BR Maths Corner-1

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SaiK
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Re: BR Maths Corner-1

Post by SaiK »

I think I jumped the gun with vague remembrance of fundamentals. I am going to take time to get more info. thx
Vayutuvan
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Re: BR Maths Corner-1

Post by Vayutuvan »

Saik I am talking about a simple answer for "is it singular?"
Rank question is a little involved (as far as I know)
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Re: BR Maths Corner-1

Post by shaardula »

rank is intimately tied to singularity sai. if the bases you have chosen cannot sufficiently model the space in which you have formulated your problem, you have a problem of non identifiability.you can go on adding and multiplying to find new bases.

imho, half the papers written on these things are about identifying the spaces which are mapped by the concerned bases.
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Re: BR Maths Corner-1

Post by Vayutuvan »

shaardula, for this particular matrix, it is easy to show that it is singular. So, it does not have full rank. I would like to point you to a small monograph. My guess is that you, being s statistician, would appreciate it. Others too - it is a nice small amusing (replacing with a better adjective) delightful book.

Random Walks and Electric Networks (Carus Mathematical Monographs) [Hardcover]
Peter G. Doyle (Author), Laurie Snell (Author)


It was out of print and then came into print when I bought it. Looks like it is out of print again and selling used for $60 (a little too high).
Last edited by Vayutuvan on 31 May 2013 09:24, edited 1 time in total.
SaiK
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Re: BR Maths Corner-1

Post by SaiK »

sorry matrimc. the answer I get is yes, but I was wanting to understand more as to why it is so.

regarding markov, question: regarding the state transition to which probability is attributed, the probability would be always one or probability given the condition to trigger, if the event is known ahead right? sorry for asking without reading much about it.
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Re: BR Maths Corner-1

Post by Vayutuvan »

saik, to prove singular try to find a simple dependency between rows (or columns - it is a symmetric matrix).

Rank question can be answered two ways after removing the singularity

1. Application of Gershgorin's Circle Theorem (this matrix has a property such that this theorem is applicable).

2. I haven't tried this out yet but think it would work out - show that the quadratic form x^t A x (where A is the matrix in question) is positive (for any general vector x) which implies that all eigenvalues are positive leading to resolution of the rank question.

But be careful - assume that the graph in question is connected. You can assume this without loss of generality.
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Re: BR Maths Corner-1

Post by Amber G. »

Two years ago, I posted a message about a way to earn $100,000 by solving a seemingly "easy" math problem. :)

Here is the link to the post for details: http://forums.bharat-rakshak.com/viewto ... l#p1112753
AmberG. wrote:There is $100,000 prize, not to mention fame, (See:http://www.math.unt.edu/~mauldin/beal.html to either find a counter example (or prove it is impossible) for an equation of the kind
x^a+y^b = z^c (a,b,c >2 and certain restrictions on x,y,z - see the above link.

So, Vinaji ( :) ) and others...try to see if that method works...
(There is a lot of interesting discussion following those pages)

Well that prize is now $1000,000 dollars according to recent time story:
Solve This Math Problem, Win a Million Bucks
Want to make a quick million? All you have to do is figure out a little math problem that goes like this: A^x + B^y = C^z. Simple algebra, right?


Oh how deceptively innocuous a few elementary variables can seem. You’re actually looking at something inspired by one of the great mysteries of mathematics, known as Fermat’s Last Theorem and named after the 17th century French lawyer and mathematician Pierre de Fermat. Fermat came up with his own theorem back in 1637, scribbling it in the margins of his copy of the Greek text Arithmetica by Diophantus and surmising that — put your math caps on and buckle up — if n were an integer greater than 2, then the equation Xn + Yn = Zn has no positive integral solutions. The note was discovered after Fermat’s death, and it took over 350 years and untold failed attempts by others for someone to prove the theorem. In 1995, British mathematician Andrew Wiles, who’d been fascinated with the theorem since he was a child, finally got the job done, having puzzled over it in secret for roughly six years.

That’s where Texas billionaire D. Andrew Beal comes in. In 1993, he posited a closely related number theory problem hence dubbed Beal’s Conjecture (that first A-B-C equation above), where the only solution is possible when A, B and C have a common numerical factor and the exponents x, y and z are greater than 2. Beal’s been trying to solve his theorem ever since, reports ABC News, offering cash rewards in steadily increasing amounts — $5,000 in 1997, $100,000 in 2000 – to anyone with the knack to get the job done.

The prize total in 2013: $1 million, which is either a sign of Beal’s magnanimity or his skepticism that it’s actually possible. (Since Beal is worth a reported $8 billion, there’s little need to worry about whether he’ll pay the winner.)

It’s apparently not just about the money for Beal, either: In a statement, he said “I’d like to inspire young people to pursue math and science. Increasing the prize is a good way to draw attention to mathematics generally … I hope many more young people will find themselves drawn into the wonderful world of mathematics.”
For those who want to try it..the challenge is sponsored by the American Mathematical Society; the rules, including where to submit proposed solutions by email or snail mail, are available at:
http://ns3.ams.org/bealprize.html

(You may publish the solution in brf's dhaga, of course..:))

PS: To prove some particular cases of Fermat's last theorem, or Beal's conjuncture etc are not very hard and very very interesting...(I have shown some beautiful ways to prove FLT for some specific cases....

