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1. eGHWT: The Extended Generalized Haar–Walsh Transform

   https://doi.org/10.1007/s10851-021-01064-w  

   Saito, Naoki; Shao, Yiqun (January 2022, Journal of Mathematical Imaging and Vision)

   Abstract Extending computational harmonic analysis tools from the classical setting of regular lattices to the more general setting of graphs and networks is very important, and much research has been done recently. The generalized Haar–Walsh transform (GHWT) developed by Irion and Saito (2014) is a multiscale transform for signals on graphs, which is a generalization of the classical Haar and Walsh–Hadamard transforms. We propose theextendedgeneralized Haar–Walsh transform (eGHWT), which is a generalization of the adapted time–frequency tilings of Thiele and Villemoes (1996). The eGHWT examines not only the efficiency of graph-domain partitions but also that of “sequency-domain” partitionssimultaneously. Consequently, the eGHWT and its associated best-basis selection algorithm for graph signals significantly improve the performance of the previous GHWT with the similar computational cost,$$O(N \log N)$$ $O(NlogN)$, whereNis the number of nodes of an input graph. While the GHWT best-basis algorithm seeks the most suitable orthonormal basis for a given task among more than$$(1.5)^N$$ ${\left(1.5\right)}^N$possible orthonormal bases in$$\mathbb {R}^N$$ $R^N$, the eGHWT best-basis algorithm can find a better one by searching through more than$$0.618\cdot (1.84)^N$$ $0.618·{\left(1.84\right)}^N$possible orthonormal bases in$$\mathbb {R}^N$$ $R^N$. This article describes the details of the eGHWT best-basis algorithm and demonstrates its superiority using several examples including genuine graph signals as well as conventional digital images viewed as graph signals. Furthermore, we also show how the eGHWT can be extended to 2D signals and matrix-form data by viewing them as a tensor product of graphs generated from their columns and rows and demonstrate its effectiveness on applications such as image approximation. 

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2. Topology and Local Geometry of the Eden Model

   https://doi.org/10.1007/s00454-022-00474-w  

   Manin, Fedor; Roldán, Érika; Schweinhart, Benjamin (February 2023, Discrete & Computational Geometry)

   Abstract The Eden cell growth model is a simple discrete stochastic process which produces a “blob” (aggregation of cells) in $$\mathbb {R}^d$$ $R^d$: start with one cube in the regular grid, and at each time step add a neighboring cube uniformly at random. This process has been used as a model for the growth of aggregations, tumors, and bacterial colonies and the healing of wounds, among other natural processes. Here, we study the topology and local geometry of the resulting structure, establishing asymptotic bounds for Betti numbers. Our main result is that the Betti numbers at timetgrow at a rate between$$t^{(d-1)/d}$$ $t^{(d-1)/d}$and$$P_d(t)$$ $P_d\left(t\right)$, where$$P_d(t)$$ $P_d\left(t\right)$is the size of the site perimeter. Assuming a widely believed conjecture, this establishes the rate of growth of the Betti numbers in every dimension. We also present the results of computational experiments on finer aspects of the geometry and topology, such as persistent homology and the distribution of shapes of holes. 

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3. Transversality of holomorphic maps into hyperquadrics

   https://doi.org/10.1007/s00208-025-03134-5  

   Huang, Xiaojun; Zhu, Weixia (March 2025, Mathematische Annalen)

   Abstract We study holomorphic mapsFfrom a smooth Levi non-degenerate real hypersurface$$ M_{\ell }\subset {\mathbb {C}}^n $$ $M_{\ell }\subset C^n$into a hyperquadric$$ {\mathbb {H}}_{\ell '}^N $$ $H_{{\ell }^{\prime }}^N$with signatures$$ \ell \le (n-1)/2 $$ $\ell \le (n-1)/2$and$$ \ell '\le (N-1)/2,$$ ${\ell }^{\prime }\le (N-1)/2,$respectively. Assuming that$$ N - n < n - 1,$$ $N-n<n-1,$we prove that if$$ \ell = \ell ',$$ $\ell ={\ell }^{\prime },$thenFis either CR transversal to$$ {\mathbb {H}}_{\ell }^N $$ $H_{\ell }^N$at every point of$$ M_{\ell },$$ $M_{\ell },$or it maps a neighborhood of$$ M_{\ell } $$ $M_{\ell }$in$$ {\mathbb {C}}^n $$ $C^n$into$$ {\mathbb {H}}_{\ell }^N.$$ $H_{\ell }^N.$Furthermore, in the case where$$ \ell ' > \ell ,$$ ${\ell }^{\prime }>\ell ,$we show that ifFis not CR transversal at$$0\in M_\ell ,$$ $0\in M_{\ell },$then it must be transversally flat. The latter is best possible. 

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4. 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)

   Abstract 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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5. 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)

   Abstract 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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