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1. Improved quantitative unique continuation for complex-valued drift equations in the plane

   https://doi.org/10.1515/forum-2022-0114  

   Davey, Blair; Kenig, Carlos; Wang, Jenn-Nan (June 2022, Forum Mathematicum)

   Abstract In this article, we investigate the quantitative unique continuation properties of complex-valued solutions to drift equations in the plane.We consider equations of the form Δ ⁢ u + W ⋅ ∇ ⁡ u = 0 {\Delta u+W\cdot\nabla u=0} in ℝ 2 {\mathbb{R}^{2}} ,where W = W 1 + i ⁢ W 2 {W=W_{1}+iW_{2}} with each W j {W_{j}} being real-valued.Under the assumptions that W j ∈ L q j {W_{j}\in L^{q_{j}}} for some q 1 ∈ [ 2 , ∞ ] {q_{1}\in[2,\infty]} , q 2 ∈ ( 2 , ∞ ] {q_{2}\in(2,\infty]} and that W 2 {W_{2}} exhibits rapid decay at infinity,we prove new global unique continuation estimates.This improvement is accomplished by reducing our equations to vector-valued Beltrami systems.Our results rely on a novel order of vanishing estimate combined with a finite iteration scheme. 

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2. A Combinatorial Proof of a Symmetry for a Refinement of the Narayana Numbers

   https://doi.org/10.37236/12681  

   Bóna, Miklós; Dimitrov, Stoyan; Labelle, Gilbert; Li, Yifei; Pappe, Joseph; Vindas-Meléndez, Andrés R; Zhuang, Yan (April 2025, The Electronic Journal of Combinatorics)

   We establish a tantalizing symmetry of certain numbers refining the Narayana numbers. In terms of Dyck paths, this symmetry is interpreted in the following way: if $$w_{n,k,m}$$ is the number of Dyck paths of semilength $$n$$ with $$k$$ occurrences of $UD$ and $$m$$ occurrences of $UUD$, then $$w_{2k+1,k,m}=w_{2k+1,k,k+1-m}$$. We give a combinatorial proof of this fact, relying on the cycle lemma, and showing that the numbers $$w_{2k+1,k,m}$$ are multiples of the Narayana numbers. We prove a more general fact establishing a relationship between the numbers $$w_{n,k,m}$$ and a family of generalized Narayana numbers due to Callan. A closed-form expression for the even more general numbers $$w_{n,k_{1},k_{2},\ldots, k_{r}}$$ counting the semilength-$$n$$ Dyck paths with $$k_{1}$$ $UD$-factors, $$k_{2}$$ $UUD$-factors, $$\ldots$$, and $$k_{r}$$ $$U^{r}D$$-factors is also obtained, as well as a more general form of the discussed symmetry for these numbers in the case when all rise runs are of certain minimal length. Finally, we investigate properties of the polynomials $$W_{n,k}(t)= \sum_{m=0}^k w_{n,k,m} t^m$$, including real-rootedness, $$\gamma$$-positivity, and a symmetric decomposition. 

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3. Internal Sequential Commutation and Single Generation

   https://doi.org/10.1093/imrn/rnaf103  

   Gao, David; Kunnawalkam_Elayavalli, Srivatsav; Patchell, Gregory; Tan, Hui (April 2025, International Mathematics Research Notices)

   Abstract We extract a precise internal description of the sequential commutation equivalence relation introduced in [14] for tracial von Neumann algebras. As an application we, prove that if a tracial von Neumann algebra $$N$$ is generated by unitaries $$\{u_{i}\}_{i\in \mathbb{N}}$$ such that $$u_{i}\sim u_{j}$$ (i.e., there exists a finite set of Haar unitaries $$\{w_{i}\}_{i=1}^{n}$$ in $$N^{\mathcal{U}}$$ such that $$[u_{i}, w_{1}]= [w_{k}, w_{k+1}]=[w_{n},u_{j}]=0$$ for all $$1\leq k< n$$), then $$N$$ is singly generated. This generalizes and recovers several known single generation phenomena for II$$_{1}$$ factors in the literature with a unified proof. 

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4. All-or-Nothing Phenomena: From Single-Letter to High Dimensions

   https://doi.org/10.1109/CAMSAP45676.2019.9022473  

   Reeves, Galen; Xu, Jiaming; Zadik, Ilias (December 2019, 2019 IEEE 8th International Workshop on Computational Advances in Multi-Sensor Adaptive Processing (CAMSAP))

   We consider the problem of estimating a $$p$$ -dimensional vector $$\beta$$ from $$n$$ observations $$Y=X\beta+W$$ , where $$\beta_{j}\mathop{\sim}^{\mathrm{i.i.d}.}\pi$$ for a real-valued distribution $$\pi$$ with zero mean and unit variance’ $$X_{ij}\mathop{\sim}^{\mathrm{i.i.d}.}\mathcal{N}(0,1)$$ , and $$W_{i}\mathop{\sim}^{\mathrm{i.i.d}.}\mathcal{N}(0,\ \sigma^{2})$$ . In the asymptotic regime where $$n/p\rightarrow\delta$$ and $$p/\sigma^{2}\rightarrow$$ snr for two fixed constants $$\delta,\ \mathsf{snr}\in(0,\ \infty)$$ as $$p\rightarrow\infty$$ , the limiting (normalized) minimum mean-squared error (MMSE) has been characterized by a single-letter (additive Gaussian scalar) channel. In this paper, we show that if the MMSE function of the single-letter channel converges to a step function, then the limiting MMSE of estimating $$\beta$$ converges to a step function which jumps from 1 to 0 at a critical threshold. Moreover, we establish that the limiting mean-squared error of the (MSE-optimal) approximate message passing algorithm also converges to a step function with a larger threshold, providing evidence for the presence of a computational-statistical gap between the two thresholds. 

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5. Optimal orthogonal group synchronization and rotation group synchronization

   https://doi.org/10.1093/imaiai/iaac022  

   Gao, Chao; Zhang, Anderson Y. (August 2022, Information and Inference: A Journal of the IMA)

   Abstract We study the statistical estimation problem of orthogonal group synchronization and rotation group synchronization. The model is $$Y_{ij} = Z_i^* Z_j^{*T} + \sigma W_{ij}\in{\mathbb{R}}^{d\times d}$$ where $$W_{ij}$$ is a Gaussian random matrix and $$Z_i^*$$ is either an orthogonal matrix or a rotation matrix, and each $$Y_{ij}$$ is observed independently with probability $$p$$. We analyze an iterative polar decomposition algorithm for the estimation of $Z^*$ and show it has an error of $$(1+o(1))\frac{\sigma ^2 d(d-1)}{2np}$$ when initialized by spectral methods. A matching minimax lower bound is further established that leads to the optimality of the proposed algorithm as it achieves the exact minimax risk. 

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