Nice video prof Though I know that this channel is dedicated to undergraduate mathematics, I would request u to cover applications of graph theory in olympiads (connectedness, use of Dirac and ores theorem etc ) and plz teach some advanced topics like Ramsey theory. 🙏🙏🙏 I couldn't find nice sources for Ramsey theory and Ur way of solving problems is very clear and understandable.
A stronger result for the 2nd problem, you can ask which numbers has a representation if you fix k, the answer is: For k>8 let T=1^2+2^2+...+k^2=k*(k+1)*(2*k+1)/6 then in the [-T,T] interval you can represent all integers with the x=+-1^2+-2^2...+-k^2 form that has the same parity as 1+2+...+k=k*(k+1)/2, with the following exceptions: x=-T+c or x=T-c, where c={4,6,12,14,16,22,24,30,36,38,44,46,48,54,56,62,64,66,86,88,94,96,120,134,144,152,184,192,216,224,256}. (so there are 2*31=62 exceptions that has no form but has good parity in the [-T,T] interval). We can prove this by induction.
@@ProfOmarMath Even bruteforcing works here, just calculate the sum for each 2^k possibilities for all +- sign choices. Observed that for k>8 we have the same exception list, but it is ofcourse provable.
@@ProfOmarMath I am well sir. Sir i have got the hang of group theory and am feeling confident now on solving questions. Also my teacher has started real analysis any tips for this chapter ??😅😅
@@ProfOmarMath hello Sir sorry for late reply I don’t use RUclips often. Basically I am from India and preparing for competitive exams. And in college we had yes you can see mix syllabus in mathematics both modern algebra and real analysis.In college in these two chapters we were taught how to qualify the university exam and score good marks but in the competitive exams it requires an analytical thinking which unfortunately I I did not do for them. So basically I just need some help in these two chapters or is there anything else you would like recommend me for better understanding. Currently I’m following Bartle for Real analysis and Joseph Gallian for modern Alegrba.
@@ProfOmarMath And sir your mathematical induction videos is very helpful in Real analysis for showing that (n+1)th term is smaller n th term vice versa. For showing some series is convergent, div
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Love that last proof. Very cool.
wow! today you were open my mind
Woww amazing !!
you are the best!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!
the best ever!!!!!!!!
Very interesting example at the end!!
Can we say that in general, {p(a(1)) & {for all n p(a(n)) => p(a(n+1))}} => p(a(n)) for all n?
Thanks Eli. This depends very much on what the a(n) values are overall. As long as somehow it covers all positive integers, you’ll be good to go
Great content
Thanks!
Nice video prof
Though I know that this channel is dedicated to undergraduate mathematics, I would request u to cover applications of graph theory in olympiads (connectedness, use of Dirac and ores theorem etc ) and plz teach some advanced topics like Ramsey theory.
🙏🙏🙏
I couldn't find nice sources for Ramsey theory and Ur way of solving problems is very clear and understandable.
Really enjoyed your explanation! Thank you! :)
Thanks @mathwithjanine
Hello profOmar, great video! One question: is there any exponencial series in the form f(x)=sum from n=0 to inf of a_n e^(nx)? Thank you.
You are so awesome 😎😎😎😎😎
Thanks Sir
Definitely!
I like your examples with n=x1+x2...+xm with restriction to xi or how many ways to write n= x1+x2.. +xm very intresting!
They’re very cool examples 😍
A stronger result for the 2nd problem, you can ask which numbers has a representation if you fix k, the answer is:
For k>8 let T=1^2+2^2+...+k^2=k*(k+1)*(2*k+1)/6 then in the [-T,T] interval you can represent all integers with the x=+-1^2+-2^2...+-k^2 form that has the same parity as 1+2+...+k=k*(k+1)/2, with the following exceptions: x=-T+c or x=T-c, where
c={4,6,12,14,16,22,24,30,36,38,44,46,48,54,56,62,64,66,86,88,94,96,120,134,144,152,184,192,216,224,256}.
(so there are 2*31=62 exceptions that has no form but has good parity in the [-T,T] interval).
We can prove this by induction.
Interesting! How are the choices for c discovered?
@@ProfOmarMath Even bruteforcing works here, just calculate the sum for each 2^k possibilities for all +- sign choices. Observed that for k>8 we have the same exception list, but it is ofcourse provable.
wow!
that was hot, especially the last one
Love it
Hello everybody! Does anyone know what happened to professor Omar?
Hi Jesus. Long time no chat! I'm still around. I haven't posted in a while but will hopefully be back soon. How have you been!
@@ProfOmarMath Great to know about you! I am fine, I hope you come back soon. Maths must go on!
Hello sir are you okay ? You haven’t posted for 1 month ?
Still here! Coming back hopefully soon. How are you?
@@ProfOmarMath I am well sir.
Sir i have got the hang of group theory and am feeling confident now on solving questions. Also my teacher has started real analysis any tips for this chapter ??😅😅
@@dulcedeleche000 Interesting what kind of course are you doing, is it a mix?
@@ProfOmarMath hello Sir sorry for late reply I don’t use RUclips often. Basically I am from India and preparing for competitive exams. And in college we had yes you can see mix syllabus in mathematics both modern algebra and real analysis.In college in these two chapters we were taught how to qualify the university exam and score good marks but in the competitive exams it requires an analytical thinking which unfortunately I I did not do for them. So basically I just need some help in these two chapters or is there anything else you would like recommend me for better understanding. Currently I’m following Bartle for Real analysis and Joseph Gallian for modern Alegrba.
@@ProfOmarMath And sir your mathematical induction videos is very helpful in Real analysis for showing that (n+1)th term is smaller n th term vice versa. For showing some series is convergent, div