BR Maths Corner-1

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Eric Demopheles
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Re: BR Maths Corner-1

Postby Eric Demopheles » 29 Aug 2014 15:32

matrimc wrote:EricD: I have not read Hardy's apology but what I read about it is that it is more of an explanation piece rather than the book length exposition which is What is Mathematics?. I will have to read it one of these days. Also I have downloaded the short article "the unreasonable effectiveness of mathematics" and syet to read it :(


Yes, sir. Courant and Robbins is far more mathematical compared to Hardy's meta-mathematical "Apology". The positive side is that Hardy is an easy read, all the more so if you skip C. P. Snow's preface. You can read it in one evening in one long sitting.

Courant and Robbins is an awesome book. No question about that. His choice of topics are also excellent, for the most part, except when he speaks of the stuff like isoperimetric problem or Steiner's problem or minimal surfaces(made using soap bubbles), maybe the sort of stuff that fetched Jesse Douglas a Fields Medal. These are important, no doubt, but this is quite an old and unfashionable topic now. This is the general problem with that book; it is too old and has older approaches and an "applied" taste all over.

The first two chapters are ok. As regards Chap. 3, Courant and Robbins made an excellent choice of topic; but his methods are archaic and currently the approach using Galois theory is preferred for proving impossibility of the geometric constructions. As regards chap. 4, he could have connected projective geometry with the rudiments of algebraic geometry and non-Euclidean geometry with modern hyperbolic geometry. Chap. 5 also has a good selection of topics; but he is far too outdated without mentioning homology, de Rham theorem, etc.. Chap 6. is ok except for the zeal in throwing out infinitesimal quantities, and he could have gone further into founding mathematical analysis properly.

The next chapter onwards, Courant focuses too much on the "analytic" approach, throwing out the geometrical or topological approaches. Also he does not go very deep into number theory in the first chapter; e. g., he skips quadratic reciprocity, "algebraic" number theory, etc..

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Re: BR Maths Corner-1

Postby Eric Demopheles » 29 Aug 2014 15:37

Amber G. wrote:
From: http://terrytao.wordpress.com/2014/08/12/avila-bhargava-hairer-mirzakhani/

Tao, BTW, is right now one of the expert in Random Matrix theory and has done lot of work... So the techneques devloped by M. Mehta is now finding applications in many fields (Not only nuclear physics but also string theory, number theory (pure math),


There is an Indian guy called Kannan Soundararajan at Stanford(called "Sound") who is an analytic number theorist and sometimes works on Random Matrices. He is from Chennai proper and went to Michigan and Princeton.

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Re: BR Maths Corner-1

Postby Neshant » 31 Aug 2014 04:57

what kind of name is Demopheles.

it is a greek name?

is that the name of some famous mathematician and if not who's name does it represent.

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Re: BR Maths Corner-1

Postby Eric Demopheles » 31 Aug 2014 06:32

Neshant wrote:what kind of name is Demopheles.

it is a greek name?

is that the name of some famous mathematician and if not who's name does it represent.


Demopheles = lover of people. It is used sparsely in some philosophical discourses etc., for example here:

https://ebooks.adelaide.edu.au/s/schope ... pter1.html

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Re: BR Maths Corner-1

Postby Vayutuvan » 05 Sep 2014 08:51

The following is from Prof. Peter Smith (I have proof-read one of his book chapters once for him) who writes very well on matters Formal Logic, Turing Machines, Recursive Function Theory (nevertheless a Philsophy professor) which I received in email

This is likely to be mainly of interest to (beginning?) grad students --
but please do pass on the word.

In good time for the new academic year/new term/new semester (depending
how things are chunked up in your neck of the woods) there is a shiny
new version of Teach Yourself Logic: a Study Guide, downloadable from

http://www.logicmatters.net/tyl/

(which always links to the latest version).

If you/your students don't know about this, it is a long (now 95pp.)
heavily annotated reading guide -- a PDF designed for onscreen reading
-- giving detailed advice about what to read, in what order, to get
from "baby logic", through the elements of the basic "mathematical
logic" curriculum, on to more advanced stuff. A previous version was
downloaded almost 3K times in four months: so although TYL (with its
Appendix and other associated webpages) is still work in progress, it
seems to be proving helpful, which is why I keep plugging away updating
and hopefully improving it.

So (i) do please spread the word to those who might find it useful, and
(ii) do feel free to send comments and suggestions for improvement.
Thanks, indeed, to FOMers for their previous input!

