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This thought popped into my head as my friend comes over to my house from about 15 miles away to put a fairly large amount of files (I think it was something like 10 GB) of data onto a file server laptop that we have set up, instead of transferring them over the Internet. I'll formalize it this way, and the footnotes at the bottom will clarify notation for those that are not already familiar:
Define F to be the set of all possible file sizes. Let S be the set of all Internet connection speed. Let W be the set containing all the points on the world map. (All three of these sets are infinite sets, if we look at this mathematically and ignore QM etc.) Let R be the set P(W), or the power set of W*.
Define a function g: F x S x W** -> R*** which takes a file size, a connection speed, and a point on the world map as input and returns the region in which it would be faster to travel (by any mechanism available at that time, such as a car or a plane; this of course means that there is a hidden "time" variable that we are ignoring for this purpose) to the destination and transfer the files directly (e.g. by USB) than to transmit them over the Internet.
How do you all think this function is behaved, roughly? I know that, for example, my friend's house is in the set g((10 GB, 50 KB/s, my house)). (My friend has slow Internet). But in general how do you think this function behaves? For example, for a given input, how circular do you expect the output region might be?
*The power set of a set is the set of all subsets of that set. (A set A is a subset of a set B if and only if the statement "for all x, if x is in A, x is in B" is true.) For example, P({1,2}) = {the empty set,{1},{2},{1,2}}. It should be noted that every element of P(A) is a set, not an ordinary object.
**The set A x B is the Cartesian product of A and B, defined as the set of all ordered pairs of the elements of A and B. In other words, it is the set {(a,b) such that a is in A and b is in B}. Cartesian products can be extended, as here, to more than 2 sets in the same way.
***The notation for functions is as follows:
f: A->B
which means that f is a function from the set A to a set B, which means that for every element of A, f associates that element to exactly one element of B (but not necessarily vice versa).
Define F to be the set of all possible file sizes. Let S be the set of all Internet connection speed. Let W be the set containing all the points on the world map. (All three of these sets are infinite sets, if we look at this mathematically and ignore QM etc.) Let R be the set P(W), or the power set of W*.
Define a function g: F x S x W** -> R*** which takes a file size, a connection speed, and a point on the world map as input and returns the region in which it would be faster to travel (by any mechanism available at that time, such as a car or a plane; this of course means that there is a hidden "time" variable that we are ignoring for this purpose) to the destination and transfer the files directly (e.g. by USB) than to transmit them over the Internet.
How do you all think this function is behaved, roughly? I know that, for example, my friend's house is in the set g((10 GB, 50 KB/s, my house)). (My friend has slow Internet). But in general how do you think this function behaves? For example, for a given input, how circular do you expect the output region might be?
*The power set of a set is the set of all subsets of that set. (A set A is a subset of a set B if and only if the statement "for all x, if x is in A, x is in B" is true.) For example, P({1,2}) = {the empty set,{1},{2},{1,2}}. It should be noted that every element of P(A) is a set, not an ordinary object.
**The set A x B is the Cartesian product of A and B, defined as the set of all ordered pairs of the elements of A and B. In other words, it is the set {(a,b) such that a is in A and b is in B}. Cartesian products can be extended, as here, to more than 2 sets in the same way.
***The notation for functions is as follows:
f: A->B
which means that f is a function from the set A to a set B, which means that for every element of A, f associates that element to exactly one element of B (but not necessarily vice versa).
