 2 x n.  Now, what I will do is when we have like this for variables x 1 x 2 x n what I am asking you corresponding to each of these variables you consider 2 nodes like this x 1 x 1 prime x 2 x 2 bar complement and so on x n x n bar likewise you consider and take a new node let me call it as a and make a triangle here for all these literals make it like this and then corresponding to each of this sum I will now tell you what do we have to do you consider a graph of following right you look at.  So, in general for each of this sum you construct a graph like this. Now, what is the important feature of this graph if you observe let me for example, give you know 0 color to all these 3 nodes what is going to happen since I am giving 0 to this this can receive 1 or 2 among the 3 colors. Since I am giving 0 here 0 here to give a proper coloring of this here for example, if I am giving 0 I have I can give 1 here since I am giving 1 here and this is 0 here I should get 2 only here. Since I am having this 1 and 2 here I should give 0 here there is no alternative because these 2 are given 0s. Since I am getting 0 here if I am giving 1 here I can give 2 if I am giving 2 here I can give here 1. So, that means let me write 2 or 1 1 of them only. So, in this triangle now it will receive 0. So, if this all these 3 nodes if we are giving 0 color you see it will force us to give 0 color to vertex v i there is no alternative. So, that is what I am making a point here if all u i j's have 0 color then any proper 3 coloring any proper 3 coloring will give you know v i color 0. And moreover if any 1 of them is relax to you know have 0 that means if 1 of them is non 0 you can always have some proper coloring 3 coloring. So, that this v i can also receive a non 0 color among the 3 0 1 2. So, that kind of graph g i is this. So, what I would suggest you this g i how it is this g i it has say something like 3 input thing. So, and say v i is here. So, this is what is u i 1 u i 2 u i 3 let me look at j graph like there are several vertex what is this inside this I can only I will only look at these 3 vertices and this. So, what I will do I take for each of this some a copy of this and put it here g 1 and so on I have k some. So, g k whatever this what are these vertices this is either x i or some x i bar whatever it is you connect it to that if it is say for example, x 2 the first one you connect it to x 2. And say for example, this x i 2 sorry u i 2 is x 1 bar you connect it to that and say for example, something else you connect it to like. So, similarly this g k whatever this literals that you have that you connect it to that whatever the connections you have here I have v 1 say in between some v i's and this is v k this is what we have. So, what is the in this construction what we have done first we have made this triangles and what are the graph that we have constructed g i you put it here and look at this literals wherever those literals are appearing in this list you connect to them. And then I am not worried about this g i I have already mentioned this graph is here in between. Now, then this v i's we will now connect to a node say some b. So, you make it like that and moreover make this a adjacent to b as a construction clear once again all these variables you have x 1 x 2 x n corresponding to which you consider 2 n vertices with the labels x 1 x 1 bar x 2 x 2 bar x n x n bar. So, that you know I can quickly see that this is having straight forward correlation with these variables. Now, if you consider any sum you see corresponding to which I want to consider the graph g i in which this let me call them as input nodes and this is an output node sort of these are the you know what are the sum that we are considering you have exactly 3 literals there these literals wherever they are appearing in this list they have to appear here you connect to them. And this v i you connect to a vertex b and make this vertices a and b are also adjacent I hope this construction is clear to you that is what is exactly I wanted to look at. Now, we will argue that the formula f is satisfiable if and only if this graph is 3 colorable how do we argue that now you cross check carefully suppose you have a truth assignment which satisfies that formula that means what is the meaning of that this in this sum one of them will receive truth value 1 one of them will receive truth value 1. That means for this graph among these 3 connections you see at least one of them will have truth value 1 that means a non 0 color. So, I will give a color here using 0 1 2 these 3 colors and whatever the truth assignment that you are giving to this you first give them straight away this x 1 x 1 bar if x 1 is receiving truth value 1 you give it to that then automatically x 1 bar will receive truth value 0. So, you make that color similarly for x 2 what are the truth value it is having you just give it to that and its complement will receive the opposite truth value that means either 0 or 1 will be you know. So, if it is 0 x 2 bar will be 1 and so on. Now, you cross check since this is formula is satisfiable each of this literal at least one of this literal in each sum you will have truth value 1 that means among these 3 connections you will have truth value I mean the color 1 here. Once you have color 1 I can always make it a proper coloring such that this v i will receive non 0 color v i will receive non 0 color when it is receiving non 0 color now what I will do I will give this I can always give 0 color to this because these are all this v i will receive non 0 colors. So, I can put 0 here and then here since x i and x i complement they are having 0 1 as colors I will give color 2 here. Now, you see this graph I can have a proper 3 coloring all right once again the truth values whatever that you are having to x i x i's you just give them as color and automatically their complements will receive the opposite truth value that means the corresponding color. So, that since each clause at least each sum since it is satisfied by this truth assignment at least one of the literals will receive truth value 1 once we have having that once we are having that this each v i can receive truth value non 0 when I am having this truth value I mean not truth value you can always give a proper 3 coloring such that each v i can receive a non 0 color. So, since we are having a non 0 values here I can now take them as colors and I can easily give coloring 0 to b since anyway these are 2 are having 0 and 1 I will give color 2 to a. So, that you see this graph can be given as a proper 3 coloring all right and then suppose you have a proper 3 coloring of this suppose you have a proper 3 coloring of this converse part I wanted to give a truth assignment which satisfies the formula f how do we do that assume you have some proper 3 coloring f to this graph. Now, what do I suggest you in this in this graph what are the proper coloring that you are considering you make a small permutation so that this f of b is receiving 0 and f of a is receiving 2 among the colors 0 1 2 why I mean how it is possible you just give a permutation what are the colors you have this is a proper coloring you just relabel them if I am receiving say for example, a equal to 0 just you call wherever 0 is there you call them as 2 just a permutation between 0 1 2 you give that will be that will still be a proper coloring fine. So, what I am asking you you just will relabel it and give the coloring color 0 to b and color 2 to a first you do that since it is a proper 3 coloring what is going to happen since a is receiving truth value sorry color 2 this x 1 and x 1 bar should receive among the truth value colors 0 and 1 because it is a triangle in this triangle to have a proper coloring here if it is receiving truth value 2 either this should be 0 and 1 or this will be 1 or 0. So, in all these triangles we will have 0 1 2 colors in all these triangles we will have 0 1 2 colors now you see let us look at b b is given color 0 and therefore, none of this v is can get color 0 none of this can get 0. Now, in this proper coloring you can argue that since this is getting a proper coloring which is having truth value non 0 and in this proper coloring each g i here in this proper coloring each g i any one of these connections at least one of these connections you will have color 1 and thus each of this sum here in f can have truth value one of these letters can have truth value 1 what are the color that we are assigning to.  So, likewise we can construct a truth assignment which satisfies f and thus we can conclude that this 3 satisfiability is can be reduced to 3 colorability. Now, polynomial time again you can argue very quickly because you look at the literals that we have the size of this and this graph construction with respect to each of this can work in constant time with respect to the input parameter a multiple of the input parameter and this was in a polynomial time and hence 3 satisfiability can be reduced to 3 colorability. So, that 3 colorability is also an NP complete problem.