An old problem ... try your hand.. to find all solutions for 3^x+4^y=5^z (x,y,z are integers)..
(or prove that there are no solutions when x,y,z > 2 etc)
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Re: BR Maths Corner-1

Post by Vayutuvan »

AmberG ji, thanks for cautioning about the $1M prize.

Following is a quote from a popular book on Four Color Theorem (FCT) - worth pondering over by people before proceeding to solve the problem on the spot. :wink:

Four Colors Suffice: How the Map Problem Was Solved [Paperback]
Robin Wilson (Author)

Chapter 8. Crossing the Atlantic

... Kempe's solution had been shown to be faulty ...

The feeling was also beginning to emerge in some quarters that a solution to the four-colour problem had not been forthcoming because no really good mathematician had worked on it. Indeed a story is told about the distinguished German number-theorist Hermann Minkowski in the first decade of the twentieth century. While lecturing on topology at Gottingen University, he mentioned the four-color problem:


'This theorem has not yet been proved, but that is because only mathematicians of the third rank have occupied themselves with it', Minkowski announced to the class in a rare burst of arrogance. 'I believe I can prove it.'

He began to work out his demonstration on the spot. By the end of the hour he had not finished. The project was carried over to the next meeting of the class. Several weeks passed this way. Finally one rainy morning, Minkowski entered the lecture hall, followed by a crash of thunder. At the rostrum, he turned towards the class with a deeply serious expression on his face.

'Heaven is angered by my arrogance', he announced. 'My proof of the FCT is also defective.' He then took up the lecture on topology at the point where he had dropped it several weeks before.
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Re: BR Maths Corner-1

Post by Vayutuvan »

SaiK wrote:sorry matrimc. the answer I get is yes, but I was wanting to understand more as to why it is so. [1]

regarding markov, question: regarding the state transition to which probability is attributed, the probability would be always one or probability given the condition to trigger, if the event is known ahead right? sorry for asking without reading much about it. [2]
SaiIK, I am splitting the asnwer into two parts.

[1] If all the rows (or columns) are added together the result is a zero. In other words, one of the rows (columns resly.) is a linear combination of rest of the rows (columns resly.).

How can one remove this linear dependency? Drop a row but to keep it symmetric (has to be square, right?) the corresponding column has to be dropped. This results in a square matrix with full rank which can be proved either by applying Gerhsgorin's Theorem or alternatively reasoning based on physical experimentation. For the latter part, please do read the reference I have given to shaardula.

[2] I am not sure I understand your question. Could you please formulate it a little better?

By the way, there is a straight forward relation between Markov chains (symmetric irreducible) and electrical networks of conductances. Again reading the book will give you some insights.

Regards
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Re: BR Maths Corner-1

Post by Vayutuvan »

shaardula, please do post your questions regarding rank etc. It would be interesting to get another POV.

regards
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Re: BR Maths Corner-1

Post by Amber G. »

matrimc wrote:AmberG ji, thanks for cautioning about the $1M prize.

Following is a quote from a popular book on Four Color Theorem (FCT) - worth pondering over by people before proceeding to solve the problem on the spot. :wink:

Four Colors Suffice: How the Map Problem Was Solved [Paperback]
Robin Wilson (Author)
Thanks! Interesting read. (I had a somewhat similar experience with FCT, I may share that story in other post)

Let me add a story abut a 6 year old Isabella who said something to the effect : "I'm good at math. Can I do that and win the prize? Then I can buy that toy I wanted."..I think she (probably with the help of her father) has posted a nice computer program to look for counter examples for the Beal's problem.
Here is the complete python program..They said that anyone can use it, small request, if you indeed win the prize please share some money so that Isabella can buy her toy.
"""Print counterexamples to Beal's conjecture.
That is, find positive integers x,m,y,n,z,r such that:
x^m + y^n = z^r and m,n,r > 2 and x,y,z co-prime (pairwise no common factor).

ALGORITHM: Initialize the variables table, pow, bases, powers such that:
pow[z][r] = z**r
table.get(sum) = r if there is a z such that z**r = sum.
bases = [1, 2, ... max_base]
powers = [3, 4, ... max_power]
Then enumerate x,y,m,n, and do table.get(pow[x][m]+pow[y][n]). If we get
something back, report it as a result. We consider all values of x,y,z
in bases, and all values of m,n,r in powers.
"""

def beal(max_base, max_power):
bases, powers, table, pow = initial_data(max_base, max_power)
for x in bases:
powx = pow[x]
if x % 1000 == 0: print 'x =', x
for y in bases:
if y > x or gcd(x,y) > 1: continue
powy = pow[y]
for m in powers:
xm = powx[m]
for n in powers:
sum = xm + powy[n]
r = table.get(sum)
if r: report(x, m, y, n, nth_root(sum, r), r)

