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Given a collection of $n$ real number arrays of length $m$, for example:

$$[r_{11},\ \dots, r_{1m}]$$ $$\vdots$$ $$[r_{n1},\ \dots, r_{nm}]$$ is it possible to transform the entire collection into one equivalent complex number array of length $m$, for example, $[c_1,\ \dots, c_m]$?

Also, the solution should be the same regardless of the row order in the collection.

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    What do you mean by "equivalent" here?2012-02-15
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    Equivalent in the sense that the information content of the real arrays is transformed but preserved.2012-02-15
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    What do you mean by "transform"? There is indeed a bijection between the sets you have indicated.2012-02-15
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    In fact, you can compress everything into a single real number by using Cantor's interlace-the-decimals trick. The result will not make much _arithmetic_ sense, but that doesn't seem to be part of your specification.2012-02-15
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    I'm looking for a function or process that can collapse those arrays into a single array. You might think of the arrays as collections of variables gathered at different times, so I want to remove the dimension of time while preserving the information in some form.2012-02-15
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    What's a "real number array"? That expression combines a mathematical term ("real number") and a computer term ("array") and doesn't make sense the way those terms are usually used in those fields, since computers can't store arbitrary real numbers. Do you mean an array on a computer or just a tuple of real numbers?2012-02-15
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    To expand upon joriki's point, you must either mean "floating point" instead of "real number", or you must mean "tuple" instead of "array".2012-02-15
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    I tend to think in computer programming terms. My meaning was a one-dimensional array of real numbers.2012-02-15
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    More accurately, I was thinking of one-dimensional arrays of floating point numbers.2012-02-15
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    I take it that you're probably then also thinking of a pair of floating point numbers when you write "complex number". In that case, the answer for $n\gt2$ is "no", since the array of "complex numbers" has only as many bits as two arrays of "real numbers", and the "information content" is given by the number of bits. For $n=2$ the answer is "yes" since you can combine $r(0,k)$ and $r(1,k)$ into $c_k$.2012-02-15
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    Thank you. I have a somewhat related question but I'll use another thread and try to be more precise in the description.2012-02-16

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