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June 30th, 2010, 02:06 AM   #1
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Infinite dimensional

Hello everyone
I read today that an equation is infinite-dimensional. Since I have to study maths quite a few years .. could you please help me what these two words mean and suggest me some resources to start reading?

Best Regards
Alex.
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June 30th, 2010, 07:14 AM   #2
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Re: Infinite dimensional

I'm not sure what you heard or what you're asking. But dimensions are a simple concept in mathematics. We say that a plane (or figures that 'live in' a plane like squares or triangles) is two-dimensional because you need precisely 2 numbers to specify its location, relative to a given origin. We say the same about the defining relation of a two-dimensional object: describes a circle, and it uses two variables and hence is two-dimensional. (There are technical points here -- you could embed a circle in three or 47-dimensional space -- but that's not relevant here.)

Example: The power radiated from an object (e.g. the sun) is where P is the power, A is the area, e is the emissivity, and T is the temperature (on an absolute scale, like Kelvins). This is (not counting the constant sigma) a four-dimensional relation.

An infinite-dimensional one is just one with infinitely many variables. It's usually expressed in terms of an infinite matrix.
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June 30th, 2010, 11:01 AM   #3
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Re: Infinite dimensional

If you provide a bit more context for your question, then you're likely to get a better answer. Dimension is a fundamnetal mathematical concept, but absolutely not a simple concept, despite the previous reply to this thread.

It's an inherent property of the object in question, but it can be difficult to define exactly what that means. There is a concept of topological dimension for manifolds (like a circle, which is one-dimensional), and an algebraic concept of dimension for vector spaces over a field. These are two different, but interrelated ideas of dimension.

Since you are asking about infinite dimensionality, the most likely applicable reference is functional analysis. This is the general subject where people most often address infinite dimensional spaces, but is pretty fancy and it takes a lot of work to get there.
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July 1st, 2010, 01:36 AM   #4
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Re: Infinite dimensional

I would like to thank you for your answers
Actually there is a function that models the spatio-temporal evolution


Integral of w(x,u)*s(u,t-1)du


According to the author this is an infinite dimensional and a common approach to reduce this dimensionallity is to emply a basis-expansion representation.
Then it proceeds by defining a basis-expansion representation.

I hope this helps somehow.
Best Regards
Alex
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July 1st, 2010, 09:53 AM   #5
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Re: Infinite dimensional

I'm not sure at all what the term "basis expansion representation" means in this context, but given the explicit reference to infinite dimensionality, I have a guess.

I imagine the integral in question is probably a functional integral, where the space that you are integrating over is a function space, like all paths connecting two points or some such. This idea started in two places (that I know of):

- the study of brownian motion and the corresponding integral, called the Weiner integral or process, has its basis in rigorous mathematics.

- quantum mechanics and the corresponding integral, called the Feynman path integral (wikipedia suspect here), DOES NOT (and cannot) have its basis in rigorous mathematics and is a purely formal idea.

This is pretty fancy math, basically in the realm of functional analysis; depending on your level of sophisification, this may be a bit much. I don't feel like I'm helping you much, but I hopefully it does some good.
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July 15th, 2010, 04:17 AM   #6
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Re: Infinite dimensional

I would like to thank you for your reply.
I am sorry I didn't reply for so long period.

Best Regards
Alex
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January 17th, 2011, 03:29 AM   #7
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Re: Infinite dimensional

Quote:
Originally Posted by CRGreathouse
We say that a plane (or figures that 'live in' a plane like squares or triangles) is two-dimensional because you need precisely 2 numbers to specify its location, relative to a given origin.

....

Example: The power radiated from an object (e.g. the sun) is where P is the power, A is the area, e is the emissivity, and T is the temperature (on an absolute scale, like Kelvins). This is (not counting the constant sigma) a four-dimensional relation.
Hello again!
I would like to ask you in your example T^4 is a four-dimensional relation. According to your simplified definition (which helped me a lot) I need two numbers(or variableS) to describe the (x,y) plane. Why T^4 depends on four variables and thus is a four dimensional plane?

I would like to thank you in advance for your help

Best Regards
Alex
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January 17th, 2011, 09:38 AM   #8
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Re: Infinite dimensional

Quote:
Originally Posted by dervast
I would like to ask you in your example T^4 is a four-dimensional relation. According to your simplified definition (which helped me a lot) I need two numbers(or variableS) to describe the (x,y) plane. Why T^4 depends on four variables and thus is a four dimensional plane?
I gave the four variables. Are you asking how it's possible for four variables to be dependent?
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January 18th, 2011, 02:18 AM   #9
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Re: Infinite dimensional

No thanks, I could not see your equation correctly so I misinterpreted it.

Thanks again
Alex
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