So we begin with a paper by Shugan Chandjain on the peculiarities of the yoga system taught in the Gyanarana of Shubhachandra. Jai Jinnendra, I use this hesitation primarily because of the topic that I'm going to discuss. You know, I am going to present this paper more as a student who has studied the text with particular reference for meditation. So I have listed here the sequence of my presentation here to you for the paper. So first of all, I'll try to talk about Shubhachandra himself because in Jain literature, the name and the chronology, the time, et cetera, become highly debatable and it causes who is first, who is follower, et cetera, et cetera. So when we talk of Shubhachandra, you know, I took three sources to see how to fix his time. The first I said is the text itself because as a matter of practice, all Acharyas, they salute their predecessors and the Jinnas. So in this text, I find that he has saluted or he has paid respects to Samantabhadra, Pujaipad, Akalank and Jinsen. I named them in chronology because Jinsen belongs to ninth century. So this brings me to the conclusion that he is after ninth century. Then I look at the historical evidence. See, history of Shubhachandra says that his father and grandfather, they were rulers and somehow when the father was sick, his uncle, you know, tried to usurp the kingdom. So he and his twin brother, Bharatari, they accepted renunciation and left for practice.  So these three things indicate, if I go historically, that Bhoj ruled Malwa in the early part of 11th century. And the last I took is the chronology of the Digambar Jain Acharyas in Jain Siddhan Kosh, which I think is a highly researched document for Jain texts. So there also, you know, all these things bring me to the conclusion that Shubhachandra's time is the early part of 11th century. This is important as I will tell you later on. Then the name of the text itself, Gyanar. Gyanar means ocean of knowledge. Most of you scholars know that in Jainism, we spend a lot of time talking about knowledge. Why? Because it's a, you know, attribute of soul only. So when you talk of Gyan, you are talking of actually the soul. And Gyanar means ocean of knowledge. So it gives you complete knowledge from a mundane soul and its process up to the liberated soul, the practices, the vows, the philosophy, et cetera. Some authors call it yogana also. This means that it emphasizes in details about yoga. And I think he uses almost 20 chapters in the book of 42 talk on the ascetic vows, how ascetic should live. So he is one, I think, who has tried to put the doctrine and the practice together in one text. And when I read this, I found that it's really a holistic presentation in which he talks of meditation. And why I say meditation? Because he says the entire process of the path of purification is through meditation. He gives prerequisites to meditation also. But ultimately, if you are thinking of liberated soul or getting liberation, then meditation is the tool. In Tattvarsut also he says, nijras chay, that is, you have to dissociate the karmas. So that's what it is. And why I say holistic? Because he talks of the four components of meditation here. The meditation itself, the dhyata, that is the one who meditates, dhyay, the object on which you meditate or you use for meditation. And the last is a result of meditation because we do everything to a certain result in mind. So he talks of all these four components at greater length with Jain perspective. The language, he uses Sanskrit language, highly poetic except one chapter where he talks of dhyata, of object of meditation, dhyay. Where he tries to refute the others. So he uses the textual type of form. And he uses a lot of adjectives to make it interesting for us. I just give you an example that in verse 16 of chapter four, he says, a human being continues his existence in the transmigration cycle by his auspicious and inauspicious activities. And the example he gives, like a silk comb veers himself through the web in the web of silk thread emitted from his own mouth continuously. So he just tries to explain to you the meaning of trans through examples. Then he has, when you read the text, he takes an iconic view of explaining things. And basically I will say he uses two aspects, the nishchana, the transcendental viewpoint and the practical viewpoint. For transcendental viewpoint, he starts with the necessity to be a monk, a yogi he calls. And from practical point of view, he talks of a shravak. That is who is Samyak Drashti and who is living a mundane life but is interested in moving. So he talks from both the aspects, but the emphasis is more or mostly on the monk, the yogi he calls it. Now I come to literature prior to his text. Because sometimes there is a lot of discussion in academia. He has accepted that or something like this. So what I felt is that there are two types, two out of the four categories in Digambal text. Digambal texts are classified in four sections. Pratvanyoga, Kannan Yoga, Chandan Yoga and Dravyan Yoga, metaphysical, ethical, mathematical and the history. So this refers to the last two, the first two, the metaphysical and the ethical. So we try to see which are the popular texts. And the Acharyas, he has mentioned who have written in Sanskrit. He does not mention Kuntkul, most probably. I feel he may not have known Prakrit or those texts may not be available to him. So we reviewed all these things and we see some references or some indebtedness, indebtedness primarily to Samant Bhadra, Puja Pad and Jinsen. So Puja Pad I particularly say because the commentary he writes on Tatvarsu Sarvasiddhi, he has taken a lot of concepts from there. And Samadhi Tantra, his own Samadhi Tantra also, he has taken some references. And Jinsen, of course, in Tantra, when he used Tantric elements, he refers to Jinsen. Then there are some other Acharyas. He mentioned only the Ambar Acharyas. He does not mention Shwetamba Acharyas, which I think was a custom in general literature. So with this, you know, we come to the, hello, yes. Now I come to his own, I have a list of references to show, but I will not use this time because of the, you know, limitation of the time here. Then I come to meditation. And here, you know, I find a very peculiar thing when he says meditation generally implies just training the wandering nature of mind, body, and speech. It's not just mind, because, you know, in Sanskrit we say chit vrati nirodh. So he talks of mind, body, and speech, all the three activities. And if we see the general literature, we talk of Manmachankaya, all the three together, you know, to be together. And then he said to control this so that you can meditate on an entity, something. And then he tries to say, he tries to define or bifurcate the meditation into two categories. The first is he says that when you have a series of steps or independent, discrete steps, like astanyu, yam niyam, you know, asan, pranayam, dharnadhyan, and samadhi, you can do and achieve what you want. But he talks of continuous, that is, you have to practice all of them together. And I will tell you later on that why, how he talks about this thing. So he talks of these meditation, where you have to keep them all together. And there he classifies the meditation into two subtypes. One he calls as unworthy meditation. That is meditation, which is not worth pursuing. And here he puts two types of meditation, artha and draudhra, because these cause pain and continues your cycle, that transmigrate cycle. And the other he calls worthy, which is prashasth, the word he uses is prashasth. And there he talks of dharnadhyan, or the auspicious meditation, or the meritorious meditation, and the pure, the shukl meditation. And in this book, he talks exclusively or exhaustively on dharnadhyan. Why he does it, we'll see later on. And shukl dharnadhyan, the pure meditation, he talks briefly in one out of the 42 chapters. So, and why he says it is important, because they result in either merit or helps you attain the ultimate objective.  And the word he uses is severe dharn. And I was really surprised when I read this word, severe dharn. Translation, some translators call it heroic meditation. But it means that you use the energy of the soul to purify itself, rather than have somebody else help you to purify. And I think this is one of the uniqueness of dharnadhyan meditation, the word he coined, severe dharn, or the heroic meditation, using the energy of the soul itself to purify.  