def initial_data(max_base, max_power):
bases = range(1, max_base+1)
powers = range(3, max_power+1)
table = {}
pow = [None] * (max_base+1)
for z in bases:
pow[z] = [None] * (max_power+1)
for r in powers:
zr = long(z) ** r
pow[z][r] = zr
table[zr] = r
print 'initialized %d table elements' % len(table)
return bases, powers, table, pow

def report(x, m, y, n, z, r):
x, y, z = map(long, (x, y, z))
assert min(x, y, z) > 0 and min(m, n, r) > 2
assert x ** m + y ** n == z ** r
print '%d ^ %d + %d ^ %d = %d ^ %d = %s' % \
( x, m, y, n, z, r, z**r)
if gcd(x,y) == gcd(x,z) == gcd(y, z) == 1:
raise 'a ruckus: SOLUTION!!'

def gcd(x, y):
while x:
x, y = y % x, x
return y

def nth_root(base, n):
return long(round(base ** (1.0/n)))

import time

def timer(b, p):
start = time.clock()
beal(b, p)
secs = time.clock() - start
return {'secs': secs, 'mins': secs/60, 'hrs': secs/60/60}
Vayutuvan
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Re: BR Maths Corner-1

Post by Vayutuvan »

Calling Singha and all other runners out there - an interesting photo of Turing.

Image
Alan Turing was also a highly accomplished runner, competing with the Walton Athletic Club in Surrey, UK. According to a blog post by Lubor Ptacek, he finished fourth in the Amateur Athletic Association Championships marathon with a time of 2 hours 46 minutes and 3 secondseven faster than Lance Armstrong’s time of 2:46:42 in the 2007 New York Marathon. Here, Turing is running a race at the National Physical Laboratory Sports Day in December 1946. (Photo courtesy of the National Physical Laboratory)
Oh, forgot to add the link to the article at Elesiever

New book spotlights Alan Turing, WWII code-breaker and ‘father of computer science’

This volume is an extract of important papers from the 4-volume Turing's collected works edited by Barry Cooper and J. van Leeuwen. Last year was Turing's birth centenary celebrations. As I understand it, this book is one of the outgrowths.
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Re: BR Maths Corner-1

Post by SaiK »

thanks for the gurus to keep this dhaaga alive.. I need to get back to my discrete times from my busy schedule. Let me plan about an hour this Sunday. I may have more questions after that.

btw: http://en.wikipedia.org/wiki/Alan_Turing
not a single desh edu tributes him. is that because, he is firang?
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Re: BR Maths Corner-1

Post by SaiK »

Vedic mathematics and its relevance
http://www.speakingtree.in/spiritual-sl ... ance/14944

"The world owes most to India in the realm of mathematics, which was developed in the Gupta period to a stage more advanced than that reached by any other nation of antiquity. The success of Indian mathematics was mainly due to the fact that Indians had a clear conception of the abstract number as distinct from the numerical quantity of objects or spatial extension." -A.L. Basham, noted Australian historian

In the year 1911, noted Vedic scholar Bharati Krishna Tirthaji, a scholar of history, mathematics, Sanskrit and philosophy. and had made a comprehensive study of the four Vedas, Rigveda, Atharvaveda, Yajurveda, and Samaveda. Studying these texts for years, he was able to reconstruct a series of mathematical formulae, called Sutras. He continued the study for 7 years and compiled them together in a major single volume, which got published five years after his death, Vedic Mathematics, in the year 1965. Repeated attempts of a few British mathematicians finally bore results duting the years 1981 to 1987, when Vedic mathematics started being taken seriously. A number of London and Indian schools took it up as subjects, and intellectuals started delving more time and energy into the same.

So, what do you do when you are to multiply, say, 78X92? You jot down the numbers hurrily on a notepad, do a quick 8X2, write down 6, carry over 1...blah blah blah...something like this 78 92 156 702X 7176 Because thats what was taught in schools. Now, what Vedic Maths asks you to do is, think beyond the obvious(am I sounding like some management Guru?) The nearest 10s base to 78 (and 92) is 100 Step 1) We subtract 100 from both numbers 78 - 100 = -22 92 - 100 = -8 Step2) Add 78 and 92 and subtract 100 from it 78+92-100=70 Step3) Multiply -22 and -8 -22 X -8 = 176 Step4) Carry the 1 from 176 and add it to the result obtained in Step 2 70+1 = 71 Step5)Concatenate the two 71 and 76=7176 And this can be done mentally, which is almost impossible in the first approach. A similar approach could be taken up for two digits or three digit numbers

What would you do if someone asks you to calculate the area of a 42m X 42 m land? Again take out your notepad and scribble? Or maybe take the above approach and solve it out much quicker? Or maybe take another better and much interesting approach? Step 1)Round of 42 to the nearest 10s, ie 40. Step 2)Now add the remaining amount, ie 2 to 42 ie 44 Step 3) Multiply 40 and 44(much easier), ie 1760 Step4)Now take the number from step 2, ie 2 and square it, ie 4. Add the number to result of Step 3, ie 1764