Peter S.

Dr Peter Smith, University of Cambridge
http://logicmatters.net


(kasturi: this is for you as well. Hopefully you are still reading the forum)

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Re: BR Maths Corner-1

Postby Neshant » 12 Sep 2014 08:16

a little something
for all you egg heads out there

[youtube]ppMF-WlvhoA&index[/youtube]

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Re: BR Maths Corner-1

Postby deejay » 12 Sep 2014 10:21

^^^ Neshant ji, Thank you, good fun.

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Re: BR Maths Corner-1

Postby Amber G. » 14 Sep 2014 07:49

Somehow this was not posted earlier ... reposting..
Eric Demopheles wrote:
Amber G. wrote:
From: http://terrytao.wordpress.com/2014/08/12/avila-bhargava-hairer-mirzakhani/

Tao, BTW, is right now one of the expert in Random Matrix theory and has done lot of work... So the techniques developed by M. Mehta is now finding applications in many fields (Not only nuclear physics but also string theory, number theory (pure math),


There is an Indian guy called Kannan Soundararajan at Stanford(called "Sound") who is an analytic number theorist and sometimes works on Random Matrices. He is from Chennai proper and went to Michigan and Princeton.

Kannan Soundararajan is a big name in Random Matrix theory, or Math for that matter (He represented India in IMO - when he was young - and won SASTRA Ramanujan Prize for example) . (As you say he was in Princeton - Where M. Mehta, Dyson and Wigner - did lot of work) and was student of Sarnak (also a big name).

I remembered ( The video archive is very nice) quite unforgetteful workshop in Random Matrix where Kanan, Mehta, and Sarnak were the gurus...
http://www.msri.org/realvideo/index2.html
Last edited by Amber G. on 14 Sep 2014 08:13, edited 1 time in total.

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Re: BR Maths Corner-1

Postby Amber G. » 14 Sep 2014 08:10

WRT to M. Mehta's work ..some may find this interesting ... The following are a few clips from Freeman Dyson's stories.. How highly Mehta was regarded in Princeton and how relevant his work is/was. Wigner and Dyson invited Mehta (who was then a grad student) to Princeton. (Each is a clip only a few minutes long and tells the story in Dyson's word)

Inviting Mehta to work on circular ensembles
My work with Mehta is now more relevant
Friends

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Re: BR Maths Corner-1

Postby Vayutuvan » 14 Sep 2014 08:54

Prof. Mehta's notes says there are only three possibilities - real, complex and quaternion.

I am going to sleep on the question of why only these three possibilities. The answer has to be simple as this is mentioned in passing in the third slide itself. Cannot think of why it is obvious.

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Re: BR Maths Corner-1

Postby Eric Demopheles » 14 Sep 2014 12:16

matrimc wrote:Prof. Mehta's notes says there are only three possibilities - real, complex and quaternion.

I am going to sleep on the question of why only these three possibilities. The answer has to be simple as this is mentioned in passing in the third slide itself. Cannot think of why it is obvious.


I haven't read these slides or for that matter, anything on Random matrices. Nevertheless, is this going to help : http://en.wikipedia.org/wiki/Frobenius_theorem_(real_division_algebras) ?

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Re: BR Maths Corner-1

Postby Amber G. » 14 Sep 2014 23:16

^^^ To add, he was talking about ensambles which follow "ordinary rules of algebra" etc..

(BTW, a NxN (self dual) matrix with quaternion elements is sort of same as 2Nx2N (ordinary Hermitian) matrix with complex elements.. but introduction of quaternion makes some parts mathematically "simpler")
(BTW Mehta's book is extremely good... He also has a book on Matrices .. as noted
Madan Lal Mehta's clear and concise presentation style is found in his two books: "Random Matrices", whose first edition was published in 1967 with a third edition in 2004, has become a classic in its field. Less well known, but giving a perfect illustration of the astonishing intuition and algebraic culture of Madan Lal Mehta, is his book on matrices, first published in India in 1977, and then in France in 1988.

if interested From Terence Tao's lecture notes...https://terrytao.wordpress.com/category/teaching/254a-random-matrices/

MIT's open course..
http://web.mit.edu/18.338/www/
http://ocw.mit.edu/courses/mathematics/18-996-random-matrix-theory-and-its-applications-spring-2004/

or Harvard
http://www.math.harvard.edu/~alexb/rm/

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Re: BR Maths Corner-1

Postby Vayutuvan » 15 Sep 2014 07:28

EicD and AmberG: Thanks for the links. I will follow the links to satisfy my curiosity and see if I can get the answer.
-------
AmberG: Self-dual == self-adjoint? I have come across the former term in graph theory but not in Linear Algebra where self-adjoint is used for real symmetric or complex hermitian (of which real symmetric matrices are a special case) matrices and also self-adjoint operators which give rise to self-adjoint matrices.

