Yo—2: The eminent bumpiness of the “non-analytic” mathematics

[Do check out the Update on 19th August 2014 near the end of this post.]

The “Yo versions” of a blog are so comfy to write. … And, what’s more, they can also instantly lend you an aura of the mathematical respectability—whether you [I mean the blogger himself] really understand[s] any of what’s going on or not. (But then, what do you expect from a “Yo version” anyway?)

Hmm…. Anyway, here is yet another “Yo” version of a blog post:

The last time, we touched upon the Taylor series and the function:
f(x) = e^{-1/x^2} for x \neq 0 and
f(x) = 0 for x = 0.

Getting a bit more complex complicated, have you ever heard of the so-called bump function?

The prominent (“canonical”) example of a bump function reads like this:

f(x) = e^{-\dfrac{1}{1-x^2}} for \vert x \vert < 1 and f(x) = 0 otherwise.

And, it really looks like a bump on the road; check out the graphs on Wiki [^] or the Wolfram MathsWorld [^].

This function is infinitely differentiable—but that fact still does not make it analytic [^]. Its support is not just bounded but also compact—but its Fourier transform isn’t so. … And that’s where I had initially found it interesting. … But then, as it turns out, thinking about the issue via the bump functions and all is what I now find to be, way (way) too complicated for me—my purposes [^].

And, yet, mathematicians call such functions mollifiers [^].

If this be a mollifier to a mathematician, then what would be an aggravator to him?

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In the meanwhile, after a bit of a college-wide “re-org,” I have been asked to increase my teaching load, and therefore further engage an additional undergraduate course on Thermodynamics. (We have two shifts for the UG programs, and so, effectively, I am teaching three courses now: Heat Transfer to two shifts and Thermodynamics to one shift.) … Obviously, therefore, even Yo-n posts should come out only once in a while during the rest of this semester… It’s just that we’ve just had a long week-end this time round, and so, I could slip this one in. … Otherwise, it’s all teaching, teaching and teaching. … All to UG students!

Update on 2014.08.19

I have to link up to a great post related to this topic (the Taylor series, the bump function, and in fact also the Lagrangian and the Taylor polynomials, etc.) by David Lowry-Duda [^]. David is a PhD student in [ahem!] maths, at Brown [^]. I came to know of his blog only after publishing this post. Also, if you wish, do check out another great post of his on Intuitive Introduction to Calculus [^]. … Ok, more, later.


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