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#1 2012-04-28 10:28:52

anonimnystefy
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Posts: 14,809

Generating Functions

This will be a thread for discussing GFs,their applications and uses,cool identities that can be proven using them and pretty much anything concerning them.Any questions about them are welcome and desirable.

Here is a starter topic:"Using GFs to represent binary trees.How can this be done?"

Last edited by anonimnystefy (2012-04-28 11:26:59)


“Here lies the reader who will never open this book. He is forever dead.

“Taking a new step, uttering a new word, is what people fear most.” ― Fyodor Dostoyevsky, Crime and Punishment

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#2 2012-04-28 14:05:38

bobbym
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From: Bumpkinland
Registered: 2009-04-12
Posts: 81,394

Re: Generating Functions

two_flips_fair_tree.png

This is a small tree but it is perfect for an example.

The resulting nodes of the tree are compactly represented by the generating function.


In mathematics, you don't understand things. You just get used to them.
I have the result, but I do not yet know how to get it.
All physicists, and a good many quite respectable mathematicians are contemptuous about proof.

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#3 2012-04-28 14:09:00

anonimnystefy
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Re: Generating Functions

I assume the T and H are tales and heads. smile

I know this will sound too much,but,what would the graph and the GF look like for a coin that has the probability of heads change according to some known formula?


“Here lies the reader who will never open this book. He is forever dead.

“Taking a new step, uttering a new word, is what people fear most.” ― Fyodor Dostoyevsky, Crime and Punishment

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#4 2012-04-28 14:12:32

bobbym
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From: Bumpkinland
Registered: 2009-04-12
Posts: 81,394

Re: Generating Functions

Sometimes that is possible with a GF but only by a trick or two. It might be a little more like a Markov chain but that can be tricky too.


In mathematics, you don't understand things. You just get used to them.
I have the result, but I do not yet know how to get it.
All physicists, and a good many quite respectable mathematicians are contemptuous about proof.

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#5 2012-04-28 14:15:35

anonimnystefy
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From: The Foundation
Registered: 2011-05-23
Posts: 14,809

Re: Generating Functions

It rhymes.

Ok. Could we get an example for that graph you posted using a Markov Chain?


“Here lies the reader who will never open this book. He is forever dead.

“Taking a new step, uttering a new word, is what people fear most.” ― Fyodor Dostoyevsky, Crime and Punishment

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#6 2012-04-28 18:56:27

bobbym
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From: Bumpkinland
Registered: 2009-04-12
Posts: 81,394

Re: Generating Functions

Hi;

Sorry about that, I do not remember how. Checked my notes and found I did not write it down.


In mathematics, you don't understand things. You just get used to them.
I have the result, but I do not yet know how to get it.
All physicists, and a good many quite respectable mathematicians are contemptuous about proof.

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#7 2012-04-28 22:52:57

anonimnystefy
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From: The Foundation
Registered: 2011-05-23
Posts: 14,809

Re: Generating Functions

I think this is the one:


“Here lies the reader who will never open this book. He is forever dead.

“Taking a new step, uttering a new word, is what people fear most.” ― Fyodor Dostoyevsky, Crime and Punishment

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#8 2012-04-28 22:55:12

bobbym
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From: Bumpkinland
Registered: 2009-04-12
Posts: 81,394

Re: Generating Functions

When you square that you get the same matrix.


In mathematics, you don't understand things. You just get used to them.
I have the result, but I do not yet know how to get it.
All physicists, and a good many quite respectable mathematicians are contemptuous about proof.

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#9 2012-04-28 23:08:54

anonimnystefy
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From: The Foundation
Registered: 2011-05-23
Posts: 14,809

Re: Generating Functions

Then it is correct.


“Here lies the reader who will never open this book. He is forever dead.

“Taking a new step, uttering a new word, is what people fear most.” ― Fyodor Dostoyevsky, Crime and Punishment

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#10 2012-04-28 23:12:33

bobbym
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From: Bumpkinland
Registered: 2009-04-12
Posts: 81,394

Re: Generating Functions

Apparently, I was trying to get it to do the calculation on the right.


In mathematics, you don't understand things. You just get used to them.
I have the result, but I do not yet know how to get it.
All physicists, and a good many quite respectable mathematicians are contemptuous about proof.

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#11 2012-04-28 23:20:45

anonimnystefy
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From: The Foundation
Registered: 2011-05-23
Posts: 14,809

Re: Generating Functions

No,squaring the matrix gives the probability of getting from the given state to another in two steps.


“Here lies the reader who will never open this book. He is forever dead.

“Taking a new step, uttering a new word, is what people fear most.” ― Fyodor Dostoyevsky, Crime and Punishment

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#12 2012-04-28 23:27:18

bobbym
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From: Bumpkinland
Registered: 2009-04-12
Posts: 81,394

Re: Generating Functions

That is what I am saying. The answer on the right needs something more than a matrix multiplication. But it is the tree answer. So my original statement that the tree can be represented entirely by a markov chain and matrix operations is not accurate.


In mathematics, you don't understand things. You just get used to them.
I have the result, but I do not yet know how to get it.
All physicists, and a good many quite respectable mathematicians are contemptuous about proof.