And one of the couplets that comes to my mind, which I repeat every day, is I can't decide it in Hindi. It says, bhava atap metavan ko, bhava atap metavan ki nizme hi chhamta samta hai, anjane me ne parime ki jhuti mamta hai. This means to remove the pains of this universe. I have the potential, I have the capability. And out of ignorance, I come to you or to somebody, please help me remove my pain. So these are the some of the things, which are getting translated into the present days. There are some puja which are demanding, there are some pujas like this, which are self-motivating. Then we come to the next point, which he talks about, aim of meditation. What is the, why do we do meditation? So he talks of different, from different views. The first point is, he says how many types of purushats, the wellers are there. And the purushats have four types, earth, a dharma, earth, kam and moksha. So he says the first three, they keep you revolving in the transmigrate cycle. And the last one, it takes you out. So from this purushat point of view, moksha is the aim of meditation. Then he says, what does my soul do? How does it manifest? How does my consciousness manifest? So in general literature, you will see manifestation of consciousness is divided into three categories. In auspicious, auspicious shub, and the last should of pure meditation. When the soul meditates on itself, it forgets everything. So from that he says that pure meditation or shuddhopya is the aim of objective. Until you achieve liberation, shuddhopya is the aim of meditation. And how does he define moksha? You know, usual elimination of pain forever, attainment of the four infinites, things like this. So from all these aspects he says, moksha is the ultimate objective of meditation. And the state in which you are to attain that objective is also the objective of meditation, shuddhopya. And then, you know, what do other scholars say? Like Kunkund, he says. Kunkund, Pujepad and Kartike, they say, moksha is the ultimate objective. And which was tradition before his time.  And there he says, shuddhopya and parmatma, supreme soul, siddhatma, and liberation. These are the highest goods. So there also he says that shuddhopya is the highest good. Jina, of course, because Tirthankara is the highest objective, siddha is later on. So he tries to, you know, identify these four. And accordingly, it comes to us is that the aim of objective should be or is, according to Shubh Chandra's, attainment of moksha or state, you know, manifestation of consciousness in itself, not in anything else. That's the true meditation. Then he talks of practitioner. And there, you know, because he is referring to pure meditation, he says the pure meditator and then what he uses is yogi. He does not use a word sadhu or money. He uses the word yogi. And he says the yogi is one who is in the sixth and seventh constan. Now, sixth and seventh constan, they are for the monk, okay? So he says you have the yogi, if you want to attain moksha or be in the state of Shuddhopya, he has to be in the sixth and seventh. So that is his criteria of a dhyata. And this he shows by devoting more than 20 chapters out of 40 on the conduct of the monks. And before that, he writes a chapter on detachment, how to develop detachment. And there, our young friends will learn, the scholars will also know, he talks exclusively on the 12 reflections, contemplation, Barabhanaya, Anuprakshas.  So with this, then we talk of the practice or the process and I will try to like to share some slides which I made for this presentation only. So there are some prerequisites for you to meditate. And the first is that you must become a yogi. Second is he says, you develop equanimity and restraint of the mental activities. You should practice these things. And then he says, practice austerities. You know, in Jainism austerities, he talks of two types, external and internal. Fasting is a very important activity. And the second are internal, where meditation is the fifth one. Before that is a self study. So he says, you should practice all these selfs, including self study, so that you can graduate to a meditation properly. After that, he talks about now, you know, here you see, he is trying to use some of the elements from Patanjali's, you know, ashtang yoga to make it practical. Because earlier than that, our texts used to say, meditate on your soul. How? So he tries to answer how? So he says the first and the most important thing is that the ambience where you meditate should be free from disturbances. In Jain code of conduct for monks, we talk of 22 andurances, preishas, you know, that is the physical elements which you must conquer.  So he says the ambience should be nice. That means it should not cause physical or mental disturbance. And then he says the place, a better place would be a place where somebody has already attained liberation.  So like this, he talks all the temple, but the place should be such, which is calm, quiet, and free from people or animal disturbances. So that you can focus, you can meditate on something. Then he talks of postures. Postures, not 101, just two in Jainism, as you would see in idols, Padmasan, or Kayot Sargal, or standing like this. And for us, he says some people find Padmasan difficult, so they can use half Padmasan, so that they feel their body does not become a hindrance in their meditation. But basically, he talks of two postures, Padmasan, and Kayot Sargal. And then he talks of breathing. So he talks of three types of breathing. One is purak, inhale deeply. Second is kumbhak, you know, keep it in, don't let it go. And the third he says, with a jerk, throw it out. So these are the type of things. And then he talks of last, the layers, or the mandals, the five layers, which we will see how he uses. But he talks that if you are a yogi, that is, you are in the sixth or Samangonstan, then fourth to sixth, these are not required. You can practice in any place, at any posture. These are, he says, I'm talking for the beginners, for us, from the practical point of view, because you need these things to start meditating. Then, you know, my yoga guru, Chappal, here, I read his paper, and he talks of these, all these, in Buddhist meditation and Upanishads. Hello, how do I go to the next slide? Yeah. Then he talks with this. Now he comes to the actual practice. And he first talks of dharmdhan, or the meritorious meditation. Okay, I'll just rest through. So he talks of these. The first three are like self-study. And the last, which is the most important, which I want to show an example, is samsthan, which is meditating on a form. Okay, and the process is, you know, padas, rupas, padas, pindas, padas, rupas, and rupati. So I will take the example of pindas, only the form. And this is the process. You see how he uses the five types, five layers.  So like this, you know, he always uses lotus. He's creating the cosmos and showing that this is a high place, so you put your soul there. Then he says, now you come to your naval point. Okay, there is a lotus there, on which alphabets are there. And on top of the alphabets, there are ram or ream emitting flames. And then on the heart, there is a lotus of six leaves, or eight leaves, each with a karma species. And these flames are burning these karmas. So when they're burning, the smoke is rising.  Then the water drops come, washes them away, and nothing is left except the big mountain on which your soul is residing, which is similar to an omniscient soul. So he takes through the step and time process. And in this, he includes colors, forms, and the mandals also. Shukul Gyan, he talks very briefly. The four types, time is running also, I will not talk about these.  