User chandrasekhar - MathOverflow

If $f$ is infinitely differentiable then $f$ coincides with a polynomial

2010-07-31T21:37:22Z

Let $f$ be an infinitely differentiable function on $[0,1]$ and suppose that for each $x \in [0,1]$ there is an integer $n \in \mathbb{N}$ such that $f^{(n)}(x)=0$. Then does $f$ coincide on $[0,1]$ with some polynomial? If yes then how.

I thought of using Weierstrass approximation theorem, but couldn't succeed.

Example of an infinite abelian but non-cyclic group whose automorphism group is cyclic.

2011-09-01T14:09:25Z

Can anyone give me an example of:

An infinite abelian but non-cyclic group whose automorphism group is cyclic.

Answer by Chandrasekhar for Old books still used

2012-12-29T11:10:55Z

My choice of books would be:

Theory of Riemann-Zeta Function by E.C. Titchmarsh, (Oxford University Press)
Theory of Functions by E.C. Titchmarsh (Oxford University Press, 1952).

Answer by Chandrasekhar for Favorite popular math book

2011-07-29T15:54:48Z

My favourite books are:

$\text{I want to be a Mathematician}$ by Paul Halmos.
Problems for Mathematicians, young and old, by the same author.
The Man who knew Infinity: : A Life of the Genius Ramanujan by Robert Kanigel. Fanastatic book. Covers almost all of Ramanujan's life and work. Certainly a must read.

Answer by Chandrasekhar for (Preferably rare) Audio/Video recordings of famous mathematicians?

2012-07-08T15:25:07Z

Michael Atiyah Beauty in Mathematics

Answer by Chandrasekhar for (Preferably rare) Audio/Video recordings of famous mathematicians?

2012-07-08T15:17:50Z

Here are some videos on Stephen Smale

Answer by Chandrasekhar for (Preferably rare) Audio/Video recordings of famous mathematicians?

2012-07-08T15:11:15Z

Timothy Gowers here on youtube lecturing a talk on: The Importance of Mathematics

Request for the proof of a result from Ramanujan's letter to Hardy.

2012-07-02T17:01:58Z

Srinivasa Ramanujan in his first letter to G.H. Hardy stated many results for which he didn't give proofs. Among them the result taken from this link seems interesting :

If $$\int\limits_{0}^{\infty} \frac{\cos{nx}}{e^{2\pi\sqrt{x}}-1} \ dx = \phi(n)$$ then $\displaystyle\int\limits_{0}^{\infty} \frac{\sin{nx}}{e^{2\pi\sqrt{x}}-1} \ dx = \phi(n)-\frac{1}{2n} + \phi\biggl(\frac{\pi^2}{n}\biggr)\sqrt{\frac{2\pi^3}{n^3}}$.

The link also mentions that $\phi(n)$ is a complicated function. The following are certain special values and shows some values.

Questions which I would like to ask here are:

Where can I find the proof of the above result?
"The following are certain special values": Whats so special about the values?

Answer by Chandrasekhar for Request for the proof of a result from Ramanujan's letter to Hardy.

2012-07-03T06:53:03Z

Thanks "@Charles Matthews". I did email Prof. Berndt (after seeing your comment) and he suggested me to look at this paper:

Some definite integrals connected with Gauss's sums, Mess. Math. 44 (1915), 75-85.

The result appears here in Page 60. with proof.

Answer by Chandrasekhar for Examples of conjectures that were widely believed to be true but later proved false

2012-07-03T07:43:50Z

Here is one which I learnt from this answer here

Answer by Chandrasekhar for Fundamental problems whose solution seems completely out of reach

2012-07-02T12:31:56Z

Sendov's Conjecture

For a polynomial $$f(z) = (z-r_{1}) \cdot (z-r_{2}) \cdots (z-r_{n}) \quad \text{for} \ \ \ \ n \geq 2$$ with all roots $r_{1}, ..., r_{n}$ inside the closed unit disk $|z| \leq 1$, each of the $n$ roots is at a distance no more than $1$ from at least one critical point of $f$.

Answer by Chandrasekhar for Fundamental problems whose solution seems completely out of reach

2012-07-02T03:20:38Z

The Bunyakovsky conjecture (or Bouniakowsky conjecture) stated in 1857 by the Russian mathematician Viktor Bunyakovsky, claims that

an irreducible polynomial of degree two or higher with integer coefficients generates for natural arguments either an infinite set of numbers with greatest common divisor (gcd) exceeding unity, or infinitely many prime numbers.

