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Mathematics
Topology and Analysis
I need a mapping from the unit Hypercube [0,1]^n to a given simplex
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[QUOTE="benorin, post: 6306604, member: 37674"] So I solved the fractional-Integral equation but with the Hadamard Left Fractional Integral Operator instead of the R-L fractional integral operator, and it turns out that post #12 has the Jacobian determinant correct. I arrived at this conclusion in a round about way. Beginning with the well known integral $$\Gamma (z) \zeta (z)=\int_{u=0}^{\infty} \frac{u^{z-1}}{e^u-1}\, du, \, \, \Re\left[ z\right] >1$$ Make the substitution ##u= \log \frac{x}{t}\Rightarrow du=-\frac{dt}{t}## and use the negative from du to flip the bounds of integration to get $$\zeta (z)=\frac{1}{\Gamma (z)}\int_{t=0}^{x} \log ^{z-1}\left( \frac{x}{t}\right) \frac{1}{\frac{x}{t}-1}\, \frac{dt}{t}, \, \, \Re\left[ z\right] >1$$ which, after setting ##x=1## in the factor of the integrand following the log factor, is exactly the Hadamard fractional integral of ##\frac{t}{1-t}## of order z and, get this, the Hadamard FI operator interpolates the n-fold integral $$\int_{0}^{x_0}\int_{0}^{x_1}\cdots\int_{0}^{x_{n-1}}f(x_n)\frac{dx_n\ldots dx_1}{x_n \cdots x_1} = \frac{1}{\Gamma (n) }\int_0^{x_0}\log ^{n-1}\left( \frac{x_0}{t}\right) f(t) \frac{dt}{t}$$ Hence the integral we were looking at in this thread should have been $$\int_{0}^{y_0}\int_{0}^{y_1}\cdots\int_{0}^{y_{n-1}}\frac{y_n}{1-y_n}\frac{dy_n\ldots dy_1}{y_{n}\cdots y_1} $$ so the Jacobian determinant of post #12 was correct after multiplying by a factor of ##\tfrac{y_n}{y_n}##. [/QUOTE]
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Forums
Mathematics
Topology and Analysis
I need a mapping from the unit Hypercube [0,1]^n to a given simplex
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