What would you do if someone asks you to find the volume of a 25X25X25 box? Consider ab=25(where a=2, b=5) Step 1)Cube a, is 8 Step 2)Divide b by a, ie 5/2=2.5 Step 3) Now write four subsequent products of a with the result of Step 2 and prepare a matrix 8 20 50 125 Step 4) Double the number at 2nd and 3rd place and add them at their subsequent places. Carry over the leftmost digit if the sum is of two digits 8 20 50 125 40 100 8 60 150 125 8 60 162 5 8 76 2 5 15 6 2 5 The answer is 15625

If the base 100 method(the first in this blog), still looks tougher to compute mentally, or if you want to explore some more of it, here is another Consider you have to multiply 12X34. Prepare a matrix of the two numbers. 1 2 3 4 Step1) Multiply ac, ie 1X3=3 Step 2)Multiply bd, ie 2X4=8 Step 3)Add ad and bc, ie 4 and 6=10 Step 4)Prepare another matrix mentally with the results of Step1, 3 and 2 respetively 3 10 8 Step 5) Carry over any extra digit to the left 4 0 8 The Answer is 408

Often people switch to Google in the blank of an eye when asked to fill up a form with weight as pounds(when they are more familiar with kilo). Or they would switch to their cellphones. Well, its much easier to do it mentally, and a better way to keep the mind healthy. If we have to convert 85 kilo to pounds, Step 1) Double the kilos, 85X2=170 Step 2) Divide the answer by 10, ie 170/10=17 Step 3) Add the two numbers, 170+17=187, and thats the answer.(to the nearest integer)

Whats the fastest way to calculate if the current time is 4:15, and someone asks you to calculate the time 2 hours 55 minutes later? You first add the minutes and then the hours, which is quite simple as such, but then what would you say about this Step1) Add 415 and 255, ie 670 Step 2) Add 40 to the answer, ie 670+40=710,and 7:10 is the answer

Two concepts highly useful in almost every practical application of maths in daily lives and even for astronomical calculations. Infinity finds its first usage in the Yajurveda. Vedic men had several terms to describe what is infinity, ananta, purnam, asamkhya being few of them. Consider the following shloka out of the Yajur Veda, पूर्णमदः पूर्णमिदम पूर्णत पूर्णमुदच्यते पूर्णस्य पूर्णमादाय पुर्नामेवावासिश्यते From infinity is born infinity. When infinity is taken out of infinity, only infinity is left. Till then, the Greeks used to distinguish between numbers with positional systems, like 27,207,270 would be represented as 27, 2 7, 27 . this system did not provide much flexibility to them to write large numbers with regular instances of 0. Similarly, Romans didnt have the concept of 0, and had to write numbers like 101,000 as 101MMM. AtharvaVeda described how the vale of a single digit number increases by 10 by writing 0 in front of it.

A fact iterated in the movie Namaste London, Trigonometry even got its name from Vedic mathematics, the word being Tri(three)-kon(angle)-mati(perimeter). Though, now the concepts have already been discovered and have been in use since years, The circular nature of Sin, Cos and Tan finds its mention in the Vedas. The fact that Sinx/Cosx = Tanx finds its mention in the Vedas.

Currently, as you are reading all of this, researches are being carried on, and undertaken, including the effects of Vedic mathematics on children. Major research is being done on how to develop more powerful and simpler application of Vedic mathematics in geometry, calculus, etc. What makes Vedic mathematics different from other schools and branches of theoretical and practical mathematics, is the flexibility of Vedic mathematics in providing them freedom with creating their own methods and breaking the "correct answer" jinx. Vedic Mathematics provides us with quick and fast one-line formulae which speeds up mental calculation. It is a mental tool which not only helps a person solve complex problems quickly, but with repeated practice, also helps a person concentrate better.

Its absolute pity, that in spite of it being such a great marvel as "Vedic Mathematics", some elitist, from within the Indian sub-continent itself, claim it to be a farce. They say that they dont find any such reference in Vedas, and that it was a result of imagination of Tirthaji himself. The Vedas, primarily, have been known to reveal the true knowledge and meaning of the verses, the hidden secrets to those who are worthy. While this may appear to be a line straight out of a Dan Brown book, it is a mighty known fact that Vedas are written in a cryptic language and a mere translation would mean nothing or may produce some relevant results. The Vedas are believed not to reveal the true meaning to the reader, but rather help the reader find certain answers, which others wont be able to see or even validate. The irony is, that while the western world is embracing and opening up itself to the charms and secrets of Vedic Mathematics, there are Indians themselves, who claim the Vedic Mathematics to be a lie.