<deleted as the link provided by EricD "Frobenius Theorem" essentially nails my doubt>

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Re: BR Maths Corner-1

Postby Amber G. » 15 Sep 2014 08:41

^^^ Yes, (self-dual is corresponding term for Hermitian/complex for quaternion) and to clearly define...
Self dual means ..a(ij)=a*(ji)
and here a* is conjugate of a

For quaternion if a = k+x.i+y.j+z.k then a* = k-x.i-y.j-z.k
(Here , of course, i^2=j^2=k^2=ijk=-1, and k,x,y,z are real numbers)

(When a is ordinary complex number , a* has the usual meaning)

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Re: BR Maths Corner-1

Postby Eric Demopheles » 15 Sep 2014 12:49

matrimc wrote:<deleted as the link provided by EricD "Frobenius Theorem" essentially nails my doubt>


Glad to read that it was useful. Did your original question get answered, or was it something else that you asked in this newer post?

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Re: BR Maths Corner-1

Postby Rahul Mehta » 15 Sep 2014 22:32

sqrt(i) + sqrt(-i) = ?? , where i = sqrt(-1)

pls note --- there are more than 1 answers

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Re: BR Maths Corner-1

Postby Amber G. » 15 Sep 2014 23:39

Nitpicking ...
(Please peek below only if you are interested as it will give solution to RM's problem)

sqrt(i) according to standard definition is +(1/sqrt(2))(1+i)
(again sqrt(2), by standard definition is about +.1.4...

(IOW, though many people call both +1.4... and -1.4 as "square root of 2". This is correct since, both values when squared gives 2. but when one talks about "sqrt(2)" or writes the symbol √ it means (again in standard definition, which most mathematician agree with) only the positive value.)

(For more, please see, for example: http://mathworld.wolfram.com/SquareRoot.html

Most of the people, to make it clear, say that square root of i is either +sqrt(i) or -sqrt(i) where
sqrt(i)=+(1/sqrt(2))(1+i)
In that case, obviously your answer will be √ 2
(And only ONE answer)

OTOH, if you allowed BOTH the values in your definition, then there are FOUR answers.
(but, most mathematicians, when asked this question will give only one answer.. for example even google agrees ...
see here .. what is sqrt(i)+sqrt(-i) :)
Last edited by Amber G. on 16 Sep 2014 07:40, edited 1 time in total.

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Re: BR Maths Corner-1

Postby Vayutuvan » 16 Sep 2014 02:14

Eric Demopheles wrote:
matrimc wrote:<deleted as the link provided by EricD "Frobenius Theorem" essentially nails my doubt>


Glad to read that it was useful. Did your original question get answered, or was it something else that you asked in this newer post?


My original question got answered.

In that post I was trying to propose why not say a base field with 8-tuple as follows:

e_0 = 1
e_i.e_i = -1, for 1 <= i <= 7
\Pi_{i=0}^{i=7}e_i = -1

and the field is defined as \Sigma_{i=0}^{i=7}a_i.e_i, a_i is real

Following your link, that Frobenius Theorem essentially cuts off generalization of that nature. I hope I am understanding the theorem right - I gave a quick scan only of the statement and the proof.

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Re: BR Maths Corner-1

Postby Eric Demopheles » 16 Sep 2014 05:03

matrimc wrote:My original question got answered.

In that post I was trying to propose why not say a base field with 8-tuple as follows:

e_0 = 1
e_i.e_i = -1, for 1 <= i <= 7
\Pi_{i=0}^{i=7}e_i = -1

and the field is defined as \Sigma_{i=0}^{i=7}a_i.e_i, a_i is real

Following your link, that Frobenius Theorem essentially cuts off generalization of that nature. I hope I am understanding the theorem right - I gave a quick scan only of the statement and the proof.


Yes, Frobenius theorem prohibits the existence of any more such fields.

But there does exist something. Note that, when we pass from Complex numbers to quaternions, we lose the commutativity of multiplication. There is a thing called: Cayley numbers or Octonions , in which we can go up to dimension 8, but we lose the associativity of multiplication too on top of losing commutativity.
Last edited by Eric Demopheles on 16 Sep 2014 09:23, edited 2 times in total.

