At the beginning was this problem asked to 11th-12th Grade students:
 

Let C a vertical cylinder of radius R_C = 10, and S a steel sphere of radius R = 4.
We place S into C and fill it with water up to the moment the water reaches the top of S.
Let V the volume of the water we used.
We then remove S and replaces it by another steel sphere S' with radius R' <> 4. Could it be that the free surface of the water reaches exactly the top of S'?
If it is so what is then the value of R'?

Mathematically does the equation (𝝅R_C²) ⨯ (2R) - (4 𝝅R_S³/3) = (𝝅R_C²) ⨯ (2R') - (4/3 𝝅R^'3), where R_C = 10 and R = 4, have other strictly positive solutions than the trivial one R' = R?

The answer here is yes: R' = 552^1/2 -2 ≅ 9.7473.. .

When I read this problem, I immediately asked myself the following question "Does a second sphere S' always exists whatever the values R_C > 0 and R ∊ (0, R_C]?".

In the attached worksheet I used two different Maple tools to answer this question:

• firstly solve+assumptions plus plots:-inequal to visualize the (R_C, R) domain where S' exists,
• next solve/parametric to present another way to get the characterization of this same domain.

The problem is that solve+assumptions and plots:-inequal both give the same correct result but solve/parametric does not.
PlotsInequal_vs_SolveParametric.mw

For this specific problem solve/parametric fails finding the correct result.
Is that a bug or did I misuse it?

Thanks in advance

I'm not sure of the Maple function I have to use to represent the modified Bessel function of the third kind with index 1.

The Bessel function of the third kind is also named the Hankel function.

I mistakenly started to use HankelH1 or HankelH2 but quickly realized they are both complex-valued functions while the paper I'm working on uses a real-valued modified Bessel function of the third kind: finally, does this latter exist in Maple 2015?

Thanks in advance

I feel that it is increasingly common for answers to generate no reaction from their authors.

Perhaps the author simply voted up... but in that case, would it be possible to indicate this (something like "XXX voted up")? 

This would avoid wasting time responding to the same author in the future, as it doesn't seem to care about commenting the answer.

I'm struggling to construct a statistical Distribution involving Product.
This is likely a question of delayed evaluation but I'm not capable to fix it.
Can you please look to this  Product_error.mw  worksheet  and help me fixing the issue?

Thanks in advance

A classical probability result says that if G₁ and G₂ are two independent Gamma random variables with same scale parameter (let's say 1 to simplify) and shape parameters a₁ and a₂ respectively, then G_k / (G₁ + G₂ ) is a Beta random variable with parameters (a_k , a_3-k ) (k=1..2).

In the attached file it is shown that (Maple 2015) function Statistics:-PDF fails in computing the PDF of G_k.
Noting strange here if you observe that even in the extremely simple case Z = X / (X+Y), where both X and Y are independent Uniform random variable with support [0, 1), Maple 2015 already fails in computing PDF(Z).

An alternative to Statistics:-PDF is to write explicitely the double integration which defines CDF(Z) (to begin with, and later PDF(G_k)) and ask Maple to do the integrations.
This approach works for Z but requires helping Maple when X and Y are still independent Uniform random variables but with respective non instanciated supports [0, a₁) and [0, a₂).

Applying to the Gamma-Gamma case the recipies I introduced in the Uniform-Uniform case does not give any result, unless in the very particular case where the shape parameters a₁ and a₂ are (strictly) positive integers.

All the details are in X_over_(X_plus_Y).mw

Do you have any idea how to prove with Maple the probability result mentioned at the head of this question?

PS: The "classical method" to do compute PDF(Z) consists in changing the integration variables < x₁, x₂ > into < x₁ = v₁v₂, x₂ = v₂ (1-v₁) > (see for instance Stack exchange)... but even after having dome it I still cannot get the desired result.

Thanks in advance.