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1. Minimizing L ₁ over L ₂ norms on the gradient

   https://doi.org/10.1088/1361-6420/ac64fb  

   Wang, Chao; Tao, Min; Chuah, Chen-Nee; Nagy, James; Lou, Yifei (May 2022, Inverse Problems)

   Abstract In this paper, we study the L 1 / L 2 minimization on the gradient for imaging applications. Several recent works have demonstrated that L 1 / L 2 is better than the L 1 norm when approximating the L 0 norm to promote sparsity. Consequently, we postulate that applying L 1 / L 2 on the gradient is better than the classic total variation (the L 1 norm on the gradient) to enforce the sparsity of the image gradient. Numerically, we design a specific splitting scheme, under which we can prove subsequential and global convergence for the alternating direction method of multipliers (ADMM) under certain conditions. Experimentally, we demonstrate visible improvements of L 1 / L 2 over L 1 and other nonconvex regularizations for image recovery from low-frequency measurements and two medical applications of magnetic resonance imaging and computed tomography reconstruction. Finally, we reveal some empirical evidence on the superiority of L 1 / L 2 over L 1 when recovering piecewise constant signals from low-frequency measurements to shed light on future works. 

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2. Projections onto L^p Bergman spaces of Reinhardt domains

   https://doi.org/10.1016/j.aim.2024.109790  

   Chakrabarti, Debraj; Edholm, Luke D (August 2024, Advances in Mathematics)

   For $$1<\infty$$, we emulate the Bergman projection on Reinhardt domains by using a Banach-space basis of $L^p$-Bergman space. The construction gives an integral kernel generalizing the ($L^2$) Bergman kernel. The operator defined by the kernel is shown to be an absolutely bounded projection on the $L^p$-Bergman space on a class of domains where the $L^p$-boundedness of the Bergman projection fails for certain $$p \neq 2$$. As an application, we identify the duals of these $L^p$-Bergman spaces with weighted Bergman spaces. 

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3. (In)equality distance patterns and embeddability into Hilbert spaces

   Alexandru Chirvasitu (October 2022, Palestine journal of mathematics)

   We show that compact Riemannian manifolds, regarded as metric spaces with their global geodesic distance, cannot contain a number of rigid structures such as (a) arbitrarily large regular simplices or (b) arbitrarily long sequences of points equidistant from pairs of points preceding them in the sequence. All of this provides evidence that Riemannian metric spaces admit what we term loose embeddings into finite-dimensional Euclidean spaces: continuous maps that preserve both equality as well as inequality. We also prove a local-to-global principle for Riemannian-metric-space loose embeddability: if every finite subspace thereof is loosely embeddable into a common R^N , then the metric space as a whole is loosely embeddable into R^N in a weakened sense. 

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4. Coarse equivalence versus bijective coarse equivalence of expander graphs

   https://doi.org/10.1007/s00209-024-03512-w  

   Baudier, Florent P; Braga, Bruno M; Farah, Ilijas; Vignati, Alessandro; Willett, Rufus (July 2024, Mathematische Zeitschrift)

   We provide a characterization of when a coarse equivalence between coarse disjoint unions of expander graphs is close to a bijective coarse equivalence. We use this to show that if the uniform Roe algebras over metric spaces that are coarse unions of expanders graphs are isomorphic, then the metric spaces must be bijectively coarsely equivalent. 

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5. The L _p chord Minkowski problem

   https://doi.org/10.1515/ans-2022-0041  

   Xi, Dongmeng; Yang, Deane; Zhang, Gaoyong; Zhao, Yiming (January 2023, Advanced Nonlinear Studies)

   Abstract Chord measures are newly discovered translation-invariant geometric measures of convex bodies in R n {{\mathbb{R}}}^{n} , in addition to Aleksandrov-Fenchel-Jessen’s area measures. They are constructed from chord integrals of convex bodies and random lines. Prescribing the L p {L}_{p} chord measures is called the L p {L}_{p} chord Minkowski problem in the L p {L}_{p} Brunn-Minkowski theory, which includes the L p {L}_{p} Minkowski problem as a special case. This article solves the L p {L}_{p} chord Minkowski problem when p > 1 p\gt 1 and the symmetric case of 0 < p < 1 0\lt p\lt 1 . 

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