Isomorphism under differentiation

[tomgotthefunk]

Is 《sinx》under differentiation a valid cyclic group.

 

[Stephen Tashi]

A group must have a binary operation involving it's elements. What are the elements of the group you are asking about? What is the binary operation?

Perhaps you are thinking that differentiation operates on a set of real valued functions. That is true, but differentiation itself is not a real valued function. (Differentiation is a function from functions to other functions.) So differentiation of a function is not a binary operation involving two real valued functions.

 

[mac_alleb]

Consider sin as a row, then differentiation is cyclic, but there is not any connection with groups.

 

[FactChecker]

You could consider differentiation operations on the set (sinx, cosx, -sinx, -cosx) where the + operation is consecutive differentiation. With a little work, I am sure you could define a cyclic group.

 

[blue_raver22]

Isomorphism under differentiation refers to the concept of two mathematical structures being equivalent under a certain operation, in this case differentiation. In order for a group to be considered cyclic, it must have an element that can generate the entire group through repeated application of the group operation.

In the case of the function 《sinx》, when considering differentiation as the group operation, it does not have a single element that can generate the entire group. This is because the derivative of 《sinx》, 《cosx》, does not have the same periodicity as 《sinx》.

Therefore, 《sinx》under differentiation is not a valid cyclic group. However, it is important to note that 《sinx》under differentiation does have certain properties that are similar to those of a cyclic group, such as closure under the operation of differentiation and the existence of an identity element (the constant function 《0》).

Overall, while 《sinx》under differentiation may exhibit some characteristics of a cyclic group, it does not fulfill the criteria for being a valid cyclic group.