Hemchandra, Rokshastra, is a very prominent, you know, text. And when you read this and compare it to Shukundra, who was 100 years before, 150 years before, you see a lot of similarities. In fact, it appears that Shukundra's text is in front when he's writing this thing. Then Tatvanu Shashan Ram Senj, which is like a summary, is also appears to be this. Then my own experience. You know, 12 years ago when I got into this, I joined a five days serious meditation program by Acharya. And now I recollect he was using this completely. And I was a totally transformed person after that, and then jumped into these studies. But now, you know, the object of meditation has changed. Okay, he's moved for physical wellness and others, and they have diluted. So like this, now it is, a lot of Acharyas are using these things. Now I conclude my session by saying, you know, Shukundra has the authority to talk on Dharam Dhyan only. He does not talk on meditation because he is capable of that. He might have practiced it. And he says there are Acharyas, or they were Acharyas in the past and they are Acharyas now also. Then he says Moksha. Now in this time era, Moksha is not attainable. You know, our scriptures say nobody can attain Moksha. So he talks of Dhamdhyan only. He does not talk too much on Moksha. He says being conscious in state of, you know, auspicious or pure consciousness and progress so that you are ready when the time is opportune so that you can move to the Moksha first. So like this now he talks of so many things. Now I can go on but I will stop here. I think it's a complete text and a lot more study is needed to understand what is really pure meditation. Thank you very much. Thank you all for your attention and I'll be perhaps addressing some material that has already been covered and it's not by mistake. I want to first of all thank Shugan for establishing the International School for Jain Studies and both Ellen and myself, Ellen Goff, have benefited from the wisdom of Professor Soghani and I wanted to just give the narrative about how this paper arose. I was traveling some time ago, I guess about three years ago with ISSJS and I was with Professor Soghani and about 20 years ago I started an inquiry into the Panchamahabhutas and I've been translating and gathering and conversing with people about this and I sat down with Professor Soghani and I said what about the Panchamahabhutas and he produced his book and we sat for several hours and he introduced me to Shubhachandra and Gyanarneva which was of course one of the many texts that is found within the Gyanpath of Purification but I had the opportunity to study it before them and what I'm proposing to do today is to give the big picture about an answer to the question about how this actually fits in with what we call Tantra and I'll be drawing from some earlier work from the Hari-Bhadras as the frame for making that perhaps sotirological leap as we heard about a little bit earlier. So first of all, I want to talk about puja and how in the early texts we don't really hear a lot about puja but in the Yoga Bindu which was compiled written by the early Hari-Bhadra of Hira-Hanka, sixth century figure there's a long description of doing puja and I would love to hear from other early descriptions of Jain Puja but in that particular text called the Yoga Bindu, we see this and then in the next Hari-Bhadra and also I want to point out that this was a shift from the usage of the term yoga which was never a good word in Tavarta Sutra it's all about ah yoga or viyoga that yoga is the negative process of the Jiva being shrouded by karmic residue and that with this Hari-Bhadra of Hira-Hanka that all of a sudden it's a catch-all term for spirituality and I think in that century a Gupta type period there was this a Boolean embrace of Patanjali in various forms or at least the word yoga as a catch-all for spiritual practice. Then the next text which was the first text that I really delved into and translated in totality the Yoga Durshti Samuchaya the next Hari-Bhadra, Hari-Bhadra-Yakini-Putra in the eighth century does really three moves that I'll share with you today. One, he takes the idea of yoga specifically of an eight-fold yoga and cloaks Jainism within a terminology that is accepting of a yoga spirituality but steeply critical of Tantra and he actually says that those who think that you can get rid of a burden by engaging the burden as happens in Tantric ritual such people are deluded and they're only taking that burden from one shoulder and putting it on the other shoulder. Yet he actually says that the Tantricas really do want to be liberated but they're just misguided and then he goes into a full embrace of goddess traditions which were another characteristic that he found quite acceptable and he retitles and reorganizes and re-describes the spiritual path taking the Gunasthanas, reordering them in an eight-fold process and giving each of those eight steps a goddess name. Okay, so it's a really remarkable subversion multiple levels in a very, very interesting text. Then we go a couple centuries hence Shubhachandra 10th century, perhaps 11th we don't really know but I appreciate the dating that was attempted and things have changed yet again and that with Shubhachandra and then his successor Haimachandra again switching over from Digambara into Svetambara we find a full embrace of this technology that is associated with Tantra. So if we think of the earlier embrace of yoga we see that in Patanjali of course embrace the vratas and retitled in the miyamas and in yoga we get the strong ethics we heard about them being presented in the early chapters of the Gyanarnava and the other part of yoga is the concentration skills. Savitarka, Nirvicharka, Savicharva, Nirvichara Savijasamadhi, Nirvijasamadhi. With Tantra we get another layer it goes beyond ethics and concentration, meditation into all of the accoutrements of geometry with Yantras of chanting with mantras with sustained focus on the elements and the whole development of Mandala. And of course these elements are perhaps incipient with Patanjali because he does briefly mention Japa. He does use the word Chakra. The details are not fully built out as they are in the later Tantric text with which we are quite familiar. But what I want to distinguish today is sort of a difference between the usages of Tantra and start out with a term that's been used by various scholars including Joel Brereton that goes all the way back. Actually this is an Aponishadic term but it translates nicely in the Tantra system which is a term that emphasizes correlation. And as we know from the Purushasukta, the small body is the big body, the microcosm is the macrocosm and it is through this process of higher and higher levels of identification that there is an ascent to the state of cosmic identity. And I've been reflecting recently about the word Atman being about the small within the breath. Expanding a birth from the verb root for Brahman is to get really large and that this whole process within the Aponishads is about breathing, meditating in order to make that connection with the cosmos. And thereby we're returned to some of these very specific details. And in the Gnarnava, we are given in chapter 29, the listing that's actually the more traditional, almost the traditional listing, but invert it slightly. And then in chapter 37 the list changes again. So let me just review that if we go back to the Markandeya Purana, we find an early accounting sort of pre-Tantric or by that times, perhaps Tantric of Pritthavi Dharana, Jal Dharana, Agni Dharana, Vayu Dharana, Akash Dharana. So earth, water, fire, air, space in that order. And in the Hindu literature, that order remains consistent in also as was mentioned on the earlier slide in the Vishuddhi Maga. The Kasuna progressive meditation also follows, Buddhagosha follows the same earth, water, fire, air, and then he does colors and then he does space. But what happens first in chapter 29 of the Gnarnava, Shubha Chandra changes this so that he starts with earth and gives the syllable, the mantra for earth, lumb. He colors the earth with gold and he says that it is symbolized by a lightning bolt and with the geometry of the square. Then water, lumb, white, and a crescent moon. So you can visualize a crescent moon above a square. Then he does, not fire, but he does wind and he describes it with a syllable yum and puts a sphere in the middle of the crescent. So you can visualize the square, the crescent, the sphere. And then for him, the crowning in chapter 29 is that on top of that sphere is to be visualized to flame. And the flame brings us full circle back to the illusion, to the ahyoga of the Tavarta Sutra that the idea is that the importance of tapas, the importance of the flame is that it's through the flame that all karmas can be burnt. And the mantra for that is rum and the color is yellow. And again, we find the triangle, the shape of the flame above the sphere. Now, going further into the text, toward the end of the text, and there's a sense of real culmination. And there was with John Court's article that was invoked earlier