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#13 2012-04-28 23:32:26

anonimnystefy
Real Member
From: The Foundation
Registered: 2011-05-23
Posts: 14,809

Re: Generating Functions

Yes,I agree.

What if we multiply by a vector?


“Here lies the reader who will never open this book. He is forever dead.

“Taking a new step, uttering a new word, is what people fear most.” ― Fyodor Dostoyevsky, Crime and Punishment

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#14 2012-04-28 23:37:38

bobbym
Administrator
From: Bumpkinland
Registered: 2009-04-12
Posts: 81,394

Re: Generating Functions

Sometimes there is an initial state vector. What vector do you want to use?


In mathematics, you don't understand things. You just get used to them.
I have the result, but I do not yet know how to get it.
All physicists, and a good many quite respectable mathematicians are contemptuous about proof.

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#15 2012-04-28 23:38:58

anonimnystefy
Real Member
From: The Foundation
Registered: 2011-05-23
Posts: 14,809

Re: Generating Functions

Yes,I know. But it doesn't help. What matrix would you like as your output?


“Here lies the reader who will never open this book. He is forever dead.

“Taking a new step, uttering a new word, is what people fear most.” ― Fyodor Dostoyevsky, Crime and Punishment

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#16 2012-04-28 23:41:06

bobbym
Administrator
From: Bumpkinland
Registered: 2009-04-12
Posts: 81,394

Re: Generating Functions

You could do a scalar multiplication by A^(1) x A^(2)


In mathematics, you don't understand things. You just get used to them.
I have the result, but I do not yet know how to get it.
All physicists, and a good many quite respectable mathematicians are contemptuous about proof.

Offline

#17 2012-04-28 23:42:11

anonimnystefy
Real Member
From: The Foundation
Registered: 2011-05-23
Posts: 14,809

Re: Generating Functions

What is A?


“Here lies the reader who will never open this book. He is forever dead.

“Taking a new step, uttering a new word, is what people fear most.” ― Fyodor Dostoyevsky, Crime and Punishment

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#18 2012-04-28 23:44:25

bobbym
Administrator
From: Bumpkinland
Registered: 2009-04-12
Posts: 81,394

Re: Generating Functions

A is your matrix. A^(1) being the first level of the tree, A^(2) being the second level.


In mathematics, you don't understand things. You just get used to them.
I have the result, but I do not yet know how to get it.
All physicists, and a good many quite respectable mathematicians are contemptuous about proof.

Offline

#19 2012-04-28 23:46:17

anonimnystefy
Real Member
From: The Foundation
Registered: 2011-05-23
Posts: 14,809

Re: Generating Functions

I don't understand what you want to do with that.


“Here lies the reader who will never open this book. He is forever dead.

“Taking a new step, uttering a new word, is what people fear most.” ― Fyodor Dostoyevsky, Crime and Punishment

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#20 2012-04-28 23:49:22

bobbym
Administrator
From: Bumpkinland
Registered: 2009-04-12
Posts: 81,394

Re: Generating Functions

Trees are computed by mutiplying nodes down the branches.


In mathematics, you don't understand things. You just get used to them.
I have the result, but I do not yet know how to get it.
All physicists, and a good many quite respectable mathematicians are contemptuous about proof.

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#21 2012-04-28 23:51:24

anonimnystefy
Real Member
From: The Foundation
Registered: 2011-05-23
Posts: 14,809

Re: Generating Functions

Ok,but what does that have to do with matrices?


“Here lies the reader who will never open this book. He is forever dead.

“Taking a new step, uttering a new word, is what people fear most.” ― Fyodor Dostoyevsky, Crime and Punishment

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#22 2012-04-28 23:52:58

bobbym
Administrator
From: Bumpkinland
Registered: 2009-04-12
Posts: 81,394

Re: Generating Functions

Instead of the usual multiplication of 2 matrices, we could use an element by element multiplication of 2 matrices.


In mathematics, you don't understand things. You just get used to them.
I have the result, but I do not yet know how to get it.
All physicists, and a good many quite respectable mathematicians are contemptuous about proof.

Offline

#23 2012-04-28 23:57:22

anonimnystefy
Real Member
From: The Foundation
Registered: 2011-05-23
Posts: 14,809

Re: Generating Functions

I don't think it would work for three steps.


“Here lies the reader who will never open this book. He is forever dead.

“Taking a new step, uttering a new word, is what people fear most.” ― Fyodor Dostoyevsky, Crime and Punishment

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#24 2012-04-29 00:00:46

bobbym
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From: Bumpkinland
Registered: 2009-04-12
Posts: 81,394

Re: Generating Functions

Sure it would. Element by element would give 1 / 8 for each branch.


In mathematics, you don't understand things. You just get used to them.
I have the result, but I do not yet know how to get it.
All physicists, and a good many quite respectable mathematicians are contemptuous about proof.

Offline

#25 2012-04-29 00:04:03

anonimnystefy
Real Member
From: The Foundation
Registered: 2011-05-23
Posts: 14,809

Re: Generating Functions

But there would be 8 branches! We only have a 2x2 matrix!


“Here lies the reader who will never open this book. He is forever dead.

“Taking a new step, uttering a new word, is what people fear most.” ― Fyodor Dostoyevsky, Crime and Punishment

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