What is a hump function?

Question

I have been in trouble with the hump function(s) What are them?

Could you give me an explicit formula for "Hump"(not bump) function.

Thanks

The mathematician's fetish equivalent of this? Or more seriously: A function used in optimisation? — Raskolnikov, Mar 22 '12 at 10:46
Could you tell us about the context in which you came across the term "hump" function? Is it possible someone meant bump function but said "hump"? You could have also included the problem behind this: "find a $\mathcal{C}^\infty$ function with analyticity set $\emptyset$" (on MO, if that is indeed you). — anon, Mar 22 '12 at 10:50
@anon, that is exactly the context I was asked this question. — honolululurahmi, Mar 22 '12 at 10:57
Tolstov uses the word "hump" here. I thought it might be related. — Dejan Govc, Mar 22 '12 at 11:25
In my neighbourhood there are speed humps. I have never understood the difference. — Martin Argerami, Mar 22 '12 at 13:47
MATLAB has something called the humps() function which it uses for demonstrating quadrature, optimization, and rootfinding routines. — J. M. ain't a mathematician, Mar 24 '12 at 14:32
With either name, it would seem something climactic is happening in a short interval of time. — zhw., Jul 19 '17 at 04:13

score 3 · Answer 1 · answered Jul 18 '17 at 22:37

i think any function $g(x)$ that is the derivative of a sigmoid function

$$ g(x) = f'(x) $$

where $f(x)$ may be any of the functions shown as examples of an S-shaped sigmoid function indicated in the Wikipedia article, any of those can be legitimately called a "hump function".

score 1 · Answer 2 · answered Jul 18 '17 at 22:25

1

If you're referring to the category of functions called humps that I'm familiar with, then of one these functions is $f$ defined below (try plotting it for $x \in [-2, 8]$, so that $f(x) \in [0, 25]$).

$$f(x) = \frac{1}{(x-3)^2+0.1} + \frac{1}{(x-2)^2+0.05} + 2$$

answered Jul 18 '17 at 22:25

and to reference @J. M. isn't a mathematician's point, I do think the name comes from MATLAB. – Jul 19 '17 at 13:28
see: https://www.mathworks.com/help/matlab/examples/function-functions.html#zmw57dd0e8359 – Jul 19 '17 at 13:29

score 0 · Answer 3 · edited Jan 23 '22 at 20:18

There are many a hump functions (so-called kernel functions): see the wikipedia.

The functions in examples above are not compact, i.e. for all $x\in(-\infty, \infty),\;\, f(x) \not= 0.$

If you want a compact function, where non-zero value is defined only in a line segment: $$ f(x)= \begin{cases} \frac{(\Delta^2 - x^2)^3}{\Delta},\quad |x|<\Delta\\ 0, \quad\quad\quad\;\;\,|x|\geq \Delta. \end{cases} $$ This function has continuous derivatives. Therefore, it $f(x)$ is a smooth function.

What is a hump function?

3 Answers3