by Alexis, this notion that somehow there's not a soteriological leap in Jainism, but these two texts, the Yogashastra, which draws from the Gyanarava, they certainly put forward moments of soteriological leap. And what I would like to do is to switch, as I've already hinted, from this notion of correlationism that is so characteristic of Hindu Tantra and even Buddhist Tantra, and say that there is what we in Jain tradition might wanna characterize as conflagrationism, okay? That there's this central importance of the burning power of austerity. And I would love to read all of it to you, but I'll just say, I'll summarize it, and then I'll read just a couple of shlokas that when I came back from India, I got our Sanskrit after three years ago, that encounter with Dr. Soghani, and I said, give me this text, give me this text. And it was just published that year in a printed form with a rough English translation. And then I found another form with commentary. So we were working from not a perfect text, but the original nonetheless. And our group spent about a year working on these two chapters that I've mentioned. And it starts with the conceptualization of karma, okay? Which is the stuff of earth. Okay, remember in Jainism, karma is substance. And it's characterized as having downward pedals and upward pedals. So the first element to be invoked is Pritavi, the stuff, which is stuff, karmic stuff. And then the second is fire. And the fire is invoked to burn the downward pedals and the upward pedals.  It's almost like a rain dance fire. And then the monsoon pours down. And as the monsoon pours down, it takes all that leftover ashen crud.  And what you're left with is the image of the pure radiant jinnah as none other than yourself. The yogi whose practice grows to uninterrupted steadiness and pitta-star meditation soon reaches unparalleled, auspicious bliss and is called the noble one. He meditates on the glistening whiteness of the new nectar on the full moon, experiencing a moment of blessed omniscience, seated on the golden mountain peak, free from all sensory outflows. He meditates on the self and the universal form, on the multitudes of teachers in the three realms as well as on the inconceivable Lord. The appendi-star meditation makes one similar to the jinnah who has crossed the great water to the other shore. Now this pronouncement, I would suggest, maps to the Tantric accomplishment of the transformation of the sadhak into the siddha, of the bodhisattva into the Buddha, and from a jiva bound by klistakarma to the jivanmukta. Notably, and perhaps at variance with a text such as the Pratibhijna Hridayam, okay, and I'll return to that, the singularity of the yogi and the autonomous nature of the individual does not disappear. And I'm speaking this to Rahul, who is part of a seminar in Germany on the status of the individual. Okay, so the individual doesn't become incinerated, but rather what's revealed is the unique and particular consciousness of that particular jiva. Yogashastra, chapter seven, verse 25, says exactly the same thing. It's virtually a copy, a plagiarism of what Shubha Chandra has said. And in 1212, it says just as rock gold turns into purified gold by coming in contact with quicksilver, and this is Uli's translation. The self obtains the state of supreme self through meditation on that supreme self. Now, my closing, yep, is that in murti puja, that is, puja in honor of an image, all of the images of the Tirtankara are identical, but as we enter a Jain temple, we see that there are 24 of them, and as we first encounter this, and then as we lead others, as we bring our students into the Jain temple, and we explain to them that the Jain temple and we explain, well, they all look the same because they're all radiant and they're all pure and they're all free, and there are 24 different ones. And I direct them to look at the throne, and the throne upon which each of the Tirtankara's sits in Padmasana, Mahavira has the lion, Parshvanath has the snakes, Adinath has the bull, et cetera, et cetera. Hence, drawing from and Shugan made reference to Shivkumar, and this past summer, our group had the good fortune of doing a meditation retreat day with Shivkumar Acharya who has been teaching exactly this meditation for many, many years, and the way that he sort of crafts it is that he directs people to pronounce first the mantra Soham, and he describes quoting verbatim this text that you visualize yourself as the purified Jinnah as really in the image as a human being having human form in the image of the liberated soul. And there's Pranayam, again as directed in the text. And then he has people switch to a Koham. And in this meditation Koham, what am I? And I'm not this, I'm not this, I'm not this. But what the meditator is left with is that it's just Puratman. It's just purely without any attributes. And this brings us back to the original sense of yoga, viyoga, ayoga, from the Tattvara to sutra. That there has to be a release systematic of any object of attachment, bringing about the expulsion of all karmas from the soul. I'm gonna end with a conversation that took place in an earlier Sanskrit translation group. And we had been reading the Pratibhaj Nya Krodhyam with a lifelong disciple of Swami Lakshmanjee. And as I think most of you know from that text, which is really considered to be the exquisite, condensed expression of Shiva Tantra, it culminates with Shivoham. And it culminates with similarly, you know, I'm not this, I'm not this, I'm not this, but what am I? I am pure consciousness. And the difference with Yanarnava and the difference with the Jain tradition is that the state of non-duality does not leave an individual without perspective. That there's still a betokening of the specificity of perspective held by particular consciousness by the designation remaining on the throne. So I'd like to put this forward as a challenge of retaining as the Jains are so good at doing, retaining their unique theology. While adopting a technology that by 9th, 10th century India had proved to be very, very successful, a technology that employs mantra, a technology that employs visualization, a technology that employs in a very particular way, the Pancha Mahabutas is an object of meditation in order to deliver a person at least momentarily into a state of moksha. So, thank you. My topic today is tantric elements in preksha meditation. For some time, I had a question. How Mahapragya, the 10th head of Tirapansekt understood Tantra? Then I found this quotation in his autobiography. And where he relates, I was surprised when I read Tantra as a means to obtain a state of personlessness. And this is a quote from Malini Vijayottar Tantra. Mahasair Narshan Saiya Hai. Meditating upon one's own body as resembling a rain cloud at dusk in autumn, one is sure to become free from person within six months. I'm indebted by Jim Malishan to get a insight of Malini Vijayottara and write direction of this translation. So, here we can get a reflection or an idea that Mahapragya studied lot more Tantra, Malini Vijayottara Tantra, Siva Sanghita and other religious and tantric texts. And his views were very positive about it. This paper aims at understanding the role of tantric elements in the development of preksha dhyan formulated by Acharya Mahapragya, an aspect of preksha meditation which has not been explored. I argue that Mahapragya's preksha meditation synthesizes tantric right hand practices, a modern development in tantric practices with element of modern science in a new approach to meditation. This paper examines the incorporation of tantric techniques such as visualization, verbalization, identification, models of the body, the assignment of colors to various parts of the body and practice of mantra nyasa. In this system of preksha meditation, all these are incorporated. In particular, it explore that these techniques map on the anthropomorphic representation of the body as a locus of the practices. Today, I will attempt to show how mainstream tantric elements are mirrored in jain preksha system, the cold power kundalini in the internal journey antryatra. Wheel and lotus, most popular in Hindutantra, are mirrored at psychic centers, chaitanya kendra preksha, and color visualization, ragdharana, which is again represented as Lashya Dhyan, the technique of preksha meditation. I argue that this is an attempt by Mahapragya to develop a new model of meditation that