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Inline MathJax bug with operatorname command? #2
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aaronsheldon committed Apr 15, 2020
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10 changes: 8 additions & 2 deletions src/vonFoersterHazards.ipynb
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"\n",
"## Methods\n",
"\n",
"In general when we are unable to observe or model every transition, such as when the integral of the hazard rate is taken over a duration larger than the smallest [characteristic dwelling time](https://en.wikipedia.org/wiki/Relaxation_%28physics%29) of the states, the probability density matrix requires calculation of the [adjoint derivative](https://en.wikipedia.org/wiki/Derivative_of_the_exponential_map) of the cumulative probability matrix, where the dot is the [currying](https://en.wikipedia.org/wiki/Currying) application of $\\operatorname{ad}_X Y=XY-YX$, the [adjoint operator](https://en.wikipedia.org/wiki/Adjoint_representation), to the parenthesized hazard rate $H$:\n",
"In general when we are unable to observe or model every transition, such as when the integral of the hazard rate is taken over a duration larger than the smallest [characteristic dwelling time](https://en.wikipedia.org/wiki/Relaxation_%28physics%29) of the states, the probability density matrix requires calculation of the [adjoint derivative](https://en.wikipedia.org/wiki/Derivative_of_the_exponential_map) of the cumulative probability matrix, where the dot is the [currying](https://en.wikipedia.org/wiki/Currying) application of the [adjoint operator](https://en.wikipedia.org/wiki/Adjoint_representation):\n",
"\n",
"$$\n",
"\\operatorname{ad}_X Y=XY-YX\n",
"$$\n",
"\n",
"to the parenthesized hazard rate $H$:\n",
"\n",
"$$\n",
"\\operatorname{\\mathbb{P}}\\left[ T_{i \\rightarrow j} = t \\right] = \\operatorname{\\left[ \\frac{e^{-\\operatorname{ad}_{\\int_0^t H ds} \\cdot} - 1}{\\operatorname{ad}_{\\int_0^t H ds} \\cdot} \\right]} \\left(H\\right) e^{- \\int_0^t H ds}\n",
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},
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"execution_count": 1,
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