Realists, as always, would claim that Vedas do not actually teach mathematics on the face of it, but the fact is Vedic mathematics is gradually making a stronghold in the education system of the entire world and is soon becoming a force to reckon.
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Re: BR Maths Corner-1

Post by ArmenT »

Amber G. wrote: Let me add a story abut a 6 year old Isabella who said something to the effect : "I'm good at math. Can I do that and win the prize? Then I can buy that toy I wanted."..I think she (probably with the help of her father) has posted a nice computer program to look for counter examples for the Beal's problem.
Here is the complete python program..They said that anyone can use it, small request, if you indeed win the prize please share some money so that Isabella can buy her toy.
...snip...
You might want to post python code with the

Code: Select all

 and [[i]/[/i]code] tags instead of [quote] tags. For one thing, the quote tag destroys indentation, which is very vital to python code.

Her dad (who wrote all the code) is quite well paid by Google and is a well known author in the field of AI research. He can easily afford to buy Isabella any toy she wants :).
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Re: BR Maths Corner-1

Post by Amber G. »

ArmenT thanks.. the link for the code is, for example here:http://norvig.com/beal.html

Meanwhile, if some one likes to try their hand on computer program, try to find (counter) example for
x^4+y^4+z^4=w^4
(Hint: example exists)
(Many thought, that there are no solution for the above kind of eq for n>3)
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Re: BR Maths Corner-1

Post by Amber G. »

Just for fun ...

I will post, once in a while, some simple, yet challenging, problems which require elementary (roughly high-school math) math background. Use of computers, Vedic math, google is okay but not necessary.

Problem:
Narad said to Bhaskara:" A number is 'pavitra' if
1. it is prime. (Let us call it p)
2. If you take half and round it up it is a perfect square. (IOW (p+1)/2 = a^2)
3. If you take half of its square and round it up, it is still a perfect square. (IOW, (p^2+1)/2 = b^2)

O Bhaskara! can you find me pavitra number(s)? ( all, or any, or as many as you can)
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Re: BR Maths Corner-1

Post by Amber G. »

The following problem has been posted for a while, and since no solution has been posted, I am posting my solution which is, in my opinion, very elegant so hope people like it.

Math is beautiful because sometimes, if you think in the right direction, some problems are very easy to solve.

Here is the problem reproduced.. (Please check out some posts after the problem (http://forums.bharat-rakshak.com/viewto ... 9#p1436969) and diagrams to understand what is meant by "line" "cut" and "domino" etc..)
Amber G. wrote:Just for fun, if some one is interested, here is some what harder (or may be very simple :) if you hit the right idea) problem (from a Russian Math competition, I was told)

You have a 6x6 board (instead of 8x8) and you want to cover it entirely. (with 18 dominoes (of the 2 x 1 size) as described above)

Prove that you can always find a straight line which divides the board into two (not necessarily equal) parts, such that no domino is cut by that line.
Solution - As said, it is simple
1. There are 10 "lines" which we are interested in this 6x6 board. (5 horizontal, 5 vertical)
2. Each domino is "cut" by one (exactly one) line.
3. If a line cuts any domino, it must cut at least two. (it must cut even number). (Why - both sides of the cut are rectangles 6x(something). So it must contain even number of squares. If only one domino is cut - you will have odd number of squares in 6xn rectangle)
4. (From 1 - there are 10 lines). So in order that every line has a "cut" (or at least two cuts, as shown above, as single one cut is not possible) - you need at least 2x10 = 20 dominoes.

But you have only 18 dominoes...

Hence .. impossible.

QED!!!!

(Crux of logic is simple arithmetic - no complex graph theory or combinatorial analysis is used here :))

(BTW: for larger boards (8x8 or larger) I was able to find solution where one can cover the board in the way described in the problem)
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Re: BR Maths Corner-1

Post by Vayutuvan »

Excellent.

I have another question to think about.

Does this property hold only for a 6x6 board? So the question would be for what other values of n other than 3 does a 2n x 2n board does admit a domino non-intersecting cut?

No, I do not have a solution may thinking on the lines of either n is a prime or some multiple of 3.
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Re: BR Maths Corner-1

Post by ArmenT »

Amber G. wrote:Meanwhile, if some one likes to try their hand on computer program, try to find (counter) example for
x^4+y^4+z^4=w^4
(Hint: example exists)
(Many thought, that there are no solution for the above kind of eq for n>3)
Just saw this. I know that there are multiple solutions (hint: counter example was given by a mathematician with the initials N.E. and he proposed a technique to find infinite solutions to the above equation) and also that the original equation was proposed by Euler. The mathematician N.E. was briefly mentioned in Simon Singh's book, Fermat's Enigma: The Epic Quest to Solve the World's Greatest Mathematical Problem. It was during the time when Andrew Wiles was trying to solve Fermat's last equation and had hit some significant difficulties in the proof and was wondering if he was on the right path or was his proof hitting difficulties because Fermat was wrong after all. Apparently, someone in the math community circulated an e-mail on April Fool's day that N.E. had found a counter-example to Fermat's last theorem (N.E. had become famous when he found the counter example to the above Euler equation a few months earlier) and the news nearly gave Andrew Wiles a heart attack before he realized that it was an April Fool's day joke.

I'll post the answer if no one else answers this in a while.
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Re: BR Maths Corner-1

Post by Amber G. »

matrimc wrote:Excellent.

I have another question to think about.