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Re: BR Maths Corner-1

Postby Rahul Mehta » 16 Sep 2014 06:08

Amber G ,

Just a request . Pls make font size very small. The question is trivial for people like you who take complex numbers with breakfast , lunch as well as dinner. But for those who have been in coding and left complex numbers after B Tech \ M Tech, the question is a good mind bender. Just a request .

===

next question : do there exist A B and R sp that A and B are irrational real numbers, and R is rational real number, and A ^ B = R? A and B can be equal.

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Re: BR Maths Corner-1

Postby Eric Demopheles » 16 Sep 2014 06:21

Rahul Mehta wrote:Amber G ,

Just a request . Pls make font size very small. The question is trivial for people like you who take complex numbers with breakfast , lunch as well as dinner. But for those who have been in coding and left complex numbers after B Tech \ M Tech, the question is a good mind bender. Just a request .

===

next question : do there exist A B and R sp that A and B are irrational real numbers, and R is rational real number, and A ^ B = R? A and B can be equal.


Sir,

In general, complex exponentials such as A^B are a priori undefined because of confusion about choice of logarithm. To be more precise, A^B is defined as e^(B * log A), and e^z is defined using the infinite series, and for real positive numbers, logarithm is defined as the inverse function of e^z, and extended to all complex numbers with Euler's formula : but note here that if w = log z, then, for any integer n, we have (2n * pi *i) + w is also a valid logarithm for z. So we have to choose a branch of logarithm beforehand, usually the main branch, to get into such things.

But your question makes sense for real A and B. There is indeed such an example with A = (sqrt 2)^(sqrt 2) and B = sqrt(2), where A^ B = 2.

If you demand in addition that A and B are algebraic numbers, then this is impossible, via the Gelfond-Schneider theorem.

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Re: BR Maths Corner-1

Postby Amber G. » 16 Sep 2014 07:57

Rahulji - Sorry, and thanks for pointing it out. I have edited my response.

I liked your other problem, and like to point out a beautiful part (which is not clear in Eric Demopheles's solution - how does one know that (sqrt(2)^(sqrt(2)) is irrational? )
next question : do there exist A B and R sp that A and B are irrational real numbers, and R is rational real number, and A ^ B = R? A and B can be equal


In anyway this is beautiful, peek only if you want to..

The method is Tail or Head - I win..
We know sqrt(2) is irrational.. (easy to prove.. how?)

Let A = sqrt(2)^sqrt(2)
Now if A is rational ... Nothing to prove we have found two such numbers.. (irrational^irrational = rational)

Now if A is irrational ... then A^(sqrt(2)) = 2 is rational and again we found such numbers...


So we prove the above, though without finding such numbers..:)
(many other examples are trivial ... eg e^(ln2))... but in all such cases, as pointed out before A or B (or both) is/are non-algebraic.

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Re: BR Maths Corner-1

Postby Eric Demopheles » 16 Sep 2014 08:03

Amber G. wrote:Rahulji - Sorry, and thanks for pointing it out. I have edited my response.

I liked your other problem, and like to point out a beautiful part (which is not clear in Eric Demopheles's solution - how does one know that (sqrt(2)^(sqrt(2)) is irrational? )


Yes, this is better and simpler.

But the Gelfond-Schneider theorem may still be of value here. It states in addition that such numbers are not only irrational, but also transcendental..

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Re: BR Maths Corner-1

Postby Vayutuvan » 16 Sep 2014 08:38

Eric Demopheles wrote:But there does exist something. Note that, when we pass from Complex numbers to quaternions, we lose the commutativity of multiplication. There is a thing called: Cayley numbers or Quaternions , in which we can go up to dimension 8, but we lose the associativity of multiplication too on top of losing commutativity.

Thanks, sir. That is interesting. I did come across Octonians but did not pursue at that time. Frobenius Theorem is simple and beautiful in its negative result.

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Re: BR Maths Corner-1

Postby Eric Demopheles » 16 Sep 2014 09:41

matrimc wrote:Thanks, sir. That is interesting. I did come across Octonians but did not pursue at that time. Frobenius Theorem is simple and beautiful in its negative result.


The Octonions can be obtained from Quaternions via the Cayley-Dickson construction. Another way of generalization is via Clifford Algebras.

Either way, some of the requirements of the Frobenius theorem will have to be dropped.