is supported by tantra and compatible with modern science. Also, it is empirical and free from superstitions and religious dogma. Some of the humanitarian aspects, like physical and mental healing is also a taste to preksha exercises. Now I'm moving, what is the idea of kundalini in tantra? A key phenomenon in the esoteric physiology of the tantric yoga is kundalini, a power, a force, which is assumed by, as located on the spinal column. It belongs to the ancient text and the history of this nexus of idea have been studied by Professor Alexanderan, Oliweli, Mechweli, and Haltli. Any many more. And some texts, like Hathayog Pradipika, Siva Sanghita, present a detailed description of kundalini and various names. These texts claims that kundalini can be activated by various prescribed procedures. Normally, kundalini rests dormant in a cold up foam at the lower edge of spine. When it is activated, however, it reaches upward towards the top of the head. This process, which is at the core of tantric path, is characterized as piercing the six wheels, shard chakrabhidan, and sometime seven chakras or nine chakras. This process is also referred to as moving the vital energy upward by the shakti-chalan mudra. Kundalini in Jain literature. Before the discussion of antaryatra, it would be better to understand whether any prior practices, which are similar to kundalini, exist within the Jaina framework. The term kundalini is not available in Jain canonical text. However, it is claimed by Mahaprakya to be an ancient idea and equivalent to the fiery radiance teju-lesha or the manifestation or attainment of fiery radiance teju-labdi, which is obtained by developing the fiery body, teju-sharir, through means of such as ascetic heat, meditation, fasting, asanas, and so on. These powers, when developed, can be utilized both for giving curses and granting boon. Mahaprakya 2007, quote from Mahaprakya. So here, the teju-lesha, teju-labdi, and teju-sharir, very important aspect of Jaina texts, and also it is a, they are discussed at great length by Ohira, Wiley, and Flughal, and many other scholars too. Here, teju-lesha, teju-labdi, and teju-s body, so how they are connected to kundalini, Mahaprakya supported his claim, the teju-lesha as kundalini on the basis of metaphysical dualism of body and self. He suggests that three bodies, gross body, fiery body, and karmic body, are connected to each other. Furthermore, that awakening of kundalini is connected with these three bodies, especially with the teju-s body, which is responsible for manifestation or attainment of fiery radiance. The audaric body is visible, which is activated by teju-s body. The teju-s body is subtle, but driven by the subtle and very subtle karmic body. However, without consciousness, these bodies cannot perform any action. Finally, consciousness is responsible for all changes which occur at the gross or subtle level of the bodies. Mahaprakya claimed that kundalini is the power of consciousness, chetna-shakti, and very, very similar term is used as chiti-shakti in Tantra literature for the kundalini. The vital energy that is available to all living beings being activated through the teju-s body. So this is Mahaprakya's comment. From historical point of view, the discussion of the idea of kundalini as teju-lesha and the practice similar to kundalini can be considered in four parts. So the first part is based on the episode of Gosalak mentioned in Bhagavati Sutra, chapter 15. This part of Bhagavati Sutra is accepted as the fifth canonical stage during the fifth century C by Ohira. The second is presented in Hemchandrasi Yogasastra, and the third part is presented in Singha Tilaksuri's Mantra Raj Rahashe, which is just discussed, and also Adhyatma Kalpadrumam, 15th century by Sundarsuri. So, and the fourth one is formulated in 19th century by Acharya Mahaprakya as Antaryatra. So these are the four main discussion which are developed or mirrored as kundalini. Bhagavati relates, the Gosalak asked Mahavira about how he could gain teju-labdi. Mahavira then instructed him about the procedure of teju-labdi as follow. One who eats only one handful of lentils, drinks one handful of water, practices two days penance, takes ascetic heat, facing the sun with his hands raised above the head at the place of ascetic heat and acquires subtle or vast, sankshipt or vast, vipul, pyriradiance at the end of six month. The procedure here includes tapas, ascetic heat, meditation, facing the sun in a specific posture. There is no information here about channel, nadi, vital force, prana, wheel, chakra or other tantric ideas. The power Gosalak acquired by this process is terrific.  Jenny noted, this may be one reason why Jainism has remained for the most part untouched by the sort of tantric practices which typified many Sahivite cults and eventually permeated the Buddhist community as well. Jenny. A later discussion of the concept similar to Kundalini in Hemchandrasa Yoga Sastra. In the context of his outline of tantric physiology, he mentioned all the three channels which are very popular, Ida Pingala and Shushumna. The Ida resides in the left side of the body and Pingala resides on the right side of the body, Abhudova sun. The Shushumna moves in the middle of the body, in the spine and the place of liberation. These are the channels which him Chandra used in his Yoga Sastra. But Hemchandra developed his version of Kundalini without using the term Kundalini. He employs mantra arhama, which is considered as essential and very important mantra in Jaina culture. The procedure he employs, the five types of sounds, mantra arhama, first uttered in a short sound, rush, then long sound, dirg, then third, lengthen, three major, pluth, and then suksma, subtle, and then finally very subtle, excessively subtle, atishukshma. These soundings or utterance of arhama act by piercing the notes, granti, from the center of the navel to heart, to the throat, as it moves through the middle part, madhya marga yayi, along the spine. Hemchandra specifies only three notes as granti, note chakras, without naming them. Noted by Kwanstown. Interpret and hear the madhya margayi. Kwanstown interprets the term madhya margayi as sushumna. In his view, the adoption of tantric element anticipated as future advantage in including material familiar to the cultural heritage of the King Kumar Pal. Though Hemchandra did not use the term kundalini and chakra, but it is very, very clear that it is a practice nearby or kind of kundalini itself. The third discussion of kundalini is presented in Sundar Surya Adhyatmakal Singh Tilkesuri, 13th century Mantra Raj Rahasya. And here he incorporated kundalini word first time in the Jain horizon. And also the 15th century Sundar Surya Adhyatmakal Mudrum or Sardasthau, where he introduced the term kundalini in a way in the Jain film, it is a jainization of Hindu tantric concept of kundalini. During the development of preksha meditation, Mahapragya coined a new term for the second limbo preksha meditation, which is referred as internal journey in the eight-limbed preksha system. These eight limbs is our khyot-sirks, swas-prikshas, sharid-prikshas, chaitanya-kendra-prikshas, leshya-dhyaana, anuprikshas, and bhavana. And this is also a part of my PhD at Savas. The question is, why the practice of antaryatra was developed as the second limb of preksha meditation? If we see the literature from 60 to 72, Acharya Mahapragya, mostly, used anuprikshas, bhavana, and khyot-sirks. But there is no antaryatra, no existence of antaryatra. But sometime he talks about chakra and kundalini. But here he make it clear, but he defined in a scientific way the movement of consciousness in the central nervous system is known as internal journey. He explained that ordinary consciousness in human being is connected with the external world, senses, mind, body, which is the outer journey. The procedure of internal journey direct towards the flow of vital energy in upward direction. This is needed to activate the deeper level of consciousness during meditation association with the psychic center and connected with the spinal cord, the pathway of the internal journey. According to Mahapragya, antaryatra is placed in the second step of preksha because it promotes vital energy, which is necessary for the subsequent development of meditational practices. And you know this antaryatra is free from any sort of jargon of any naming. If we see Hatha Yoga Pradipika, there