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#appliedmathematics

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A few days back, I posted some #AnimatedGifs of the exact solution for a large-amplitude undamped, unforced #Pendulum. I then thought to complete the study to include the case when it has been fed enough #energy to allow it just to undergo #FullRotations, rather than just #oscillations. Well, it turns out that it is “a bit more complicated than I first expected” but I finally managed it.

Continued thread

Here we see three identical pendulums, oscillating independently. The red and purple ones are vibrating with small amplitudes and so their periods are nearly the same. But the blue one is undergoing what would be considered to be very large amplitude oscillations and has a significantly longer period. In fact, as the amplitude approaches π radians, the period increases without bound and approaches infinity.

Continued thread

A subtlety probably difficult to spot in the animation is that the interaction of the two waves leaves them phase shifted, with the taller wave gaining position, while the shorter one loses it. This further animation shows the interactions again (purple) but I’ve also shown what would happen if each wave moved without interaction with the other (red and blue).

Continued thread

When you start looking at #NonlinearWaves, some of these principles no longer apply. For example, in the Korteweg-de Vries equation, which has #Soliton or solutions, you can no longer simply add two solutions together as the resulting function would not be a solution of the governing equation. Waves of different heights travel at different velocities, with the taller waves moving faster than the shorter ones. Instead, they interact #nonlinearly.

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“My PhD was all about understanding what happens to blood flow in collapsible blood vessels like the giraffe jugular vein. In my postdoc I was investigating how to optimise ventilator settings for patients in ICU and then how to deliver inhaled therapies into the lungs. Since then, my focus has been in trying to understand how diseases like Asthma and other respiratory diseases originate and then progress. This involves incorporating biology and physics into mathematical and computational models, using approaches from different areas of applied maths. More recently I have started to look into the mechanisms that could lead to a rare lung disease called lymphangioleiomyomatosis (LAM) and Long Covid.” - Bindi Brook

➡️ hermathsstory.eu/bindi-brook/

“Since my goal was to solve a problem no one had ever solved before, it required a creative and flexible approach, one that emphasized the exploration, experimentation, and steady refinement of ideas. But perhaps the most important lesson I learned was that there is no single “correct” way to be a mathematician.” - Katy Micek

➡️ Find the full story at hermathsstory.eu/catherine-mic

Really happy with this new preprint with @profmjsimpson on
‘A profile likelihood-based workflow for identifiability analysis, estimation, and prediction with mechanistic mathematical models’
Brings together and builds on a bunch of work we’ve been doing with a great group of folks on inference for mechanistic models
biorxiv.org/content/10.1101/20 #Statistics #AppliedMathematics

bioRxivA profile likelihood-based workflow for identifiability analysis, estimation, and prediction with mechanistic mathematical modelsInterpreting data using mechanistic mathematical models provides a foundation for discovery and decision-making in all areas of science and engineering. Key steps in using mechanistic mathematical models to interpret data often include: (i) identifiability analysis; (ii) parameter estimation; and (iii) model prediction. We present a computationally efficient workflow that addresses these steps in a general likelihood-based framework. We first summarise the theoretical background of our workflow for efficient parameter identifiability, parameter estimation, and predictive confidence set construction. A central aspect of the workflow is using profile likelihood to construct profile-wise prediction intervals that propagate confidence sets for model parameters forward to predictions in a way that explicitly isolates how different parameter combinations affect model predictions. We also show how to combine these prediction confidence sets to give an overall prediction confidence set that accounts for all parameters and approximates the full likelihood-based prediction confidence set well. We demonstrate practical aspects of the workflow via three case studies focusing on ordinary differential equation (ODE) mechanistic models with both Gaussian and non-Gaussian noise models. While the three case studies focus on ODE-based models, the workflow applies to other classes of mathematical models, such as partial differential equations and simulation-based stochastic models. We provide open-source software on GitHub to replicate the case studies ### Competing Interest Statement The authors have declared no competing interest.