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#1
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functions of uncountably many variables
is there such a thing? If there is could someone point me in the right direction
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#2
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Re: functions of uncountably many variables
A functional is essentially a function of a function, which is sort of what you're talking about.
http://en.wikipedia.org/wiki/Functio...mathematics%29 I suppose you could also consider a function of an infinite sequence of discrete variables, but I don't know the name of such a thing. |
#3
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Re: functions of uncountably many variables
[ QUOTE ]
is there such a thing? If there is could someone point me in the right direction [/ QUOTE ] If you're familiar with them, the derivative and integral are examples. |
#4
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Re: functions of uncountably many variables
[ QUOTE ]
is there such a thing? [/ QUOTE ] GDP growth, interest rates, industrial output, stock prices, etc etc etc Do you mean uncountably infinite in a strict mathematical sense, as in having no equivalence relation to the natural numbers? If so, I don't understand how the derivative or integral satisfy this property. |
#5
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Re: functions of uncountably many variables
Yes uncountably many independent variables as in infinately many but not in one to one correspondence qith the natural numbers. Yes i did not see how the der or integral were examples either although I can see that the der/int take into account infinately many values ( but they are not variables or independent). Right?
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#6
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Re: functions of uncountably many variables
It's possible to think of real functions as (uncountably) infinite-dimensional vectors. As such any function that maps real functions to something else is a function of infinitely many variables.
Another (admittedly trivial) example of a function of infinitely many variables is a function that takes any ordered set of real numbers with the cardinality of the continuum, and is constant. |
#7
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Re: functions of uncountably many variables
[ QUOTE ]
It's possible to think of real functions as (uncountably) infinite-dimensional vectors. As such any function that maps real functions to something else is a function of infinitely many variables. [/ QUOTE ] so the function g(x)=x+1 is actually a function of infinately many variables, not one? lame. |
#8
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Re: functions of uncountably many variables
[ QUOTE ]
so the function g(x)=x+1 is actually a function of infinately many variables, not one? lame. [/ QUOTE ] Not really. But you could say that g(f(x))=f(x)+1 is a function of infinitely many variables. Functions that operate on real functions rather than operating on real numbers are effectively functions of uncountably many variables. I don't know that you'd ever want to do it but the characteristic function of continuity g:f->{0,1} which is 1 if f is continuous everywhere, and 0 otherwise. |
#9
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Re: functions of uncountably many variables
Consider the vector v = (3,1,7,9). Normally, we use subscript notation to denote its components:
v_1 = 3 v_2 = 1 v_3 = 7 v_4 = 9 But sometimes we use parentheses instead: v(1) = 3 v(2) = 1 v(3) = 7 v(4) = 9 When we use parentheses, it makes it clear that the vector v is really just a function whose domain is {1,2,3,4}. Conversely, any real-valued function whose domain is {1,2,3,4} is really just a vector in R^4. Now consider the infinite sequence a = {2,4,8,16,32,...}. This can be regarded as a vector with a countably infinite number of components. Normally, we use subscript notation for the elements of the sequence: a_1 = 2 a_2 = 4 ... a_n = 2^n ... But sometimes we use parentheses instead: a(1) = 2 a(2) = 4 ... a(n) = 2^n ... When we use parentheses, it makes it clear that the sequence a is really just a function whose domain is {1,2,3,...}. Conversely, any function whose domain is {1,2,3,...} is really just an infinite sequence. In other words, an infinite sequence can be regarded either as a vector with countably many components (one component for every natural number) or as a function whose domain is {1,2,3,...}. Mathematically, they are the same thing. There is no mathematical difference between functions and vectors. A function is just a vector that has one component for every point in its domain. Extending this reasoning, any function whose domain is R is really just a vector that has one component for each real number. That is, it is a vector that has uncountably many components. A function of 4 variables is a function whose domain consists of vectors in R^4. That is, a function of 4 variables is a function of functions on {1,2,3,4}. A function of countably many variables is a function whose domain consists of sequences. That is, a function of countably many variables is a function of functions on {1,2,3,...}. Likewise, an example of a function of uncountably many variables would be a function of functions on R. rufus has given several examples. |
#10
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Re: functions of uncountably many variables
Thanks. Good explanation.
What would happen to Q if you adjoined uncountably many variables to it? Anything interesting? |
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