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1. Infinite measure renewal theorem and related results: INFINITE MEASURE RENEWAL THEOREM AND RELATED RESULTS

   https://doi.org/10.1112/blms.12217  

   Dolgopyat, Dmitry; Nándori, Péter (November 2018, Bulletin of the London Mathematical Society)
2. Geometric scattering on measure spaces

   https://doi.org/10.1016/j.acha.2024.101635  

   Chew, Joyce; Hirn, Matthew; Krishnaswamy, Smita; Needell, Deanna; Perlmutter, Michael; Steach, Holly; Viswanath, Siddharth; Wu, Hau-Tieng (May 2024, Applied and Computational Harmonic Analysis)

   The scattering transform is a multilayered, wavelet-based transform initially introduced as a mathematical model of convolutional neural networks (CNNs) that has played a foundational role in our understanding of these networks’ stability and invariance properties. In subsequent years, there has been widespread interest in extending the success of CNNs to data sets with non- Euclidean structure, such as graphs and manifolds, leading to the emerging field of geometric deep learning. In order to improve our understanding of the architectures used in this new field, several papers have proposed generalizations of the scattering transform for non-Euclidean data structures such as undirected graphs and compact Riemannian manifolds without boundary. Analogous to the original scattering transform, these works prove that these variants of the scattering transform have desirable stability and invariance properties and aim to improve our understanding of the neural networks used in geometric deep learning. In this paper, we introduce a general, unified model for geometric scattering on measure spaces. Our proposed framework includes previous work on compact Riemannian manifolds without boundary and undirected graphs as special cases but also applies to more general settings such as directed graphs, signed graphs, and manifolds with boundary. We propose a new criterion that identifies to which groups a useful representation should be invariant and show that this criterion is sufficient to guarantee that the scattering transform has desirable stability and invariance properties. Additionally, we consider finite measure spaces that are obtained from randomly sampling an unknown manifold. We propose two methods for constructing a data-driven graph on which the associated graph scattering transform approximates the scattering transform on the underlying manifold. Moreover, we use a diffusion-maps based approach to prove quantitative estimates on the rate of convergence of one of these approximations as the number of sample points tends to infinity. Lastly, we showcase the utility of our method on spherical images, a directed graph stochastic block model, and on high-dimensional single-cell data. 

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3. How to measure natural selection

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   Stinchcombe, John R.; Kelley, Joanna L.; Conner, Jeffrey K.; Freckleton, Robert (June 2017, Methods in Ecology and Evolution)
4. Empirical Approaches to Measure Connectivity

   https://doi.org/10.5670/oceanog.2019.311  

   White, J. Wilson; Carr, Mark; Caselle, Jennifer; Palumbi, Stephen; Warner, Robert; Menge, Bruce; Milligan, Kristen (September 2019, Oceanography)
5. Hausdorff Dimension of Caloric Measure

   https://doi.org/10.1353/ajm.2025.a954648  

   Badger, Matthew; Genschaw, Alyssa (April 2025, American Journal of Mathematics)

   abstract: We examine caloric measures $$\omega$$ on general domains in $$\RR^{n+1}=\RR^n\times\RR$$ (space $$\times$$ time) from the perspective of geometric measure theory. On one hand, we give a direct proof of a consequence of a theorem of Taylor and Watson (1985) that the lower parabolic Hausdorff dimension of $$\omega$$ is at least $$n$$ and $$\omega\ll\Haus^n$$. On the other hand, we prove that the upper parabolic Hausdorff dimension of $$\omega$$ is at most $$n+2-\beta_n$$, where $$\beta_n>0$$ depends only on $$n$$. Analogous bounds for harmonic measures were first shown by Nevanlinna (1934) and Bourgain (1987). Heuristically, we show that the \emph{density} of obstacles in a cube needed to make it unlikely that a Brownian motion started outside of the cube exits a domain near the center of the cube must be chosen according to the ambient dimension. In the course of the proof, we give a caloric measure analogue of Bourgain's alternative: for any constants $$0<\epsilon\ll_n \delta<1/2$ and closed set $$E\subset\RR^{n+1}$$, either (i) $$E\cap Q$$ has relatively large caloric measure in $$Q\setminus E$$ for every pole in $$F$$ or (ii) $$E\cap Q_*$$ has relatively small $$\rho$$-dimensional parabolic Hausdorff content for every $$n<\rho\leq n+2$$, where $$Q$$ is a cube, $$F$$ is a subcube of $$Q$$ aligned at the center of the top time-face, and $$Q_*$$ is a subcube of $$Q$$ that is close to, but separated backwards-in-time from $$F$$: \begin{gather*} Q=(-1/2,1/2)^n\times (-1,0),\quad F=[-1/2+\delta,1/2-\delta]^n\times[-\epsilon^2,0),\\[2pt] \text{and }Q_*=[-1/2+\delta,1/2-\delta]^n\times[-3\epsilon^2,-2\epsilon^2]. \end{gather*} Further, we supply a version of the strong Markov property for caloric measures. 

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