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The same question was asked on mathematics stack exchange https://math.stackexchange.com/questions/5126824/representing-vector-field-in-anyway-i-like-around-regular-points but has not received any response or comments.
Let $M$ be a smooth $n$-dimensional manifold and $X$ be a smooth vector field on it. Consider a chart $(U, x^1, \ldots, x^n)$ and another set of coordinates $(y^1,\ldots,y^n)$ on the same chart (or on an open subset of $U$ in which case we will just...

I've been trying to read the paper of G. Mockenhaupt, A. Seeger & C. Sogge, Wave Front Sets, Local Smoothing and Bourgain's Circular Maximal Theorem, 136 Ann. Math. 207 (1992).
In estimating a square function in $L^4(\mathbb{R}^2)$, the authors rely on a certain geometric "overlap lemma"....

I've seen it written that the existence of the Joyal model structure on simplicial sets is one of the foundational results on which the theory of $\infty$-categories (or, rather, the theory of quasi-categories as a model of $\infty$-categories) rests...

Is it consistent that there exists a $\subseteq^*$-decreasing chain $(X_\alpha)_{\alpha<\omega_1}$ of infinite subsets of $\omega$ such that for all ordinals $\alpha_0 < \alpha_1 < \cdots < \alpha_\omega$ with $\alpha_\omega = \sup_{n...