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Recently, Large Language Models (LLMs) have been increasingly able to solve advanced mathematical problems, including many that have been open for decades. This opens the door to expansion of mathematical knowledge at unprecedented scale. Yet, while LLMs may be able to conjecture and prove more and more theorems, it re...
Weighted integrated observability inequalities for heat equations usually involve a small-time factor of the form $e^{-\gamma/t}$. We prove that this scale is not an artefact of Carleman or spectral methods: it is forced by the geometry of the observation set. Let $A$ be a nonnegative self-adjoint operator on sections...
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