are so many names of kundalini. But Mahapragya used it very simplistic way and the modern physiology, rather than esoteric tantric physiology. He claims that the channels of three tantra channels are compatible with the physiology of nervous system. Further, he equates the right channel ida with sympathetic nervous system, then left channel pingala with parasympathetic nervous system, and the final, the middle channel, which is kendriyanadi sansthan, that is susumna. He explained why this is so significant. The nervous system is a location of knowledge and consciousness. It prevails throughout the body, the core place of consciousness. And this is very important part of the soul, psyche, mind and senses. And all these chakras, which are used as psychic center in preksha meditation are also located in this place. Further, he explains the upward movement of vital force from the lower psychic center is a way to enter into the world of spirituality. Normally, a human being's energy flow downward. And he does not know how to make the flow upward. When the direction change, the internal power flow towards the upside. And this is very important. The bhautik suk, if the energy is downside, then it is giving bhautik suk, kam suk. And when the energy change its direction, it gives a dhyatmik suk, spiritual space. Mahapragya 2004. The explaining the concept of antryatra, Mahapragya based his exposition on acharang sutra, which is very important to understand that these sutras, none of these aphorisms, directly support the interpretation of the antryatra. In fact, Mahapragya adduces a homology between the concept of Kundalini from the hatha yoga and in a way is looking towards the, through the lens of hatha yoga. Acharang sutra used the term maha viti, the great path which is commonly translated path of non-violence. This is considered by Mahapragya to refer to Kundalini, drawing upon the hatha yoga pradipika. Mahapragya states that there are many synonyms of Kundalini, among them great path is one, it is known as maha path and very similar meaning and synonym meaning to the Kundalini. He controversially claims that a great path, which is mentioned as Kundalini and he argues how he developed his interpretation and understanding. During the study of hatha yoga pradipika, I came across a term maha path, which is used for Kundalini. Immediately, I remembered an aphorism from Acharang sutra, panayavira maha viti, brave men, are those who worked on the great path. I did not understand the meaning clearly. The great path means highway, rajmarg. They walked on the highway as well as on the narrow way, pagdandhi. It is quite clear that the literal meaning of this word, maha viti has its hidden meaning, sanketik meaning. Here, maha viti means Kundalini. The one who walked on the maha viti of the spiritual practice is brave. If you look at the book on Tantra, you will find that whose vital force has not moved into Susumna is not free from desires. Mahapragya's quote, within the Jain tradition, the concept of Kundalini as maha path were unknown prior to Mahapragya. These terms do not appear in any of the Jain scripture. Thus, it is likely that Mahapragya drew inspiration from manual of Hindutantra and Hatiyuga, where these concept features prominently to form the second part of his eight-limbed pricksha meditation. According to Mahapragya, antryatra purifies negative instinct when one is connected with the Kendriya Nadi Sansthan, the path of spiritual development. And this is noted in the article by Quastown and Birch also. Anyone practices, Mahapragya again reinforces it. Anyone practicing the deep breathing, the spinal column with concentration and commitment, the serenity will experience a vibration of spinal cord because how he experience and he put his experience on the top of his practices, going deeper into the practice, he will feel something creeping along the spinal cord. Within this happen, he will feel a strange kind of peace and stillness. The whole body gets cool as if he is a hot and tired traveler relaxing under the shadow of a tree. After the practice, just 10 or 15 minutes, the slow breathing will free the mind of thoughts and ambiguities and he enter a state of no mind, nirvicharata. This is from Mahapragya's own experience. Furthermore, he expounds the another aphorism whose general meaning is one who is dispassionate of devoid of attachment and aversion of one who lives in equanimity. Translation for Mahindra Kumar. The term madhyastha normally means dispassionate or neutral and nirvicharata pixi denotes one who wants to cast karma. Mahapragya raised a question whether the writer means to refer only the general meaning of dispasses and interpretation he considers inadequate. So here again, Mahapragya presents a new interpretation of this aphorism. He argues that the process of setting karma is the flow of the vital force in Susumina. So this is again very important sutra from Acharang sutra and this is Mahapragya's new interpretation for supporting antaryadra. While the main objective of such internal journey is to initiate an upward movement of energy and get benefit of the peace and a change of personality, but Mahapragya never talked about the meeting of any union like hatha yog talks. So there is no such description and he never talked about any kind of liberating state through the antaryadra keeping very simple and using for the common man's practice. These include the cultivation of power of self-restraint, boosting the vital energy and the creation of a strong and sound basis for the meditation because mind moves inward. In this way, it enhances the concentration on the object what is to be achieved. In addition to establish the concept of antaryadra, Mahapragya states that all psychic centers, chaitanya-kendra of the body connected with the vital force and linked with the spinal column as we see similarly in the hatha yog and tantra, the old chakras are on the spinal code. So very similar aspect, but side by side, Mahapragya used some more chakras and those chakras are not directly located on the spinal column. The vital energy enters into the brain through the two pathways, inner and outer. So the inner is related to ida, inner is related to shishumna, but outer is related to ida and pingala. And Mahapragya used the sympathetic and parasympathetic nervous system. The vitality and normal strength of the body depends on the inflow of vital energy through the outer channel. These activities, he relates the jain concept of ten pranas located in the body and keeps the organism alive. A special kind of vital force is created in the body when vital force begin to flow into the inner channel. Mahapragya does not explain about the supernatural power of antryatra. He approaches the more pragmatic. According to Mahapragya, nervous system is considered to be essential for functioning the conscious activity. It permits the entire body through the network of nervous system. The spinal cord and brain are the two most important component of the bodily system. The spine extends from the shakti kendra, which is the store of energy and reaching towards the gyan kendra top of the head. Similarly, the sensation and motor nerves are also connected to the seredient head. The practice of antryatra. So in this way, Mahapragya used the scientific area to, I think, convince everybody, not for jain, not for Hindu, not for any religious committee, but common man can understand the scientific explanation of sympathetic, parasympathetic nerve system and the central nerve system. So in this way, I'm not going in the detail, but just using this picture, he used this body locus and the old psychic centers are here and side by side, he used the glendary system. So according to Mahapragya, the neuro-endocrinology is a very important concept and side by side, he used many exercises which are very much helpful for the common masses, for the health-related issues. And there is a big discussion about various aspect of health-related issue through antryatra and psychic center. But he presented in a very, very simple manner and concluding, Mahapragya has developed antryatra through a new interpretation of the acharang sitra and modern physiology. It remains very close to the practice of Tantric Kundalini Yoga, but also maps on the center of human nerve system and concept from neuroendocrinology, so which is very important to understand everybody. Thank you. And side by side, yes.