Answer by Chandrasekhar for Fundamental problems whose solution seems completely out of reach

2012-07-02T03:15:28Z

Schinzel-Sierpinski Conjecture

Taken from this MathOverflow link.

Melvyn Nathanson, in his book Elementary Methods in Number Theory (Chapter 8: Prime Numbers) states the following:

A conjecture of Schinzel and Sierpinski asserts that every positive rational number $x$ can be represented as a quotient of shifted primes, that $x=\frac{p+1}{q+1}$ for primes $p$ and $q$. It is known that the set of shifted primes, generates a subgroup of the multiplicative group of rational numbers of index at most $3$.

Answer by Chandrasekhar for Not especially famous, long-open problems which anyone can understand

2012-06-24T06:53:06Z

The Happy Ending Problem

Says that any set of five points in the plane in general position has a subset of four points that form the vertices of a convex quadrilateral. More generally, Erdös and Szekeres proved that for any positive integer $N$, there is a minimal integer $f(N)$ such that any set of $f(N)$ points in the plane in general position has a subset of $N$ points that form the vertices of a convex polygon, and it is known that $f(N)$ is at least $1+2^{N-2}$.

An open question is: does $f(N)=1+2^{N-2}$ hold?. Taken from this MO link.

Answer by Chandrasekhar for Good books on problem solving / math olympiad

2012-06-24T08:17:32Z

These are some of the books / links which I would recommend:

Functional Equations and How to solve them by Christopher G. Small. This book especially discusses techniques for solving functional equations which appear in the Olympiads.
Geometry Unbound by Kiran Kedlaya.
The Math problems notebook by Louis Funar and Valentin Boju.
Komal, I think is a Hungarian Magazine which contains Olympiad level problems. The archived set of problems along with their solutions can be found at this link.
International Mathematics Competition for University students has problems more or less like the Putnam.
Vojtech-Jarnik is again a Undergraduate Mathematical Competition whose archived problems and solutions can be found at this link.
Problems in Elementary Number Theory by Hojoo Lee and Peter Vandendriessche has nice collection of problems in Number Theory.

Answer by Chandrasekhar for Not especially famous, long-open problems which anyone can understand

2012-06-23T02:34:37Z

Sendov's Conjecture

For a polynomial $$f(z) = (z-r_{1}) \cdot (z-r_{2}) \cdots (z-r_{n}) \quad \text{for} \ \ \ \ n \geq 2$$ with all roots $r_{1}, ..., r_{n}$ inside the closed unit disk $|z| \leq 1$, each of the $n$ roots is at a distance no more than $1$ from at least one critical point of $f$.

Answer by Chandrasekhar for Not especially famous, long-open problems which anyone can understand

2012-06-23T04:18:49Z

Schinzel-Sierpinski Conjecture

Taken from this MathOverflow link.

Melvyn Nathanson, in his book Elementary Methods in Number Theory (Chapter 8: Prime Numbers) states the following:

A conjecture of Schinzel and Sierpinski asserts that every positive rational number $x$ can be represented as a quotient of shifted primes, that $x=\frac{p+1}{q+1}$ for primes $p$ and $q$. It is known that the set of shifted primes, generates a subgroup of the multiplicative group of rational numbers of index at most $3$.

Answer by Chandrasekhar for math circles video lectures for school children?

2012-06-23T09:47:16Z

Mathvids.com has some interesting videos on Algebra, Trigonometry and Statistics.
MIT Open Course Ware has a nice collection of lectures on Single-Variable Calculus.

Answer by Chandrasekhar for Not especially famous, long-open problems which anyone can understand

2012-06-22T07:12:08Z

Here is one which I found at this MO link:

$$ \frac{24}{7\sqrt{7}} \int_{\pi/3}^{\pi/2} \log \left| \frac{\tan( t)+\sqrt{7}}{\tan(t)-\sqrt{7}}\right|\ dt = \sum_{n\geq
1} \left(\frac n7\right)\frac{1}{n^2}, $$ where $\displaystyle\left(\frac n7\right)$ denotes the Legendre symbol. Not really my favorite identity, but it has the interesting feature that it is a conjecture! It is a rare example of a conjectured explicit identity between real numbers that can be checked to arbitrary accuracy. This identity has been verified to over 20,000 decimal places. See J. M. Borwein and D. H. Bailey, Mathematics by Experiment: Plausible Reasoning in the 21st Century, A K Peters, Natick, MA, 2004 (pages 90-91).

Why is this theorem attributed to Serre?

