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1. Spectral Stability of the $${\overline{\partial }}-$$Neumann Laplacian: Domain Perturbations

   https://doi.org/10.1007/s12220-021-00769-z  

   Fu, Siqi ; Zhu, Weixia ( January 2022 , The Journal of Geometric Analysis)

   We study spectral stability of the$${\bar{\partial }}$$$\overline{\partial }$-Neumann Laplacian on a bounded domain in$${\mathbb {C}}^n$$$C^n$when the underlying domain is perturbed. In particular, we establish upper semi-continuity properties for the variational eigenvalues of the$${\bar{\partial }}$$$\overline{\partial }$-Neumann Laplacian on bounded pseudoconvex domains in$${\mathbb {C}}^n$$$C^n$, lower semi-continuity properties on pseudoconvex domains that satisfy property (P), and quantitative estimates on smooth bounded pseudoconvex domains of finite D’Angelo type in$${\mathbb {C}}^n$$$C^n$.

    

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2. Spherical convex hull of random points on a wedge

   https://doi.org/10.1007/s00208-023-02704-9  

   Besau, Florian ; Gusakova, Anna ; Reitzner, Matthias ; Schütt, Carsten ; Thäle, Christoph ; Werner, Elisabeth_M ( August 2023 , Mathematische Annalen)

   Consider two half-spaces$$H_1^+$$$H_1^{+}$and$$H_2^+$$$H_2^{+}$in$${\mathbb {R}}^{d+1}$$$R^{d+1}$whose bounding hyperplanes$$H_1$$$H_1$and$$H_2$$$H_2$are orthogonal and pass through the origin. The intersection$${\mathbb {S}}_{2,+}^d:={\mathbb {S}}^d\cap H_1^+\cap H_2^+$$$S_{2,+}^d:=S^d\cap H_1^{+}\cap H_2^{+}$is a spherical convex subset of thed-dimensional unit sphere$${\mathbb {S}}^d$$$S^d$, which contains a great subsphere of dimension$$d-2$$$d-2$and is called a spherical wedge. Choosenindependent random points uniformly at random on$${\mathbb {S}}_{2,+}^d$$$S_{2,+}^d$and consider the expected facet number of the spherical convex hull of these points. It is shown that, up to terms of lower order, this expectation grows like a constant multiple of$$\log n$$$logn$. A similar behaviour is obtained for the expected facet number of a homogeneous Poisson point process on$${\mathbb {S}}_{2,+}^d$$$S_{2,+}^d$. The result is compared to the corresponding behaviour of classical Euclidean random polytopes and of spherical random polytopes on a half-sphere.

    

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3. Fractional regularity, global persistence, and asymptotic properties of the Boussinesq equations on bounded domains

   https://doi.org/10.1007/s00028-025-01057-x  

   Aydın, Mustafa_Sencer ; Jayanti, Pranava_Chaitanya ( February 2025 , Journal of Evolution Equations)

   We address the long-time behavior of the 2D Boussinesq system, which consists of the incompressible Navier–Stokes equations driven by a non-diffusive density. We construct globally persistent solutions on a smooth bounded domain, when the initial data belongs to$$(H^k\cap V)\times H^k$$$(H^k\cap V)\times H^k$for$$k\in \mathbb {N}$$$k\in N$and$$H^s\times H^s$$$H^s\times H^s$for$$0$0<s<2$. The proofs use parabolic maximal regularity and specific compatibility conditions at the initial time. Additionally, we also deduce various asymptotic properties of the velocity and density in the long-time limit and present a necessary and sufficient condition for the convergence to a steady state.

    

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4. A note on polydegree (n, 1) rational inner functions, slice matrices, and singularities

   https://doi.org/10.1007/s00013-022-01812-3  

   Sola, Alan ( December 2022 , Archiv der Mathematik)

   We analyze certain compositions of rational inner functions in the unit polydisk$$\mathbb {D}^{d}$$$D^d$with polydegree (n, 1),$$n\in \mathbb {N}^{d-1}$$$n\in N^{d-1}$, and isolated singularities in$$\mathbb {T}^d$$$T^d$. Provided an irreducibility condition is met, such a composition is shown to be a rational inner function with singularities in precisely the same location as those of the initial function, and with quantitatively controlled properties. As an application, we answer ad-dimensional version of a question posed in Bickel et al. (Am J Math 144: 1115–1157, 2022) in the affirmative.

    

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5. An Efficient, Memory-Saving Approach for the Loewner Framework

   https://doi.org/10.1007/s10915-022-01800-3  

   Palitta, Davide ; Lefteriu, Sanda ( March 2022 , Journal of Scientific Computing)

   The Loewner framework is one of the most successful data-driven model order reduction techniques. IfNis the cardinality of a given data set, the so-called Loewner and shifted Loewner matrices$${\mathbb {L}}\in {\mathbb {C}}^{N\times N}$$$L\in C^{N\times N}$and$${\mathbb {S}}\in {\mathbb {C}}^{N\times N}$$$S\in C^{N\times N}$can be defined by solely relying on information encoded in the considered data set and they play a crucial role in the computation of the sought rational model approximation.In particular, the singular value decomposition of a linear combination of$${\mathbb {S}}$$$S$and$${\mathbb {L}}$$$L$provides the tools needed to construct accurate models which fulfill important approximation properties with respect to the original data set. However, for highly-sampled data sets, the dense nature of$${\mathbb {L}}$$$L$and$${\mathbb {S}}$$$S$leads to numerical difficulties, namely the failure to allocate these matrices in certain memory-limited environments or excessive computational costs. Even though they do not possess any sparsity pattern, the Loewner and shifted Loewner matrices are extremely structured and, in this paper, we show how to fully exploit their Cauchy-like structure to reduce the cost of computing accurate rational models while avoiding the explicit allocation of$${\mathbb {L}}$$$L$and$${\mathbb {S}}$$$S$. In particular, the use of thehierarchically semiseparableformat allows us to remarkably lower both the computational cost and the memory requirements of the Loewner framework obtaining a novel scheme whose costs scale with$$N \log N$$$NlogN$.

    

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