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Differentiation under the integral sign
In calculus, Leibniz's rule for differentiation under the integral sign states that, modulo precise regularity assumptions, $$\frac{d}{dx} \int_{a(x)}^{b(x)} f(x,t)\, dt = f(x,b(x))b'(x) - f(x,a(x))a'(x) + \int_{a(x)}^{b(x)} \frac{\partial}{\partial x}f(x,t)\, dt.$$ There is a nice generalization of this result to the case of integrating a time-dependent differential $k$-form over a time-varying $k$-submanifold with boundary.
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SICB 2018
In January 2018, we had the opportunity to present and attend at SICB 2018 held in San Francisco, CA.
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A fun fact about the Euler characteristic
There are many equivalent definitions of the Euler characteristic $$\chi(Y)$$ of a space $$Y$$. I recently learned another of these, valid in the case that $$Y$$ is a compact, orientable, smooth manifold.
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BigANT
Often, the time, cost, required tooling, and technical expertise associated with the design and fabrication of mechanical components often inhibit the prototyping of robots. To address this issue, we have developed BigANT -- a robot with chasis built from less than $20!
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Rigorous Enclosures of a Slow Manifold
Slow-fast dynamical systems have two time scales and an explicit parameter representing the ratio of these time scales. Locally invariant slow manifolds along which motion occurs on the slow time scale are a prominent feature of slow-fast systems. This paper introduces a rigorous numerical method to compute enclosures of the slow manifold of a slow-fast system with one fast and two slow variables.