2012-06-18T16:43:20Z

Page $117$ of Atiyah, MacDonald's Introduction to Commutative Algebra text has the following theorem. Let $P(M,t)$ denote the Poincare- series of $M$.

$\textbf{Theorem.}$ $\bigl(\mathsf{Hilbert-Serre}\bigr)$. $P(M,t)$ is a rational function in $t$ of the form $f(t)/\prod_{i=1}^{s} (1-t^{k_i})$, where $f(t) \in \mathbf{Z}[t]$.

This theorem appears in the section of the book called Hilbert-Functions (page 116), so one understands that it could have possibly been discovered by Hilbert.

But why is the above theorem attributed to Serre? References about when Serre was credited to the above theorem would be helpful.

Answer by Chandrasekhar for Interesting mathematical documentaries

2012-06-20T11:43:33Z

Here is a short film on Paul Halmos.

The 44-minute film contains a rare interview with Paul Halmos by Peter Renz, revealing his thoughts on mathematics, and how to teach it and write about it. Five bonus features include comments by mathematicians Robert Bekes, David Eisenbud, Jean Pedersen, and Donald Sarason about their experiences with Halmos. Interviews with Halmos by Don Albers and Halmos's own writings are included as PDF documents

Answer by Chandrasekhar for Examples of interesting false proofs

2012-06-16T16:08:04Z

Here is an interesting false proof as to how to multiply $2 \cdot 2$. Taken from this link.

$\Large\textbf{Another example}$:

Answer by Chandrasekhar for Blackbox Theorems

2012-06-14T10:32:35Z

Fundamental lemma (Langlands program) which Ngô Bảo Châu proved and got the Fields medal in 2010.

Doubt in the proof of $\lim_{s \to 1^{+}} L(s,\chi) = L(1,\chi)$

2012-02-19T14:12:55Z

Theorem 12 of the following link asserts the following:

$\textbf{Theorem.}$ Let $\chi \in X_{N}$ with $\chi \neq \epsilon$. There exists $C > 0$ such that $$L(s,\chi) = L(1,\chi) + O(s-1)$$ as $s \to 1^{+}$. In particular, $$\lim_{s \to 1^{+}} L(s,\chi) = L(1,\chi).$$

The proof is as follows: Let $1< s < 2$. From the proof of $\textbf{Theorem 9}$ we have $$L(s,\chi) - L(1,\chi) = \sum\limits_{n=1}^{\infty} a_{n} \Biggl[\biggl(\frac{1}{n^s} - \frac{1}{(n+1)^{s}}\biggr) - \biggl(\frac{1}{n} - \frac{1}{n+1}\biggr)\Biggr]$$ where the sequence $\{a_{n}\}$ is bounded. Applying the mean value theorem to the function $s \mapsto n^{-s} - (n+1)^{-s}$ gives a sequence $\{s_{n}\}$ with $1 < s_{n} < s$ and $$L(s,\chi) - L(1,\chi) = (s-1) \sum\limits_{n=1}^{\infty} a_{n} \Biggl[\frac{\log\:(n+1)}{(n+1)^{s_n}} - \frac{\log\:(n)}{n^{s_n}} \Biggr] \qquad \qquad \cdots\cdots (1)$$

I don't understand how $(1)$ is derived. When I applied the Mean-Value-Theorem to the function $f(s)=x^{-s} - (x+1)^{-s}$ on $[n,n+1]$ i get $$f'(s) = -x^{-s}\log\:(x) + (x+1)^{-s}\log\:(x+1).\hspace{40pt}(\ast)$$ So by the Mean-Value-Theorem i get an $s_{n} \in (n,n+1)$ such that $$f'(s_{n}) = -n^{-s_n}\log\:(n) + (n+1)^{-s}\log\:(n+1) - (n+1)^{-s_n}\log\:(n+1) + (n+2)^{-s_n}\log\:(n+2)$$ which gives \begin{align*} f'(s_{n}) &= \frac{\log(n+2)}{(n+2)^{s_n}} - \frac{\log\:(n)}{n^{s_n}} \\ &= \frac{f(b)-f(a)}{b-a} = (n+1)^{-s} - (n+2)^{-s} - (n+1)^{-s} + n^{-s} \\ &= \frac{1}{n^s} - \frac{1}{(n+2)^s} \end{align*}

Am I making a mistake. I am not able to see how the author get's to that step.

Are there any other nice proofs of the above theorem which you people would like to recommend?

Answer by Chandrasekhar for An example of a beautiful proof that would be accessible at the high school level?

