Many of the students are asking their doubts by sending a picture in whatsapp. Here I post the Hints for the answers, so that others also can use the problems. As I have too little time to type all in Latex, here simply I post the photo.
Any more suggestion, comments are welcome.
Also I post Problems and Hints, that I found in some groups asked and solved by others. On behalf of myself and readers of this post, I express my sincere thanks to them.
*Problems will be updated regularly*
Warning: ALL THE ANSWERS AND HINTS NEED NOT BE CORRECT. VERIFY THOROUGHLY. ALSO IF YOU FIND ANY MISTAKES PLEASE WARN ME. I TRY TO CORRECT.
Hint: https://math.stackexchange.com/questions/697738/dimension-of-the-vector-space-of-homogeneous-polynomials
Hint: Use M-l inequality in complex integration. Also Here the question is incorrect. Instead of |z|=1, take |z|=2.
*End of Problem*
Hint: Use binomial expansion for numerator. find the coefficient of x^2n. From that find the coefficient of 1/z in the series of given function
*End of Problem*
Hint: A similar problem: https://math.stackexchange.com/questions/811391/f-nx-nx1-xn-determine-whether-the-sequence-f-n-converges-uniformly-o
*End of Problem*
See Question 11
Solution:
*End of Problem*
If both graph G and its complements are trees, then the number of vertices is ________
Hint: Winding number https://en.wikipedia.org/wiki/Winding_number
*End of Problem*
Number of automorphisms of Z, Zn, Sn and Klien four group.
Hint: A automorphism of a cyclic group is completely determined by the image of a generator of the cyclic group.
In Z, the generators are $\pm 1$. So by finding the image of 1, the problem can be solved. Therefore two automorphisms, namely $1 \mapsto 1$ and $1 \mapsto -1$ .
In Zn, consider the generator 1. suppose totally m(=???) generators we have, then 1 can be mapped to m values.
But Sn and Klien four group are not cyclic. Anyway here we can do a similar work by taking a "generator set". In Sn two elements generate the group, namely a 2-cycle $\tau$ and a 3-cycle $\sigma$. Keeping in mind the fact "Automorphism preservs orders" here too we can find the number of ways to map $\tau$ and $\sigma$.
The case of Klein four group is left to the reader.
https://math.stackexchange.com/questions/103390/follow-up-to-question-autg-for-g-klein-4-group-is-isomorphic-to-s-3
https://math.stackexchange.com/questions/1538681/finding-the-automorphisms-of-s-3-by-looking-at-the-orders-of-the-elements
Hint: A automorphism of a cyclic group is completely determined by the image of a generator of the cyclic group.
In Z, the generators are $\pm 1$. So by finding the image of 1, the problem can be solved. Therefore two automorphisms, namely $1 \mapsto 1$ and $1 \mapsto -1$ .
In Zn, consider the generator 1. suppose totally m(=???) generators we have, then 1 can be mapped to m values.
But Sn and Klien four group are not cyclic. Anyway here we can do a similar work by taking a "generator set". In Sn two elements generate the group, namely a 2-cycle $\tau$ and a 3-cycle $\sigma$. Keeping in mind the fact "Automorphism preservs orders" here too we can find the number of ways to map $\tau$ and $\sigma$.
The case of Klein four group is left to the reader.
https://math.stackexchange.com/questions/103390/follow-up-to-question-autg-for-g-klein-4-group-is-isomorphic-to-s-3
https://math.stackexchange.com/questions/1538681/finding-the-automorphisms-of-s-3-by-looking-at-the-orders-of-the-elements
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