Both of these generalizations apparently have applications into mathematical physics. Unfortunately I am ignorant in that direction.

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Re: BR Maths Corner-1

Postby Rahul Mehta » 16 Sep 2014 10:12

Amber G. wrote:Rahulji - Sorry, and thanks for pointing it out. I have edited my response.


No worry. You are keeping this thread alive as such.

Rahul Mehta: do there exist A , B and R so that A and B are irrational real numbers, and R is rational real number, and A ^ B = R? A and B can be equal


Amber G: I liked your other problem, and like to point out a beautiful part (which is not clear in Eric Demopheles's solution - how does one know that (sqrt(2)^(sqrt(2)) is irrational? )

In anyway this is beautiful, peek only if you want to..

The method is Tail or Head - I win..
We know sqrt(2) is irrational.. ...


Dear All,

Yes. In fact, I use this example to explain the difference between EXISTENTIAL PRROFS and NON-EXISTENTIAL PROOFS aka Constructive proofs. An existential proof ends after proving that "one or more solutions exist" but gives no clue about a specific solution. Technically , they are valid , but many mathematicians dont like it. while a constructive proof actually gives a solution.

For many years, only existential proof to "do there exist A , B and R so that A and B are irrational real numbers, and R is a rational real number, and A ^ B = R? " existed. The existential proof is not so easy to derive but easy to understand. For years, none had actual A, B and R . Much later, A, B and R were found (see above posts).

====

Next question : There are 100 persons in a circle, numbered 1 to 100 clockwise. P1 has coin. The rule is --- A person with coin will pass the coin to next to next person and the next person walks out of the game. Sp P1 passes coin to P3 and P2 walks out. And P99 will pass coin to P1 and P100 will walk out. And if only two persons are left, then one without coin walks out. So which person from 1 to 100 will have coin in the end?

Those who have lost touch with maths can write code and find out. Also run code for N = 1000, 10000 . And write code to calculate f(m) for all m from 3 to N in O(N) steps only !! f(m) = index of the person who will have coin when m persons start the game.

The original problem statement is very violent. I am stating the Jain version of this problem.

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Re: BR Maths Corner-1

Postby Amber G. » 16 Sep 2014 21:06

For many years, only existential proof to "do there exist A , B and R so that A and B are irrational real numbers, and R is a rational real number, and A ^ B = R? " existed. The existential proof is not so easy to derive but easy to understand. For years, none had actual A, B and R . Much later, A, B and R were found (see above posts).


I don't know, exactly what you mean, or what is the basis of saying "for years only existential proof existed... and "none had actual A,B and R.."... and "Much later".. were found. What time period are you talking about?

Sure the method (what you call " existential proof") is fun/interesting and often is an example of such method, but if some one asks this question , most will be able to find actual values... and method can be so easy that it is hard to believe that no one knew about it "for years".

For example, e^(ln2)=2 is so obvious that it is hard to believe that NO one can find them...and one can prove that both e and ln(2) are irrational. (okay, proving ln(2) is irrational may be a little harder than proving e is irrational and may require advance math)..

I just found a few more examples where one can easily find A and B and prove, (by very elementary methods) that R is rational but both A and B are irrational.

(Challenge Problem: Can any one find this and supply an elementary (requires high school math only) proof.

===

Next question : There are 100 persons in a circle, numbered 1 to 100 clockwise. P1 has coin. The rule is --- A person with coin will pass the coin to next to next person and the next person walks out of the game. Sp P1 passes coin to P3 and P2 walks out. And P99 will pass coin to P1 and P100 will walk out. And if only two persons are left, then one without coin walks out. So which person from 1 to 100 will have coin in the end?


Interesting problem...Hint: Found a solution which is quite easy.
Mega Hint: (Assume you are a thinking binary computer).

If you want to check your answer, for 100 people, last one is 73, for 1000 it is 977

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Re: BR Maths Corner-1

Postby Vayutuvan » 18 Sep 2014 09:12

Rahul Mehta wrote:Dear All,
Yes. In fact, I use this example to explain the difference between EXISTENTIAL PRROFS and NON-EXISTENTIAL PROOFS aka Constructive proofs.


RM ji: The above is not correct. NON_EXISTENTIAL proofs are not same as Constructive. In fact the set of constructive proofs are contained in the set of existential proofs. Constructive proofs are in fact stronger proofs of existence - they give an algorithm to construct a mathematical object with the required properties.