2011-12-21T15:26:37Z

Well I personally liked the Euler's Theorem when I first saw, and I feel one can easily understand it at the High-School level.

Transcendence of PI

2010-07-31T20:05:21Z

Can anyone suggest me an ingenious proof of the transcendence of $\pi$. I have seen Lindemann's proof but it appears intricate.

Idea behind choosing $\small f(x)$ as $c^{s}x^{p-1} \frac{[\theta(x)]^{p}}{(p-1)!}$ in the proof that $\pi$ is transcendental.

2011-10-14T20:28:28Z

I am going through the article at this link, where the author proves that: "$\pi$ is $\text{transcendental}$ over $\mathbb{Q}$". Although, I understand the proof, I have some doubts.

At page $6$, the author defines a new function $f(x)$ as $$f(x) = c^{s}x^{p-1} \frac{[\theta(x)]^{p}}{(p-1)!}$$ Can anyone tell me what is the motivation behind defining $f(x)$ in this manner.

Example of a Group which has $\text{SL}_{n}(\mathbb{Z})$ as the automorphism group

2011-09-10T13:58:46Z

For the past one week, I have been trying to learn more about Automorphism groups of different groups. Very recently one of my friend asked this question to me:

What is the Autmorphism group of $(\mathbb{Q}^{\ast},\times)$. In short, what is $\text{Aut}(\mathbb{Q}^{\ast})$?

I emailed couple of friends and got the answer as:

$\text{Aut}(\mathbb{Q}^{\ast})$ is isomorphic to the automorphism group of a free abelian group of countable rank. In particular, it will contain $\text{GL}(n,\mathbb{Z})$ for all $n$.

My question would be :

Can we realize $\text{SL}_{n}(\mathbb{Z})$ to be the automorphism group of some group?
Are there groups which are which are "very difficult" to be realized as the Automorphism group of a certain group.

So suppose someone comes and asks me: Is $S_{3}$ or $\text{GL}_{2}(\mathbb{Z})$ the Automorphism group of some group, then how can i answer the question. I am particularly interested in seeing how to think for a solution.

Answer by Chandrasekhar for A good book of functional analysis

2011-08-09T03:05:33Z

Some of the good books are:

Elements of the Theory of Functions and Functional Analysis by Kolmogorov, Fomin.

Functional Analysis, by F.Riesz and Nagy.

Functional Analysis : Spectral Theory by V.Sunder. Freely available here.

Analysis now. By Gert Kjeargård Pedersen. (As suggested by Theo Buehler at $\textbf{Math.SE}$

Stein, Elias; Shakarchi, R. (2011). Functional Analysis: An Introduction to Further Topics in Analysis. Princeton University Press. ISBN 0691113874.

Functional Analysis, Sobolev Spaces and Partial Differential Equations (Universitext) by Haim Brezis.

Elementary Functional Analysis by Georgi E. Shilov.

Introductory Functional Analysis with Applications by Erwin Kreyszig.

Notes on Functional Analysis by Rajendra Bhatia. (Hindustan Book Agency.)

Functional Analysis by S.Kesavan. ( Hindustan Book Agency.)

Elementary functional analysis By Barbara D. MacCluer

Functional analysis: an introduction By Yuli Eidelman, Vitali D. Milman, Antonis Tsolomitis (AMS)

Principles of functional analysis By Martin Schechter. (AMS)

You may also want to see this thread: Problem books in Functional Analysis

$\textbf{Note.}$ The books which are written in Italics are the ones which I have read partially. The ones which are not in Italics are the ones which I have come to know (by friends, teachers) are good books in Functional Analysis. Also, I really don't know which publisher actually publishes the book in foreign edition written by Kesavan and Bhatia.

Answer by Chandrasekhar for Counterexamples in Algebra?

2011-06-18T08:25:12Z

Does $R[x] \cong S[x]$ imply $R \cong S$? ( Taken from this link. )
Here is a counterexample. Let $$R=\displaystyle\frac{\mathbb{C}[x,y,z]}{(xy(1-z^2))}, \quad \ S= \displaystyle\frac{\mathbb{C}[x,y,z]}{(x^2y(1-z^2))}$$ Then, $R$ is not isomorphic to $S$ but, $R[T]\cong S[T]$. In many variables, this is called the Zariski problem or cancellation of indeterminates and is largely open. Here is a discussion by Hochster (problem 3)
http://www.math.lsa.umich.edu/~hochster/Lip.text.pdf

Excellent Counterexamples.