Does this property hold only for a 6x6 board? So the question would be for what other values of n other than 3 does a 2n x 2n board does admit a domino non-intersecting cut?

No, I do not have a solution may thinking on the lines of either n is a prime or some multiple of 3.
Any square board (2nx2n) greater than 6, can be tiled to have non-intersecting cut. No solution for 6x6 or smaller board.

A rectangular board of the type (mxn) has interesting solutions, specially when m is even and n is odd. (case when n is odd is different) For example one can find a solution for a 6x5 board.

Added later: for 2n x 2n board to have the solution we need...
Number of dominoes = (1/2)(2n)^2 = 2n^2
Number of lines = 2 (2n-1)

So to have a solution we need:
2n^2 >= 2*2(2n-1)
or n^2 >= 4n -2
or (n-2)^2 >=2


or n>3
Last edited by Amber G. on 01 Jul 2013 03:23, edited 2 times in total.
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Re: BR Maths Corner-1

Post by Amber G. »

ArmenT wrote:
Amber G. wrote:Meanwhile, if some one likes to try their hand on computer program, try to find (counter) example for
x^4+y^4+z^4=w^4
(Hint: example exists)
(Many thought, that there are no solution for the above kind of eq for n>3)
Just saw this. I know that there are multiple solutions (hint: counter example was given by a mathematician with the initials N.E. and he proposed a technique to find infinite solutions to the above equation) and also that the original equation was proposed by Euler. The mathematician N.E. was briefly mentioned in Simon Singh's book, Fermat's Enigma: The Epic Quest to Solve the World's Greatest Mathematical Problem. It was during the time when Andrew Wiles was trying to solve Fermat's last equation and had hit some significant difficulties in the proof and was wondering if he was on the right path or was his proof hitting difficulties because Fermat was wrong after all. Apparently, someone in the math community circulated an e-mail on April Fool's day that N.E. had found a counter-example to Fermat's last theorem (N.E. had become famous when he found the counter example to the above Euler equation a few months earlier) and the news nearly gave Andrew Wiles a heart attack before he realized that it was an April Fool's day joke.

I'll post the answer if no one else answers this in a while.
Please post the answers... when you get time..

Very interesting. (Now I am very interested in reading Simon Singh's book.!)
N.E (I assume Elkies), from what I know used an elliptic curve with a lucky rational point...but one smaller (smallest) example was found (or can be found) by just brute-force from computer ! (all numbers are less than a million)..
***
Interestingly counter example of Euler's equation (as you called it) for 3, is simple...

3^3+4^3+5^3 = 6^3

(This also fascinated Ramanujuan quite a bit... His famous "taxi cab" story deals with this...

1729=10^3+9^3 = 12^3 + 1^3 which is similar to above (just extending it to negative numbers
we have 10^3+9^3+(-1)^3 = 12^3)

Most books which tells the taxi cab story do not tell how Ramanujan knew to generate all such numbers, but the trick (or method) to generate all such numbers are not hard to understand.
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Re: BR Maths Corner-1

Post by Vayutuvan »

Amber G. wrote:...
AmberG ji, yes. As soon as I posted (on an iPod :(), I left for work and on the way reached the same conclusion as above. We need n > 0, integer and 4n -2 > n^2, which is true for n = 1, 2, or 3
(only now am able to get to a keyboard at home).
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Re: BR Maths Corner-1

Post by Amber G. »

ArmenT wrote: Just saw this. I know that there are multiple solutions (hint: counter example was given by a mathematician with the initials N.E. and he proposed a technique to find infinite solutions to the above equation) and also that the original equation was proposed by Euler. The mathematician N.E. was briefly mentioned in Simon Singh's book, Fermat's Enigma: The Epic Quest to Solve the World's Greatest Mathematical Problem. It was during the time when Andrew Wiles was trying to solve Fermat's last equation and had hit some significant difficulties in the proof and was wondering if he was on the right path or was his proof hitting difficulties because Fermat was wrong after all. Apparently, someone in the math community circulated an e-mail on April Fool's day that N.E. had found a counter-example to Fermat's last theorem (N.E. had become famous when he found the counter example to the above Euler equation a few months earlier) and the news nearly gave Andrew Wiles a heart attack before he realized that it was an April Fool's day joke.
Just to add to above, yes N.E. (Noam Elkies)'s counter example to Euler equation is quite note worthy but Noam was quite famous before that. At the age of 14 (!!!) he got a gold medal (and a perfect score! (youngest to ever have done so!) in International Math Olympiad. He was also a Putnam fellow (really very big deal) at age of 16 (again perhaps youngest) and PhD at the age of 20 from Harvard. (according to wiki) and youngest full professor later, in the history of Harvard!
(The timing of the result ArmenT talks about - NE was just a very young grad student/post_doc)

(BTW, he is also extremely good at Chess, and Music)

For those interested, here is a popular article about this in 2008 Science Daily.
http://www.sciencedaily.com/releases/20 ... 145039.htm

****
For those, who are little more interested and are familiar with some advance math here is how he tackled the problem... (ArmenT - please add, I am curious how Singh's book explains this..)

