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How do I explain Operator Semigroups, in particular, positive operator semigroups to someone who hasn't studied math beyond high school?

I just want to give a vague idea/analogy to someone to let them know a bit about my a project I am working on.

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  • $\begingroup$ It depends on what you are doing with them. I don't think just explaining what they are is particularly motivating. $\endgroup$ – Don Thousand Jan 22 at 18:09
  • $\begingroup$ @DonThousand Right now, I am just studying Positive Operator Semigroups. However, I just want to give someone from a non-math background a rough idea of what I'm studying. How do I do that without getting technical? $\endgroup$ – Mark Jan 22 at 18:12
  • $\begingroup$ Wikipedia provides a good starting point. $\endgroup$ – Don Thousand Jan 22 at 18:19
  • $\begingroup$ Were you likely to understand positive operator semigroups when you were in High School? $\endgroup$ – DisintegratingByParts Jan 23 at 7:30
  • $\begingroup$ @DisintegratingByParts I don't really want them to completely understand. Something like a real-world application or analogy of sorts. Just give a really basic idea. $\endgroup$ – Mark Jan 23 at 10:12
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How do I explain Operator Semigroups to someone who hasn't studied math beyond high school? I just want to give a vague idea/analogy. I don't really want them to completely understand.

Maybe, a possible analogy is the exponential function: you are studying a generalization of $f(t)=e^{a t}$ which allow ''matrix exponents''.

  • But why would anyone want to study things like that?

Because, as we know that the said function is the solution of some important problems, we expect that the said generalization is the solution of some important generalized problems.

  • What are these problems?

They are the functional equation $$f(x+y)=f(x)f(y)$$ and the differential equation $$f'(x)=af(x).$$ If we assume that $f$ is real-valued, then a solution is the exponential function $f(t)=e^{at}$. If we assume that $f$ is matrix-valued, then a solution will be given by a ''matrix exponential''. If we want go one step further (which have important applications), we will need semigroup of operators. Here is where your project starts.

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