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1. On the Classification of Normal Stein Spaces and Finite Ball Quotients With Bergman–Einstein Metrics

   https://doi.org/10.1093/imrn/rnab120  

   Ebenfelt, Peter ; Xiao, Ming ; Xu, Hang ( June 2021 , International Mathematics Research Notices)

   null (Ed.)

   Abstract We study the Bergman metric of a finite ball quotient $\mathbb{B}^n/\Gamma $, where $n \geq 2$ and $\Gamma \subseteq{\operatorname{Aut}}({\mathbb{B}}^n)$ is a finite, fixed point free, abelian group. We prove that this metric is Kähler–Einstein if and only if $\Gamma $ is trivial, that is, when the ball quotient $\mathbb{B}^n/\Gamma $ is the unit ball ${\mathbb{B}}^n$ itself. As a consequence, we characterize the unit ball among normal Stein spaces with isolated singularities and abelian fundamental groups in terms of the existence of a Bergman–Einstein metric. 

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2. Free Stein irregularity and dimension

   https://doi.org/10.7900/jot.2019aug29.2271  

   Charlesworth, Ian ; Nelson, Brent ( January 2021 , Journal of operator theory)

   null (Ed.)

   We introduce a free probabilistic quantity called free Stein irregularity, which is defined in terms of free Stein discrepancies. It turns out that this quantity is related via a simple formula to the Murray–von Neumann dimension of the closure of the domain of the adjoint of the non-commutative Jacobian associated to Voiculescu’s free difference quotients. We call this dimension the free Stein dimension, and show that it is a ∗-algebra invariant. We relate these quantities to the free Fisher information, the non-microstates free entropy, and the non-microstates free entropy dimension. In the one-variable case, we show that the free Stein dimension agrees with the free entropy dimension, and in the multivariable case compute it in a number of examples. 

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3. Universal Hypothesis Testing with Kernels: Asymptotically Optimal Tests for Goodness of Fit

   Zhu, Shengyu ; Chen, Biao ; Yang, Pengfei ; Chen, Zhitang ( April 2019 , International Conference on Artificial Intelligence and Statistics)

   We characterize the asymptotic performance of nonparametric goodness of fit testing. The exponential decay rate of the type-II error probability is used as the asymptotic performance metric, and a test is optimal if it achieves the maximum rate subject to a constant level constraint on the type-I error probability. We show that two classes of Maximum Mean Discrepancy (MMD) based tests attain this optimality on Rd, while the quadratictime Kernel Stein Discrepancy (KSD) based tests achieve the maximum exponential decay rate under a relaxed level constraint. Under the same performance metric, we proceed to show that the quadratic-time MMD based two-sample tests are also optimal for general two-sample problems, provided that kernels are bounded continuous and characteristic. Key to our approach are Sanov’s theorem from large deviation theory and the weak metrizable properties of the MMD and KSD. 

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4. Complete Kähler–Einstein Metric on Stein Manifolds With Negative Curvature

   https://doi.org/10.1093/imrn/rnaa142  

   Lee, Man-Chun ( June 2020 , International Mathematics Research Notices)

   null (Ed.)

   Abstract We show the existence of complete negative Kähler–Einstein metric on Stein manifolds with holomorphic sectional curvature bounded from above by a negative constant. We prove that any Kähler metrics on such manifolds can be deformed to the complete negative Kähler–Einstein metric using the normalized Kähler–Ricci flow. 

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5. A non-increasing tree growth process for recursive trees and applications

   https://doi.org/10.1017/S0963548320000073  

   Eslava, Laura ( January 2021 , Combinatorics, Probability and Computing)

   null (Ed.)

   Abstract We introduce a non-increasing tree growth process $((T_n,{\sigma}_n),\, n\ge 1)$ , where T n is a rooted labelled tree on n vertices and σ n is a permutation of the vertex labels. The construction of ( T n , σ n ) from ( T n −1 , σ n −1 ) involves rewiring a random (possibly empty) subset of edges in T n −1 towards the newly added vertex; as a consequence T n −1 ⊄ T n with positive probability. The key feature of the process is that the shape of T n has the same law as that of a random recursive tree, while the degree distribution of any given vertex is not monotone in the process. We present two applications. First, while couplings between Kingman’s coalescent and random recursive trees were known for any fixed n , this new process provides a non-standard coupling of all finite Kingman’s coalescents. Second, we use the new process and the Chen–Stein method to extend the well-understood properties of degree distribution of random recursive trees to extremal-range cases. Namely, we obtain convergence rates on the number of vertices with degree at least $c\ln n$ , c ∈ (1, 2), in trees with n vertices. Further avenues of research are discussed. 

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