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Re: BR Maths Corner-1

Postby Rahul Mehta » 19 Sep 2014 12:21

An ant starts its walk on a 100 cm rubber rod. The ant walks 1 cm in 1 second. And then ant will rest for a very very very very small duration of time. In that very small duration, someone pulls the rod by exactly 1 cm. The length increases uniformly. Please note - the length increases uniformly. And so the ant automatically gets moved forward by (N+1)/N cm, where N was the length of rod when the 1 cm expansion occurred.

eg 1st second . Ant walks 1cm, and then rod streches from 100 cm to 101 cm. So ant is 1.01 cm from origin

Then ant moves 1 cm again. So ant is at 2.01 cm. Now rod length increases from 101 cm to 102 cm. So ant is now situated at 2.01 * 102/101 = 2.02990099 cm from origin.

So will ant reach the end? If yes, then after how many seconds, will ant reach the end? And what will be the length of rod at that time? And how much distance ant traveled oh his own, and how much lead he got due to mere extension

Hint : use excel :mrgreen:

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Re: BR Maths Corner-1

Postby Amber G. » 20 Sep 2014 01:07

Rahulji and others - I think it is, often, more fun not to know what the method is called, just enjoy the beauty..

Rahulji - As I said, to find A and B (both irrational - and prove that they are irrational using elementary methods) such that A^B is rational (or even an integer) is not too hard.. were you able to find such an example? If so, you can give that example (and method to prove - which is fun and beautiful). ... (If there is an interest, I will post my example here)

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Re: BR Maths Corner-1

Postby Amber G. » 20 Sep 2014 01:19

^^^ Interesting ..

First, to be clear let us assume, when one talks about "pulling the rubber rod" - it means -
One end (say from where ant started) remains fixed, and the other end is pulled and the whole rod is expanded uniformly..

Let me modify this problem a little, what if the other end of the rubber rod is pulled 1meter(=100 cm) every second... (Yes it is VERY flexible rod, and the pull is 100 cm instead of Rahulji's original 1cm).. In other word the problem is following ..

An ant starts its walk on a 100 cm rubber rod. The ant walks 1 cm in 1 second. And then ant will rest for a very very very very small duration of time. In that very small duration, someone pulls the rod by exactly 100 cm. The length increases uniformly. Please note - the length increases uniformly. And so the ant automatically gets moved forward by (N+100)/N cm, where N was the length of rod when the 100 cm expansion occurred.

eg 1st second . Ant walks 1cm, and then rod stretches from 100 cm to 200 cm. So ant is 2 cm from origin.. (But now the rod is 200 cm long and the ant still has to go 198 cm)

Then ant moves 1 cm again. So ant is at 3 cm. Now rod length increases from 200 cm to 300 cm. So ant is now situated at 3 * 300/200 = 4.5 cm from origin.

So will ant reach the end?

Hint : use excel, but it may not help :) :mrgreen:


Hint: Some say answer is surprising.

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Re: BR Maths Corner-1

Postby Amber G. » 20 Sep 2014 09:09

Rahulji - BTW, your problem is some what classic in Physics.. change ant to a photon, and the rubber rod as fabric of space and consider "expanding universe" (thanks to Einstein) and we have similar to your problem..:)

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Re: BR Maths Corner-1

Postby Rahul Mehta » 20 Sep 2014 15:46

Rahul Mehta wrote:An ant starts its walk on a 100 cm rubber rod. The ant walks 1 cm in 1 second. And then ant will rest for a very very very very small duration of time. In that very small duration, someone pulls the rod by exactly 1 cm. The length increases uniformly. Please note - the length increases uniformly. And so the ant automatically gets moved forward by (N+1)/N cm, where N was the length of rod when the 1 cm expansion occurred.

eg 1st second . Ant walks 1cm, and then rod streches from 100 cm to 101 cm. So ant is 1.01 cm from origin

Then ant moves 1 cm again. So ant is at 2.01 cm. Now rod length increases from 101 cm to 102 cm. So ant is now situated at 2.01 * 102/101 = 2.02990099 cm from origin.