I am basically giving the simplified portion, details can be seen in any good reference (just google -"Elkies" + Euler + sum of powers etc..)

A particular case of Elkies' solution can be
reduced to the simple identity:

(85v^2+484v-313)^4 + (68v^2-586v+10)^4 + (2u)^4 = (357v^2-204v+363)^4

where,

-22030-28849v+56158v^2-36941v^3+31790v^4 = (+/-)u^2

This is, of course, an elliptic curve, with one solution (for the - case)
as v_1 = -31/467... (okay don't ask me here how he got this value) From this rational point, one can then find an infinite number of v_i. (This part is hard and require some advance math, but is a standard technique)

From this he found,
2682440^4 + 15365639^4 + 18796760^4 = 20615673^4



(Note that the smallest known solution found by computer
searching is: 95800^4 + 217519^4 + 414560^4 = 422481^4)
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Re: BR Maths Corner-1

Post by Amber G. »

I am giving one solution to my own problem (posted below for convinience( referemce) as I have not seen any solution here. The answer is very simple and illustrate the concept of "invariant property" .. hence the solution here.

The problem.
Amber G. wrote: Armen T - Thanks for mentioning Singh's book, I have not read it, will do when I get a chance.
Invariant concept is very powerful in Physics too.. Many difficult problems, (including everyday problems) can be solved this way....The trick is to find the right invariant property..

For example, if our dominoes are 4x1 (instead of 2x1) can one cover a 6x10 board with them? (The board is to be fully covered)
Answer: Color the board with colors Row 1 = (1,2,1,2,1,2..) Row 2 = (3,4,3,4,3,4,..), All odd rows same as row 1, all even rows - same as row 2.. (see figure below)..

(There are 60 total squares, 15 each of one color)

Rest is simple, since each (4x1) dominoes has any color even number of times.. It is impossible to cover 15 squares of any one color.

Image
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Re: BR Maths Corner-1

Post by SaiK »

contd.. from the gizmo dhaaga:

so, mathrimc guru, on the DAG : if the nodes represents the various CS in N threads, would not the DAG itself be a replicated entity for each thread?
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Re: BR Maths Corner-1

Post by Vayutuvan »

SaiK garu, I meant the rank question. and I need to post on the tiling problem(s) and NP-complete etc.

Not the CS/threads stuff.
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Re: BR Maths Corner-1

Post by SaiK »

looking forward to learn.
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Re: BR Maths Corner-1

Post by Nilesh Oak »

All Math Gurujan,

Need your help....
----------
I was reading about complex numbers (in a context of my research- Astronomy) and found that on Wikipedia it lists Gerolamo Cardano as its inventor.

http://en.wikipedia.org/wiki/Gerolamo_Cardano

The Wikipedia entry o Gerolamo Cardano says that complex numbers came into the picture (need for them) in solving certain trigonometric functions/solutions etc.

This entry (on Gerolamo Cardano) as it related to complex numbers.
------------
In his exposition, he acknowledged the existence of what are now called imaginary numbers, although he did not understand their properties (described for the first time by his Italian contemporary Rafael Bombelli, although mathematical field theory was developed centuries later).
--
AmberJi, Matrimc and others..

Do we any evidence for existence, usage or inkling of complex numbers in ancient Indian mathematics/astronomy/trigonometry..etc., anytime before say 1500 CE ?

Appreciate your help,
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Re: BR Maths Corner-1

Post by ArmenT »

Nilesh Oak wrote:All Math Gurujan,

Need your help....
----------
I was reading about complex numbers (in a context of my research- Astronomy) and found that on Wikipedia it lists Gerolamo Cardano as its inventor.

http://en.wikipedia.org/wiki/Gerolamo_Cardano

The Wikipedia entry o Gerolamo Cardano says that complex numbers came into the picture (need for them) in solving certain trigonometric functions/solutions etc.

This entry (on Gerolamo Cardano) as it related to complex numbers.
------------
In his exposition, he acknowledged the existence of what are now called imaginary numbers, although he did not understand their properties (described for the first time by his Italian contemporary Rafael Bombelli, although mathematical field theory was developed centuries later).
--
AmberJi, Matrimc and others..

Do we any evidence for existence, usage or inkling of complex numbers in ancient Indian mathematics/astronomy/trigonometry..etc., anytime before say 1500 CE ?

Appreciate your help,
Indians did publish treatises on negative numbers, however I don't think anyone dealt with the concept of complex numbers. The Bakshali manuscript from 400 AD shows -ve numbers and Brahmagupta also wrote about it in 620 AD and also established the rules of addition/subtraction/multiplication and division for +ve and -ve number combinations. On the question of square roots, he touches on it briefly and IIRC says that the square root of -ve numbers is unanswered.

