![]() ![]() His favorite theorem appears in this video from the MOOC. Ghrist is a dedicated teacher, and this theorem is a favorite of his from his massive online open course (MOOC), available here. ![]() When we do this, it turns out the exponential of the differentiation operator is the shift operator, a bridge between the continuous and discrete worlds.ĭr. (These polynomials are in turn called Taylor series.) By using the Taylor series for the exponential function, we can give meaning to the idea of taking e to the power of the differentiation operator. Taylor’s theorem allows us to approximate more difficult functions with polynomials. In fact, Taylor’s theorem is the best way to make sense of raising e to irrational or imaginary powers. ![]() While the idea of raising a number to a functional power doesn’t immediately seem meaningful, the magic of Taylor’s theorem allows us to define the process into something meaningful. Ghrist’s favorite theorem connects the shift operator to a procedure in continuous calculus of exponentiating a function, that is, of taking e to the power of a function. If we call the shift operator E, as Ghrist does, E(f(x))=f(x+1). The shift operator acts on a function by taking an input to the output of the function at the next time step. The derivative is a measure of change in a system over time, so the discrete analog of the derivative is a shift. In continuous calculus, the derivative is a central object of study. (For example, think of the birth of a new organism or generation of organisms.) Some tools from calculus can be used to study these systems as well, and Ghrist’s favorite theorem is one way to form a link between the two subjects. Calculus is the study of continuous change over time, but many important systems change discretely. Ghrist chose a theorem (one that doesn't have a name of its own, so we’ll just have to call it Ghrist’s favorite theorem) that to him sums up a deep relationship between discrete analogs of calculus. ![]()
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