CSIR-TIFR-ISI-NBHM - 2019-01-08

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N. Maneesh

14:59

What is the largest possible $c$ in $||a_1x_1+a_2x_2+...+a_nx_n||\ge c(|a_1|+|a_2|+...+|a_n|)$. if $X = \mathbb R^2 $and $x_1 = (1, 0)$, $x_2 = (0, I)$? If $X = \mathbb R^3$ and $x_1 = (1, 0, 0), x_2 = (0, 1,0), x_3 = (0, 0, I)$?
we have $||a_1x_1+a_2x_2+...+a_nx_n||\ge c(|a_1|+|a_2|+...+|a_n|)$
what we want to find is $c=\inf\{||a_1x_1+a_2x_2||:|a_1|+|a_2|=1,a_1,a_2 \in \mathbb R\}$
So this value will be the largest value for $c$.

@vidyarthi

Do you have any idea?

vidyarthi

15:38

@N.Maneesh I am unable to get your question. Could you state it more clearly?

N. Maneesh

15:53

okay

we have $||a_1x_1+a_2x_2+...+a_nx_n||\ge c(|a_1|+|a_2|+...+|a_n|)$
what we want to find is $c=\inf\{||a_1x_1+a_2x_2||:|a_1|+|a_2|=1,a_1,a_2 \in \mathbb R\}$
So this value will be the largest value for $c$.

@vidyarthi

vidyarthi

16:24

@N.Maneesh I think the answer is $\sqrt{2}$ for $X=\mathbb{R}^2$ and $\sqrt{3}$ for $X=\mathbb{R}^3$, by using the Cauchy-Schwarz inequality

N. Maneesh

How do you apply Cauchy Schwartz inequality?

can you write?

@vidyarthi

vidyarthi

We have $\|(\alpha_1,0)+(\alpha_2,0)\|=\sqrt{\alpha_1^2+\alpha_2^2}\le(\frac{|\alpha_1|+‌|\alpha_2|}{\sqrt{2}}$ by Cauchy-Schwarz. Hope you could do the next one similarly

@N.Maneesh hope you got it?

I am sorry the $\le$ in the inequality I stated must be replaced by $\ge$

N. Maneesh

$||a_1x_1+a_2x_2||\ge c(|a_1|+|a_2|)$

but you proved $\|(\alpha_1,0)+(\alpha_2,0)\|=\sqrt{\alpha_1^2+\alpha_2^2}\le(\frac{|\alpha_1|+‌‌|\alpha_2|}{\sqrt{2}}$

vidyarthi

16:40

@N.Maneesh yes, so sorry, replace $\sqrt{2}$ by $\frac1{\sqrt{2}}$ and $\sqrt{3}$ by $\frac1{\sqrt{3}}$.

hope you got it?

N. Maneesh

yes

Thank you

vidyarthi

The inequality is not $\le$, but $\ge$

N. Maneesh

yes

vidyarthi

I think this lemma is somewhat resembling Hölder or Minkowski inequality, isnt it?

N. Maneesh

nope

we need to find $\inf\{||a_1x_1+a_2x_2||:|a_1|+|a_2|=1,a_1,a_2 \in \mathbb R\}$

which is the maximum value for $c$.

right?

vidyarthi

16:44

which book is this from?

N. Maneesh

Kreyzig functional analysis

are you comming for NBHM?

vidyarthi

I guessed it right!, even your previous functional analysis problem regarding norms was also from there, right?

N. Maneesh

IISc bangalore?

vidyarthi

yes, I am coming

N. Maneesh

@vidyarthi yes

@vidyarthi I am also having the same centre.

vidyarthi

16:46

At first, the NBHM people did not mention my roll number in the list of candidates. Then, after repeated emails, they put my name in the list

yes. Yours in the first page of candidates, right?

N. Maneesh

where is the list?

vidyarthi

So this time, have you advanced booked chennai, so that you could attend the interview easily?!

N. Maneesh

bro I couldn't prepare well.

I losed my interest in mathematics

I am forcing me to do so. I have no other option.

vidyarthi

@N.Maneesh the list is here

N. Maneesh

I wish to move to theoretical physics.

vidyarthi

16:50

Hey, but you are solving advanced problems in functional analysis

anyways, quantum mechanics contain a plethora of advanced analysis, lie algebras and dynamical system theory

N. Maneesh

@vidyarthi I am doing because I don't have no other option. want to obtain some job. then want to shift the subject.

Yes quantum mechanics is full of operator theory

vidyarthi

so are you planning to write JEST, by any chance? It is a theoretical physics exam. I think you can write it

@N.Maneesh Also, try to write CSIR physical sciences in June session

N. Maneesh

@vidyarthi I know only bit classical and quantum mechanics

I have to extend my knowledge.

I hope if i obtain some job, I can.

I lost interest in lecturing. students complaining. they don't understand my class :(

very few understand my class :(

vidyarthi

do not trust students/others. Trust yourself. Are you really enthusiastic about Mathematics?

Its not whether you can clear an exam or not. Do you like to solve tough Mathematics problems is what you have to ask yourself

N. Maneesh

I loved math. I loved solving problems. Studying without applying making me disappointing.

@vidyarthi Yes I like to solve problems

vidyarthi

16:59

Now, if you ever shift to theoretical physics, again maybe some other thing would interesting, like, say engineering

@N.Maneesh Its ok, then. See, there are many applied Mathematics institutes. There, the Mathematics is application oriented.

The Computational Science department and interdisciplinary Mathematics program at IISc is very application oriented

N. Maneesh

I think no chance. I wish to study about universe.

vidyarthi

They are accepting Mathematics students for their admission

N. Maneesh

mmm.

vidyarthi

There are lot of Physics and astronomical projects they are working on

N. Maneesh

I need to come top 100 right? in GATE

vidyarthi

17:03

You can surely try there. They called me also for interview. So no, you need not have a high rank in GATE for those departments

N. Maneesh

Thank you for your guidelines. I will apply next time. I will work for it :)

vidyarthi

Also, you could have applied for TIFR-CAM in Bangalore. It is wholly application oriented

and ICTS Bangalore is also physics and application oriented

And some IITs also accept Mathematics students for their applied programmes

N. Maneesh

okay. We are away from the problem. Let's try.

vidyarthi

Ok. Post the problem. I will try later. Have to leave now. Happy problem night

N. Maneesh

Problem is minimizing the function $\sqrt{a_^2+a_2^2}$ under constraint $|a_1|+|a_2|=1$

right?

@vidyarthi

vidyarthi

17:08

yes, you can look at it through that way also. But, Cauchy-Schwarz is a good tool here

@NManeesh In fact, it gives the condition when equality occurs

@N.Maneesh ok, good bye

N. Maneesh

good bye

but cauchy-schwartz inequality is not working it is $(\sum x_iy_i)^2\leq (\sum x_i^2) (\sum y_i^2)$

yes got it

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