Everybody welcome back to another Adobe After Effects tutorial guys today. I'm gonna show you how to change colors We're gonna do color transitions with shape layers And I'm also gonna show you how to do it using typography or changing the color of Your font or your type and that's actually kind of tricky And these are really cool for those of you that are new to After Effects and you want to add some color and you don't want everything to be the same color You want to add some transitions? So let's just dig in and get it rolling. I've opened up after effects here I'm using CC 2019 the latest version whatever version you're using. It's probably this one But if it's not well, just follow along because it hasn't changed in the last few years. So we've opened it up I'm gonna go ahead click new composition and This doesn't really matter. I'm just gonna do 1920 by 1080 and I'm gonna hit okay now The composition is here. It's loaded, but there's nothing here So the next step is I'm just gonna create a new layer. I'm gonna create a background. So you want to go in layer New and I'm just gonna do a solid and I'm just gonna do black solid. So we've got something in the background there We've got a black solid. We are on our way. Let's do the Let's do a shape layer first I'll show you how to do that transition or that color change and then I'll show you how to do the typography next So the first one is the shape layer and I'm just gonna create a nice Oh shoot. I made a mistake there actually you see what I did This is actually really really interesting for newer people if you click on the solid and you draw out It draws a mask, but if you're not clicked on the solid you're clicked on nothing here where my cursor is I'm not clicked on it. I'm clicked off of it. It will draw a shape layer This is very frustrating for for me when I first started out because I didn't know what the hell I was doing wrong So if you caught me there and we're and you've struggled with that that is possibly the problem So again, I've done a shape layer. It's purple. I'm just gonna align it to the center here in the align section and Presto there we go guys. We've got our shape layer now The playhead is at zero seconds and I want to change this from purple to red. Let's say I want to go from purple to red and Let's say I want to do it in like three seconds Ish or something like that So the way to do that is you want to go down to the shape layer and start dropping these triangles Really simply here. You're gonna see rectangle one and then you're gonna see fill one This is where you go ahead and make the color change in particular. It's set to purple just like we want I'm gonna click on this little stopwatch and That basically says well actually you know what I'm gonna command Zed that I'm gonna would or control Zed if you're on a PC So I've done nothing so we've got nothing on all we've got is this open You want to click on the stopwatch here on the color and then you'll see here that I can go ahead now and adjust the color So at zero seconds, I want it to be purple and then let's say at around three seconds almost three seconds Yeah, something like that. I want it to be Red let's go with bright red. We want to show that transition by doing that I will show you I'll go to the start click on it I'll hit space bar to just run the composition or just run it through and You're gonna see here that it over those three seconds it went from red from purple to red Which is exactly what we wanted This is extremely useful for those of you guys that do motion graphics and things like that And you want to have some colors or some adjustments on there. That's all there is to it I'm gonna delete this shape layer and now I'm gonna show you how to do it with Typography or with font. This is a little trickier. It's not straightforward. So let's just dig right in I'm gonna go ahead and click on my little horizontal type tool up here and I'm gonna write my name My name is Curtis. Ah, ah, there we go. And that's a pretty good size. Let's go ahead and make it a little bigger and Let's align it in the middle. Why not? Okay, good. So we've got my name Curtis it's in red But I want to make the color go from red to let's say white that makes sense, right? That was a nice high contrasty color now normally you would go over here You click on this and you see it's set to red That's great, but there's no stopwatch here. You can't make any color changes on that over time Which is what I want to show you so that's not actually how you do it.  This is where we do the work In particular, you're going to see fill color and then if you go across you'll see RGB Which means red green blue. This is what we're looking for left click on that now when you do that it opens up Automatically and you're going to see fill color and you're gonna see the stopwatch. This is what we're looking for So I am going to click the stopwatch Here at zero seconds and it is on red and then let's say at like around almost three seconds It's gonna go. What did we say we wanted white? This is how you change it. You'll notice here again stopwatch is clicked It's gonna go from red as you can see here. Let's hit play. I'm just gonna click off and hit spacebar red To white over those three seconds This is how you do a change in typography your font or whatever you want to call it guys This is how you do a color transition. I hope this tutorial helped. I've got a ton more stuff coming up Thank you for watching


If you read the basic map of one's prana, you know their past, present and future, everything about that person. Because in a way your life is just a manifestation of your energies, how they function. So with the yogic system, what you're trying to do is to change how it functions from an unconscious repetition of things or it is right now being heavily manipulated by the memory bank that one has or the karmic structure that one has, accordingly the energy, energy is moving. Now, if we look at this, let's forget whose image this is, okay? This is like a… an energy x-ray for immediate purposes. We can make a much elaborate one, but that would take much more involvement. This is for immediate. So, now if you look at this map, what you will see is Mooladhara is slightly imbalanced, needs little… little… little more work. Swadhisthana, not very strong, but okay, Manipuraka is good. Anahata, confused. You know what these are? You know which is which? Anahata is confused and a little bit of like that. This should be nothing might as happened as you were born, it is. Agna, okay, stable, nothing phenomenal, but it is potential. I think because of a certain continuous emotional pattern, it's… the system has become slightly skewed, which must be corrected. Hatha yoga is the right thing to do. If you do proper Hatha yoga, these slightly skewed positions that your energy system has taken will naturally get corrected. In the occult processes, if they want to destroy somebody, just deny him the prana, he'll just crumble. If you read the basic map of one's prana, you know their past, present and future. Right now, this is Anahata, not necessarily.  If one has some mastery over his Vishuddhi, his whole aura will turn electric blue. It is this type of aura which allows you to function in the world in a way that other people think is superhuman. And that is why Shiva sits there. There are certain systems of tantra, particularly in Central Asia, these were certain groups of gypsies. In a matter of two-and-a-half hours to three hours, they will plant an apple seed and make the tree grow and bear fruit and ease that dimension. One should not meddle with it unless they're truly, truly competent.