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#26 2006-06-21 07:59:11

luca-deltodesco
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Re: The Golden Ratio by Mario Livio. Anybody read it?

MathsIsFun wrote:

And we could all be simulations in that very computer, and not even know it!

perhaps, but im pretty sure theyd have been getting bored of watching us after the billions upon billions of years, and cut off the program, and even so, i think they would be having, well fun with us, rather than just letting existance get on with it, that is, unless theyre preoccupied with the other side of our universe.

The Beginning Of All Things To End.
The End Of All Things To Come.

#27 2006-06-21 08:05:27

mikau
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Re: The Golden Ratio by Mario Livio. Anybody read it?

or maybe they're busy playing un updated version of reality.

A logarithm is just a misspelled algorithm.

#28 2006-06-21 18:04:24

George,Y
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Re: The Golden Ratio by Mario Livio. Anybody read it?

I guess if x and y are not independent and have some certain kind of relation that can be described by complex numbers, they will be very useful.

for example, negative numbers, dot product, and matrices initially seem a waste of time. But when they just fit the certain problems, they become very useful indeed!

X'(y-Xβ)=0

#29 2006-06-23 04:38:41

mikau
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Re: The Golden Ratio by Mario Livio. Anybody read it?

I think the use of matrices is pretty clear in the computer world. It requires no judicious rearranging techniques and solves its general form. Matrics I guess are sort of like the quadratic formula for all linear equations. But perhaps these are the "certain problems" you are refering to.

A logarithm is just a misspelled algorithm.

#30 2006-06-23 12:30:19

George,Y
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Re: The Golden Ratio by Mario Livio. Anybody read it?

I think it's  mainly Gaussian-Elimination that simplifies linear equation solving.
And it's vector combination that underpins theory for multi-solution linear equations.
Becaus if you throw away matrices multiplication and make small adjustments in the form of the two theories (or a method and a theory) above, you will probably find little difficulty in linear equations dealing.

Then why complicated matrices combining multiplication and addition together?(Actually it took me a long time to accept that concept)

Matices multiplication and determinants are some kind of winners , as  there are actually many other competing definations that are not as poplular as them.(A lot of mathematicians are inventing new definations to arrange numbers)  The reason why they win over might be that their usefulness are not limited to linear equations solving - matrices multiplication is good at handling data, especially in statistics, and determinant is popular among physicians.

And I guess determinant's application in physics is quite lucky- the inventor wouldn't have ever dreamed about it!

Last edited by George,Y (2006-06-23 18:28:06)

X'(y-Xβ)=0

#31 2006-06-23 18:34:11

George,Y
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Re: The Golden Ratio by Mario Livio. Anybody read it?

Actually there are competitions in math theories!
for example, staticians are debating whether classical or Bayesian for decades! Econometritians are debating whether (Ax)'=A or (Ax)'=A' for a long time!

So I would rather put existing theories as the competitive theories that win their opponents or at least due than being the truth.

But my view looks tricky, lol~

X'(y-Xβ)=0

#32 2006-06-23 23:35:00

Ricky
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Re: The Golden Ratio by Mario Livio. Anybody read it?

for example, staticians are debating whether classical or Bayesian for decades!

As far as I know, Bayesian theory is a lame attempt to gather a probability in which scientific theory is correct.  It amounts to going around and asking people, "How much do you believe in theory X?"

Unless you are talking about something else the Bayesians did?

"In the real world, this would be a problem.  But in mathematics, we can just define a place where this problem doesn't exist.  So we'll go ahead and do that now..."

#33 2006-06-24 13:08:30

George,Y
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Re: The Golden Ratio by Mario Livio. Anybody read it?

Bayesians are good at helping court. In that case, only Bayesians could help.

I discover that many people believe in Classical and are against Bayesian just because they think they understand Classical. So may I ask you a question: what does 95% in confidence interval mean? (what does it stand for)

X'(y-Xβ)=0

#34 2006-06-24 13:33:31

Ricky
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Re: The Golden Ratio by Mario Livio. Anybody read it?

Not really.  Like I said, I only know about the Bayesians as a group going around trying to figure out how much people believe in certain theories.  What do you mean by classical?

"In the real world, this would be a problem.  But in mathematics, we can just define a place where this problem doesn't exist.  So we'll go ahead and do that now..."

#35 2006-06-24 14:10:59

George,Y
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Re: The Golden Ratio by Mario Livio. Anybody read it?

After a tiring investigation I find Classical is not as solid as many people think. Probablity is a hard subject, and neither classical nor Bayesian can make progress without sacrificing something.

95% in confidence interval is the probablity generated by samples. Actually it's the X-bar probablity. If you want expectation(parameter) probability, (if you want more), you have to make assumptions that expectation u couid be everywhere and initially its probablity in everywhere is equal. By adding new knowledge of samples you can get conditional (treamed) probility of u. And this time whole distribution, better than just interval. Though the distribution is fake due to assumption.

Luckily, they get a same answer with classical until now. But Bayesians'competitive advantage is to change the equal assumption for expert prejudice. By doing that they can make the predicted interval narrower.

The difference is like if you allow a forecaster to add his personal intuition according to his experience beyond given theory and compution or not. If you allow him to do so, he may get a narrower forecast but more subjective one.

And it's a philophical debate whether a person's experience is more subjective than staticians' theory. Is their theory qualified enough to be truth?

And there's some time that a narrower but more risky forecast overweighs the less risky but broader one.

Last edited by George,Y (2006-06-24 14:21:37)

X'(y-Xβ)=0

#36 2006-06-24 14:49:20

George,Y
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Re: The Golden Ratio by Mario Livio. Anybody read it?

On fibonacci series: to get a general formula for Sn

arrange it-

this gives us an inspiration, can we change the equity into this form:
?

since the coefficient for Sn-1 and Sn-2 are 1, l and k should satisfy:

thanks to the equation

we get l and k are
. Which is bigger? We can only leave this question here.

On the other hand,
,
Thus

Here we use a trick-we set S1=0 and S2=1
then

since
and so on,

The "bigger" question does not matter here, so we get the final formula:

Last edited by George,Y (2006-06-24 22:20:52)

X'(y-Xβ)=0

#37 2006-06-24 22:31:29

George,Y
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Re: The Golden Ratio by Mario Livio. Anybody read it?

Luckily S1=0 and S2=1 satisfy this formula too.

for example
0,1,1,2,3,5,8,13,21,34
S10=34 here

and using the formula approximation
(1.6189+0.6189)/√5=33.99
for n a large number

Sn≈Golden Ration-1 /√5,

This could explain why
Sn/Sn-1≈Golden Ratio

X'(y-Xβ)=0

#38 2006-07-12 16:02:35

All_Is_Number
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Re: The Golden Ratio by Mario Livio. Anybody read it?

I've read Livio's book and found it very interesting. For even more information on Phi and the Fibonacci sequence, I highly recommend checking out this site.

You can shear a sheep many times but skin him only once.

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