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Cuchulainn
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Re: exp(5) = [$]e^5[$]

November 5th, 2016, 5:32 pm

Indeed!  Of course, to get |M| strictly less than one, the scaling term needs to be increased by some epsilon.  

The deeper issue is whether ln(c*A) = ln(c)+ln(A) for matrix, A?
A compiler would throw an error; right side  is a scalar + Matrix.
But it could be programmed to make it true without too much effort.
This might be true in some other syntax systems, but not is the context of Python that I was discussing. Mixed type operations are defined in many programming languages and hence do what they are designed to do. Another example of a mixed type syntax in Python is multiplying a string with an integer! eg

'Cuch' * 42

Since Python also doesn't need a compiler, you are probably thinking about something else? It can't be math (like in the e^A) because math doesn't need a compiler either.
Anything is possible with software. Mathematics is less compromising.
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outrun
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Re: exp(5) = [$]e^5[$]

November 5th, 2016, 5:52 pm

Both are symbol manipulation systems, good software design uses math to proof its correctness, and that mathematical proof is in return executed by software.

http://securityaffairs.co/wordpress/270 ... orrow.html

It's all about having clear definitions and rules. There is nothing wrong or exiting about 'Cuch'*42, it's just a valid operator that manipulates symbols.
 
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Paul
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Re: exp(5) = [$]e^5[$]

November 5th, 2016, 8:00 pm

Make Mathematics Great Again!

P
 
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Re: exp(5) = [$]e^5[$]

November 5th, 2016, 8:27 pm

[..]
One of the most powerful lessons I learned in math was when I learned that all "the algebra" I had studied was just "an algebra".  One coud define different algebras on different sets or with variations in operator behavior (within strict limits, of course).   As both Paul and outrun noted, there's the issue of non-explicit knowledge about which definition of exp() we are using which, IMO, depends on the meaning of A and whether it really is a matrix in the mathematical sense or an array in the computational sense.
If you study analytic solution of systems of ODEs at university then you know exactly what the matrix exponential is,  For algebra, Gil Strang is a cool teacher.
There is nothing non-explicit in mathematician's mind. if you give a seminar to mathematicians and you don't give definitions they will leave the room.

In this thread it is oscillating between two different definitions. It's a mess. 
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outrun
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Re: exp(5) = [$]e^5[$]

November 5th, 2016, 8:54 pm

That's because math is incomplete!

"Gödel's incompleteness theorems are two theorems of mathematical logic that establish inherent limitations of all but the most trivial axiomatic systems capable of doing arithmetic. "
 
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Paul
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Re: exp(5) = [$]e^5[$]

November 5th, 2016, 9:58 pm

It's [$]\sqrt{1+\sqrt{1+\cdots}}[$] again.

P
 
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outrun
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Re: exp(5) = [$]e^5[$]

November 5th, 2016, 10:18 pm

Haven't seen that one before. It's the Golden ratio!
 
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Paul
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Re: exp(5) = [$]e^5[$]

November 5th, 2016, 10:30 pm

You haven't seen that thread?!?!?! It's the craziest thread of all time!!!!

P
 
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Re: exp(5) = [$]e^5[$]

November 5th, 2016, 10:43 pm

You haven't seen that thread?!?!?! It's the craziest thread of all time!!!!

P
Well, at least on this thread we all agree(d) on 148.413. 
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Paul
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Re: exp(5) = [$]e^5[$]

November 5th, 2016, 10:47 pm

Or rather it was any thread involving dunrewpp. He got very upset if we were loose in our notation/definitions!

P
 
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outrun
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Re: exp(5) = [$]e^5[$]

November 5th, 2016, 11:12 pm

No. Don't know it!
I remember having a very long discussion that ended in trying to proof things based on internally inconsistent axioms. something about the solutions of abs(x)=-1,

..and I've heard stories about arctan.

I'll search for it!
 
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outrun
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Re: exp(5) = [$]e^5[$]

November 5th, 2016, 11:39 pm

Found it, it's arctan(1 + arctan(1 +
and 3rd post by T4A starts with the troubling "Interesting!

viewtopic.php?f=26&t=78676&hilit=Arctan
 
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Paul
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Re: exp(5) = [$]e^5[$]

November 6th, 2016, 9:34 am

It's coming back to me now...

It all started with the [$]\sqrt{1+\sqrt{1+\cdots}}[$], viewtopic.php?f=26&t=78619&p=505617 

Then it moved on to arctan. Then stuff like abs(x)=-1. 

There were a few threads on definitions in mathematics.

Then lots of disagreement between you (outrun) and dunrewpp.

And finally dunrewpp's head exploded.

P
 
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outrun
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Re: exp(5) = [$]e^5[$]

November 6th, 2016, 12:50 pm

I wasn't alone, 
but this happens with me when I run into certain character traits. It awakes an irresistible urge in me to turn it into a lesson and be confrontational, create an experience that in principle should changes future behaviour. What the benefit of that is, is anyones guess. I'm now working on learning to let things go earlier on an move on to better things. As I grow older I get this feeling that I have to spend my time more wisely.
 
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Cuchulainn
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Re: exp(5) = [$]e^5[$]

November 6th, 2016, 12:56 pm

Nothing to do with pdes or numerics but simple relationship between credit ratings over one period and ratings over another period results in exponential of a transition prob matrix.

So we can now blame the credit crisis on software engineers!

P
Back to topic.
Yes, did they use the Python/APL approach? I wonder in which language they do MBS. A little knowledge is a dangerous thing.
Your remark is a very good example; From the infinitesmial generator matrix [$]Q[$] we can compute the 1-step transition probability matrix [$]P[$]:

[$]P(t, t + dt) = I + dtQ + o(dt)[$]

Now compute to [$]s = t + mdt[$] (m-period) to get a matrix polynomial (or is it polynomial matrix ;)) and then let [$]m[$]  go very big to get

[$]P(t, s) = e^{(s-t)Q}[$]

This can now be computed by one of the 19 methods of Moler and Van Loan.

I have not given any definitions of the italicized words; for maximum flexibility you can fill in your own.

https://en.wikipedia.org/wiki/State-transition_matrix
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