So will ant reach the end? If yes, then after how many seconds, will ant reach the end? And what will be the length of rod at that time? And how much distance ant traveled oh his own, and how much lead he got due to mere extension

Hint : use excel :mrgreen:


my solution


f(n) = distance between origin and ant at nth second in cm
f(0) = 0
f(1) = 1.01
f(2) = 2.01 * 102/101 = 2.02990099
etc

f(n+1) = f(n) * (100+n+1)/(100+n) + 1

f(n+1)/(100+n+1) - f(n)/(100+n) = 1/(100+n+1)
f(n)/(100+n+1) - f(n-1)/(100+n) = 1/(100+n)
...
..
f(1)/101 - f(0)/100 = 1/100

adding all, we get
f(n+1)/(100+n+1) = 1/100 + 1/101 + .... + 1/(100+n+1) = H(100+n+1) - H(99)
H(n) = harmonic sum
now ln(n) < H(n) < ln(n+1)

ant reaches end , iff f(n+1)/(100+n+1) > 1

f(n+1)/(100+n+1) = H(100+n+1) - H(99) > 1
H(100+n+1) > H(99) + 1
ln(100+n+1) > ln(99) + 1

n = approx 170


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Re: BR Maths Corner-1

Postby Rahul Mehta » 20 Sep 2014 16:27

Rahul Mehta wrote:An ant starts its walk on a 100 cm rubber rod. The ant walks 1 cm in 1 second. And then ant will rest for a very very very very small duration of time. In that very small duration, someone pulls the rod by exactly 1 cm. The length increases uniformly. Please note - the length increases uniformly. And so the ant automatically gets moved forward by (N+1)/N cm, where N was the length of rod when the 1 cm expansion occurred.

eg 1st second . Ant walks 1cm, and then rod streches from 100 cm to 101 cm. So ant is 1.01 cm from origin

Then ant moves 1 cm again. So ant is at 2.01 cm. Now rod length increases from 101 cm to 102 cm. So ant is now situated at 2.01 * 102/101 = 2.02990099 cm from origin.

So will ant reach the end? If yes, then after how many seconds, will ant reach the end? And what will be the length of rod at that time? And how much distance ant traveled oh his own, and how much lead he got due to mere extension

Hint : use excel :mrgreen:


Amber G. wrote:^^^ Interesting ..

First, to be clear let us assume, when one talks about "pulling the rubber rod" - it means -
One end (say from where ant started) remains fixed, and the other end is pulled and the whole rod is expanded uniformly..

Let me modify this problem a little, what if the other end of the rubber rod is pulled 1meter(=100 cm) every second... (Yes it is VERY flexible rod, and the pull is 100 cm instead of Rahulji's original 1cm).. In other word the problem is following ..


no ji please :)


f(n) = distance between origin and ant at nth second in cm
f(1) = 1
f(2) = 1 * 200/100 + 1 = 3
f(3) = 3 * 300/200 + 1 = 5.5
f(4) = 5.5 * 400/300 + 1 = .....
etc

f(n+1) = f(n) * (100*(n+1))/(100*n) + 1
f(n+1) = f(n) * (n+1)/n + 1
f(n+1)/(n+1) - f(n)/n = 1/(n+1)
f(n)/n - f(n-1)/(n-1) = 1/n
...
f(2)/2 - f(1) = 1/2
f(1) = 1

adding all, we get

f(n) = n*H(n)
ant has reached end iff f(n) = 100n
f(n) = nH(n) = 100n
H(n) = 100
now H(n) = ln(n)
n = e ^ 100 !!

so after (e^100 + 1) seconds , ant reaches end of bar


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Re: BR Maths Corner-1

Postby Vayutuvan » 27 Sep 2014 01:45

Some interesting stuff for Math anecdotes/apocrypha aficionado :) (Attribution: John Cook's G+ timeline)

John Cook on google+ wrote:Someone at the Heidelberg Laureate Forum asked Gerd Faltings a few minutes ago what he thought of Shinichi Mochizuki's proof of the abc conjecture.

"Shinichi Mochizuki was a student of mine. ... I read 500 pages of his notation, then I realized I would have to read another 500 pages, and I gave up."

Mochizuki's work is so idiosyncratic that nobody else can read it. Faltings said "He's a serious mathematician and not a crank" but he's also "headstrong" and unwilling to make concessions for other mathematicians :lol: .


If one goes to the Wiki page on abc conjecture, there are some other interesting facts about the history of the conjecture. Also follow the link to Akshay Venkatesh - these are the kinds of people for whom the tag genius should be used IMHO.

I was naturally interested in the computational aspects of the conjecture.