Bhaskaracharya, in his Bijaganita (around 1150 AD) was the first text book to recognize that a square root can have two numbers (+ve and -ve), however he also does not really deal with square roots of -ve numbers, only saying that they are absurd. Later on, one of the commentators of Bijaganita was one Krishna Daivajna, who wrote Bijankura. In there he attempts to present a proof that a negative number cannot have a square root (see here for more details)
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Re: BR Maths Corner-1

Post by Vayutuvan »

Nilesh, did not see your question till now. I will go through ArmenT's post and see if I can add any more material. Here is hoping that AmberG hasn't withdrawn from this thread (speaking for self - not really concerned about other threads, though).
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Re: BR Maths Corner-1

Post by Vayutuvan »

SaiK,

Here is an example set of small size

s = {3,9}

Prop1: "Set cardinality is 2" implies "s is not empty" (<--> "set cardinality > 0")
Prop2: "\forall x \in s odd(x)" which is same as "any x \in s is odd" which is also same as "every x \in s is odd"
Prop3: "\exists x \in s prime(x)". To prove all we have to do is to produce such a prime - obviously it is 3.

Now think about two statements - "every x \in s odd(x)" vs. "\exists x \in s odd(x)"

and compare it with the two statements - "every x \in s prime(x)" vs. "\exists x \in s prime(x)"

Here is a rule I can state (assume that s is non-empty)

"every x \in s has some property p" implies "there exists an x \in s that has property p". This is true as any member from s will serve the purpose of the witness for existence.

But the the implication turned around will not work.

Let us look at the Implication "there exists an x \in s that has property p" implies "every x \in s has property p".

Let us disprove the contra-positive "not every x \in s has property p" implies "there does not exists an x \in s that has property p". A simple reflection for a minute shows that when "not every x \in s has property p" is TRUE, the "there does not exists an x \in s that has property p" could be TRUE or FALSE. Since contra-positive is FALSE, the original Implication is FALSE also.

Now contrast the above line of thought process for set s2 = {3,5}.
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Re: BR Maths Corner-1

Post by SaiK »

mat guru, i was only thinking to the logical functional equivalence (path) of using the list.. (search function let us assume). any or there exists qualifier can terminate a search or select function at the property find, whereas for all search function, must go through the complete list of entities in the list/set.

but if the same search is for an even number in your given set, it will answer to a for all search equivalence satisfying that it touches all elements.

on the set theory and cardinality, pranam for re-education. thanks. sometimes, i need to revisit these again and again to get back to basics.
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Re: BR Maths Corner-1

Post by SaiK »

mat guru,

http://en.wikipedia.org/wiki/Regression_analysis

need some help: MS Excel is peachy and spits out a summary report, doing a regression analysis on two columns, dependent and the independent one. now, i am trying to understand the summary report. you or anyone can help me understand the various stats output especially what those coefficents mean, intercepts and x var ?

tia
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Re: BR Maths Corner-1

Post by SaiK »

well, i should have researched it a better.. found a link
http://www.unf.edu/~dtanner/dtch/dt_ch6.htm
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Re: BR Maths Corner-1

Post by SaiK »

trying to understand a little bit more on the algo and applied math used at: http://en.wikipedia.org/wiki/Akamai_Technologies

one thing is it is a network - hierarchical nodes, with each node has stats on monitored data - traffic, access, content types, etc.
i can get the congestion spots, but i don't get how they are resolving ?
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Re: BR Maths Corner-1

Post by Vayutuvan »

SaiK ji, look at research of two people - Professor Tom Leighton of MIT and his student Dr. Satish Rao of NEC. Also Prof. Shang Hua Teng, head of CS at USC, Prof. Leiserson (one of the authors of "the" Algorithms book by Cormen, Leiserson and Rivest). None of them work for Akamai right now the most interesting would be what Dr. Satish Rao is doing right now. His work on Mult-icommodity flows probably is most relevant.

Added later
Another student of Prof. Leighton (I don't remember the name) was the founder. He was in one of the planes that hit WTC. Looking at his thesis might give some details.
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Re: BR Maths Corner-1

Post by SaiK »

Thanks for the inputs.. mat guru, could you pl give some 2 min brush on derivatives.. eons have past.

say pi = yq^2 + (x - c) q, and i want to know pi changes by varying q.

first derivative -> dpi/dq = ?
second derivative?

some steps. tia
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Re: BR Maths Corner-1

Post by SaiK »

never mind.. i got some links http://derivative-functions.cours-de-math.eu/
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Re: BR Maths Corner-1

Post by Amber G. »

matrimc wrote: ....
Another student of Prof. Leighton (I don't remember the name) was the founder. He was in one of the planes that hit WTC. Looking at his thesis might give some details.
Some may be interested in:

Visualizing Global Internet Performance with Akamai (see link below)

http://www.akamai.com/html/technology/v ... kamai.html

Looks like Akamai is a cool place. Only company I know who has unlimited vacation as their official policy!

My son (A Physics PhD) who really loves Physics/math switched from academic world, actually recruited by one of his prof. He joined Akamai recently and is very happy.. his group is mostly Physics and Math PhD's.

(I meanwhile have learned a lot about internet network traffic, security etc..:)

Another map, some may find interesting ...

Internet use in the world - By time of day ..

Image
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