Let $G$ be a group and let $\mathscr{S}(G)$ denote the group of Inner-Automorphisms of $G$.

The only isomorphism theorem I know, that connects a group to its inner-automorphism is: $$G/Z(G) \cong \mathscr{S}(G)$$ where $Z(G)$ is the center of the group. Now, if $Z(G) ={e}$ then one can see that $G \cong \mathscr{S}(G)$. What about the converse? That is if $G \cong \mathscr{S}(G)$ does it imply that $Z(G)=\{e\}$? In other word's I need to know whether there are groups with non-trivial center which are isomorphic to their group of Inner-Automorphisms. That is if $G \cong \mathscr{S}(G)$ does it imply that $Z(G)= \{e\}$?

The answer is yes there are groups with non-trivial center which are isomorphic to $\mathscr{S}(G)$. The answer is given at this link

Next one:

Does there exists a finite group $G$ and a normal subgroup $H$ of $G$ such that $|Aut(H)|>|Aut(G)|$

Arturo Magidin posed this question some time ago at MATH.SE

Question. Can we have a finite group $G$, normal subgroups $H$ and $K$ that are isomorphic as groups, $G/H$ isomorphic to $G/K$, but no $\varphi\in\mathrm{Aut}(G)$ such that $\varphi(H) = K$?

Answer was provided by Vipul Naik. Link is given here.

Question was posed by Zev Chonoles at $\textbf{MATH.SE}$

I know it is possible for a group $G$ to have normal subgroups $H, K$, such that $H\cong K$ but $G/H\not\cong G/K$, but I couldn't think of any examples with $G$ finite. What is an illustrative example?

Answer from this link: Take $G = \mathbb{Z}_4 \times \mathbb{Z}_2$, $H$ generated by $(0,1)$, $K$ generated by $(2,0)$. Then $H \cong K \cong \mathbb{Z}_2$ but $G/H \cong \mathbb{}Z_4$ while $G/K \cong \mathbb{Z}_2 \times \mathbb{Z}_2$.

Comment by Chandrasekhar

2013-05-06T11:44:49Z

@Noah: Dear Noah, the link which you mention doesn't seem to work. Could you please rectify the link

Comment by Chandrasekhar

2013-04-15T10:00:03Z

@FF: Dear FF, please read the FAQ. I think this question is not suitable for MO, you might want to try posting it a math.stackexchange.com

Comment by Chandrasekhar

2013-03-01T07:39:25Z

@Victor: Hmmm..I would rather consider $\Gamma$-functions as part of real analysis than Functional Analysis.

Comment by Chandrasekhar

2013-03-01T07:34:27Z

@Victor: Does proving $\pi$ to be transcendental involve functional analysis?

Comment by Chandrasekhar

2013-01-02T17:41:12Z

@Hollowdead: This uestion is not meant to be asked here.

Comment by Chandrasekhar

2013-01-01T14:55:57Z

For me, its Prof. Zhiwei Yun being awarded with the SASTRA-Ramanujan Prize. It's special because it's Ramanujan's 125th birth anniversary :)

Comment by Chandrasekhar

2012-12-26T08:13:57Z

A related question mathoverflow.net/questions/53736/…

Comment by Chandrasekhar

2012-10-09T12:12:52Z

@S.Carnahan: Thanks for pointing out the mistake :)

Comment by Chandrasekhar

2012-07-08T15:04:00Z

@Todd: Yes I completely agree with you. I got confused with "recent results" and recent trends and thought both were the same. I delete my answer. – Chandrasekhar

Comment by Chandrasekhar

2012-07-08T15:01:18Z

@Todd: Yes I completely agree with you. I got confused with "recent results" and recent trends and thought both were the same. I delete my answer.

Comment by Chandrasekhar

2012-07-08T03:55:58Z

@ecb: Please pose this question at math.stackexchange.com

Comment by Chandrasekhar

2012-07-08T03:55:34Z

I think this is an elementary Linear Algebra problem and hence does not fit here.

Comment by Chandrasekhar

2012-07-07T07:36:36Z

The left hand side is also the Legendre symbol $\displaystyle \biggl(\frac{n}{7}\biggr)$

Comment by Chandrasekhar

2012-07-06T08:01:57Z

I suggest trying this question at Math.stackexchange.com

Comment by Chandrasekhar

2012-07-06T02:13:18Z

@Akhil: Dear Akhil there is a typo in your statement "*; I recall that one of them did not $\text{now}$ linear algebra*"