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Abstract the Question an Observation Abstract There are many notions of means, the arithmetic mean and geometric mean are quite familiar to many. There are also a few generalizations of these concepts, including the mean developed by V. M. Tikhomirov. However, these means fail to capture the general nature of means. A new definition is presented and it is shown that the Tikhomirov mean is a form of this mean. The Question While a mean is a powerful mathematical structure, it is not well defined. One common definition of a mean is a measure of center or a middle. V. M. Tikhomirov defines a generalized mean as follows. Given a continuous strictly monotone function ϕ a nd its inverse ψ, the mean of a ϕ(x1)+ϕ(x2)+...+ϕ(xn) sequence of values {x1, x2, ..., xn}, M n(x1, x2, ..., xn) = ψ( n ). As Tikhomirov explains, the arithmetic mean, the geometric mean, the harmonic mean, and the root-mean-square are all forms of this generalized mean1. However, this definition does make the essence of the operation clear. An Observation To try to understand what is happening when we take the mean of a collection of values, consider the arithmetic mean. To calculate the mean of a sequence X = {x1, x2, ..., xn} we add all the values of the sequence together, and divide by the number of elements in the sequence: n ˉ 1 X = n ∑ xi. i=1 What property does this value have? One property is that if we sum the elements of X, we end up with the same result that we would get if we replaced each element of X with Xˉ . In other ˉ words, if we were to construct a new sequence, Y = {y1, y2, ..., yn} s.t. yi = X, the sum of the elements of X is equal to the sum of the elements of Y. Essentially, if we operate on some sequence, or more generally some net, the mean of that net, w.r.t. to that operation, is the value which can be substituted in for every element of the net, without changing the result of the operation. 1 Tikhomirov, V. M. "On the Notion of Mean." Selected Works of A. N. Kolmogorov, 1991, 144-46. doi:10.1007/978-94-011-3030-1_17. Rigorous Definition While the above analysis is insightful, a rigorous definition is needed. Let A and B be sets. Let I ˉ be an index set, {ai} be a net from I to A, and f be a function mapping {ai} to B. A is a mean of ˉ {ai} iff f({ai}) = f({mi}), where {mi} is a net from I to A s.t. mi = A. Relation to Generalized If the abstract mean is valid, the generalized mean described by Tikhomirov should be an example of the abstract mean. Consider a continuous strictly monotone function ϕ a nd its n inverse ψ. It will be shown that M n is a mean w.r.t. f = ∑ ϕ(xi). Consider a sequence i (x )+ (x )+...+ (x ) ∈ ⊂ ℝ ϕ 1 ϕ 2 ϕ n X = {x1, x2, ..., xn}, xi [a. b] . Then M = ψ( n ). While X is a set, a set is a type of net. Next, consider the sequence {mi}, s.t. mi = M. This sequence is also a net. Now n n n n ϕ(x1)+ϕ(x2)+...+ϕ(xn) ϕ(x1)+ϕ(x2)+...+ϕ(xn) f({mi}) = ∑ ϕ(M) = ∑ ϕ(ψ( n )) = ∑ n = ∑ ϕ(xi) = f(X). Thus M is a i=1 i=1 i=1 i=1 mean of X w.r.t. f as was desired. Discussion It has been shown that the concept of a mean can be abstracted to some generic operation on a net and that the generalized mean described by V. M. Tikhomirov is a form of this abstract mean. While the operation is generally implied by the type of the mean in question, such as n f = ∑ xi for the arithmetic mean, the abstract mean does not have any meaning without i=1 indicating the operation being applied to the net. Finally, while the abstract mean is very similar to Tikhomirov’s mean, it is less restrictive, as f can be from any set to any other set. This condition however does not necessarily imply the existence or uniqueness of a mean. Bibliography Tikhomirov, V. M. "On the Notion of Mean." Selected Works of A. N. Kolmogorov, 1991, 144-46. doi:10.1007/978-94-011-3030-1_17. .
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