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Re: BR Maths Corner-1

Postby Amber G. » 27 Sep 2014 21:48

Rahul Mehta wrote:
Rahul Mehta wrote:An ant starts its walk on a 100 cm rubber rod. The ant walks 1 cm in 1 second. And then ant will rest for a very very very very small duration of time. In that very small duration, someone pulls the rod by exactly 1 cm. The length increases uniformly. Please note - the length increases uniformly. And so the ant automatically gets moved forward by (N+1)/N cm, where N was the length of rod when the 1 cm expansion occurred.

eg 1st second . Ant walks 1cm, and then rod streches from 100 cm to 101 cm. So ant is 1.01 cm from origin

Then ant moves 1 cm again. So ant is at 2.01 cm. Now rod length increases from 101 cm to 102 cm. So ant is now situated at 2.01 * 102/101 = 2.02990099 cm from origin.

So will ant reach the end? If yes, then after how many seconds, will ant reach the end? And what will be the length of rod at that time? And how much distance ant traveled oh his own, and how much lead he got due to mere extension

Hint : use excel :mrgreen:


Amber G. wrote:^^^ Interesting ..

First, to be clear let us assume, when one talks about "pulling the rubber rod" - it means -
One end (say from where ant started) remains fixed, and the other end is pulled and the whole rod is expanded uniformly..

Let me modify this problem a little, what if the other end of the rubber rod is pulled 1meter(=100 cm) every second... (Yes it is VERY flexible rod, and the pull is 100 cm instead of Rahulji's original 1cm).. In other word the problem is following ..


no ji please :)


f(n) = distance between origin and ant at nth second in cm
f(1) = 1
f(2) = 1 * 200/100 + 1 = 3
f(3) = 3 * 300/200 + 1 = 5.5
f(4) = 5.5 * 400/300 + 1 = .....
etc

f(n+1) = f(n) * (100*(n+1))/(100*n) + 1
f(n+1) = f(n) * (n+1)/n + 1
f(n+1)/(n+1) - f(n)/n = 1/(n+1)
f(n)/n - f(n-1)/(n-1) = 1/n
...
f(2)/2 - f(1) = 1/2
f(1) = 1

adding all, we get

f(n) = n*H(n)
ant has reached end iff f(n) = 100n
f(n) = nH(n) = 100n
H(n) = 100
now H(n) = ln(n)
n = e ^ 100 !!

so after (e^100 + 1) seconds , ant reaches end of bar



Yes, Harmonic series is divergent so no matter how fast the rubber rod expands, ant will always reach the other side. (Yes the result may be surprising to many)

But it is a very slow divergent series..!!!

For example, Rahulm's e^100 sec (see above) is a HUGE number...
so e^100 secs is billion billion times the age of universe.!!! (Ant may or may not live that long :) )
and (e^100 cm is much much bigger than diameter of known universe..)

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Re: BR Maths Corner-1

Postby Rahul Mehta » 10 Oct 2014 12:28

copy-paste

You visit a remote desert island inhabited by one hundred very friendly dragons, all of whom have green eyes. They haven’t seen a human for many centuries and are very excited about your visit. They show you around their island and tell you all about their dragon way of life (dragons can talk, of course). They seem to be quite normal, as far as dragons go, but then you find out something rather odd. They have a rule on the island which states that if a dragon ever finds out that he/she has green eyes, then at precisely midnight on the day of this discovery, he/she must relinquish all dragon powers and transform into a long-tailed sparrow. However, there are no mirrors on the island, and they never talk about eye color, so the dragons have been living in blissful ignorance throughout the ages.

Upon your departure, all the dragons get together to see you off, and in a tearful farewell you thank them for being such hospitable dragons. Then you decide to tell them something that they all already know (for each can see the colors of the eyes of the other dragons). You tell them all that at least one of them has green eyes. Then you leave, not thinking of the consequences (if any). Assuming that the dragons are (of course) infallibly logical, what happens? If something interesting does happen, what exactly is the new information that you gave the dragons?

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Re: BR Maths Corner-1

Postby Amber G. » 11 Oct 2014 07:56

^^^ I think this puzzle (a version of this puzzle) has been asked here in this dhaga. It is a classic puzzle, ( For example: Here it is home work for a Physics class:https://www.physics.harvard.edu/uploads/files/undergrad/probweek/prob2.pdf

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Re: BR Maths Corner-1

Postby Rahul Mehta » 11 Oct 2014 08:00

:oops: Its the same puzzle. I didnt know it was already asked on BRF before !!

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Re: BR Maths Corner-1

Postby Amber G. » 11 Oct 2014 08:40

^^^No problem, it is a good problem, and can be asked/discussed/commented again.


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