x Im tc

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These are questions asked by x Im tc

I am trying to figure out the width-at-half-maximum for a speical case of the difference of Guaussians.  In this scenario, I start with a standard formula for a Gaussian:

f(x, x0, S, Gmax) = -Gmax*(exp(S*(x - x0)^2*(-1/2)) - 1)*heaviside(x - x0)

 

where x0, is the location of the peak, S is the spread, and Gmax is the amplitude.  However, then I take a difference of two, assuming that x0 is 0 for both, that Gmax is unity for both, and the only thing free to vary between them is the spread.

fDOG(x, S_a, S_d) = f(x, 0, S_a, 1) - f(x, 0, S_d, 1)

 

Here I also assume that S_a > S_d, and that all values (including x) are real positive numbers.  In this case, I (believe) I always get a peak function that rises from y = 0 at x = 0 to some peak, and then falls back to y = 0 at infinity.

Differentiating fDOG and solving for y = 0, I can find the time of the peak of this function:

tpeak(S_a, S_d) = (sqrt(2)*sqrt(ln(S_a/S_d))/sqrt(S_a - S_d))

 

I can then find the amplitude at the peak by substituting tpeak for x in fDOG. However, what I would like to do now is is find the (two) points on x where y = fDOG(tpeak, S_a, S_d)/2.  Is this possible?

Hello all,

I am very new to Maple but it looks like a wonderful symbolic computing tool.  I am hoping to gain some familiarity with it and one of my first questions is this:  I have identified a nice formula refered to as a modified inverse gamma function.  This is a "peak function" with a couple nice features.  Firstly, it can describe both positively and negatively skewed distributions, and secondly its mode and amplitude are easily recognized as (X0,Ym).  I have entered it into a Maple 2019 notebook as so:

Y = Ym*(c/((d + 1)*(X - X__0) + c))^(d + 1)*exp((X - X__0)*(d + 1)^2/((d + 1)*(X - X__0) + c))

What I would like to do is calculate the positions, X, of the two points where it is half-maximal.  That is, where Y=1/2Ym.  I would like to assume that all numbers are real, etc.  I should point out that in the real world, I will have optimized the values of Ym, X0, c and d by first fitting to actual data, so this may helpfully constrain the problem.

Can someone explain how I might go about this?  If you assume I know nothing, I will not be offended.

Thank you!

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