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1. The extended generalized Haar-Walsh transform and applications

   https://doi.org/10.1117/12.2528923  

   Shao, Yiqun; Saito, Naoki (September 2019, Wavelets and Sparsity XVIII)

   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. Our previous Generalized Haar-Walsh Transform (GHWT) is a multiscale transform for signals on graphs, which is a generalization of the classical Haar and Walsh-Hadamard Transforms. This article proposes the extended Generalized Haar-Walsh Transform (eGHWT). The eGHWT and its associated best-basis selection algorithm for graph signals will significantly improve the performance of the previous GHWT with the similar computational cost, O(N log N) where N is the number of nodes of an input graph. While the previous GHWT/best-basis algorithm seeks the most suitable orthonormal basis for a given task among more than 1.5^N possible bases, the eGHWT/best-basis algorithm can find a better one by searching through more than 0.618 ⋅ (1.84)^N possible bases. This article describes the details of the eGHWT/basis-basis algorithm and demonstrates its superiority using several examples including genuine graph signals as well as conventional digital images viewed as graph signals. Keywords: Multiscale basis dictionaries, wavelets on graphs, graph signal processing, adapted time-frequency analysis, the best-basis algorithm 

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2. 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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3. Fast Generalized DFTs for all Finite Groups

   https://doi.org/10.1109/FOCS.2019.00052  

   Umans, Chris (November 2019, Proceedings of FOCS 2019)

   null (Ed.)

   For any finite group G, we give an arithmetic algorithm to compute generalized Discrete Fourier Transforms (DFTs) with respect to G, using O(|G|^{ω/2+ε}) operations, for any ε > 0. Here, ω is the exponent of matrix multiplication. 

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4. Unitary groups and augmented Cuntz semigroups of separable simple Z-Stable C∗-algebras

   https://doi.org/10.1142/S0129167X22500185  

   Lin, Huaxin (February 2022, International journal of mathematics)

   Let A be a separable simple exact Z-stable C∗-algebra. We show that the unitary group of \tilde{A} has the cancellation property. If A has continuous scale then the Cuntz semigroup of A has strict comparison property and a weak cancellation property. Let C be a 1-dimensional noncommutative CW complex with K1(C) = {0}. Suppose that λ : Cu∼(C) → Cu∼(A) is a morphism in the augmented Cuntz semigroups which is strictly positive. Then there exists a sequence of homomorphisms φn : C → A such that limn→∞ Cu∼(φn) = λ. This result leads to the proof that every separable amenable simple C∗-algebra in the UCT class has rationally generalized tracial rank at most one. 

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5. Gyroscopic polynomials

   https://doi.org/10.1016/j.jcp.2023.112268  

   Ellison, Abram C.; Julien, Keith (September 2023, Journal of Computational Physics)

   Gyroscopic alignment gives rise to highly spatially anisotropic columnar structures that in combination with complex domain boundaries pose challenges for efficient numerical discretizations and computations. We define gyroscopic polynomials to be three-dimensional polynomials expressed in a coordinate system that conforms to rotational alignment. We remap the original domain with radius-dependent boundaries onto a right cylindrical or annular domain to create the computational domain in this coordinate system. We find the volume element expressed in gyroscopic coordinates leads naturally to a hierarchy of orthonormal bases. We build the bases out of Jacobi polynomials in the vertical and generalized Jacobi polynomials in the radial. Because these coordinates explicitly conform to flow structures found in rapidly rotating systems the bases represent fields with a relatively small number of modes. We develop the operator structure for one-dimensional semi-classical orthogonal polynomials as a building block for differential operators in the full three-dimensional cylindrical and annular domains. The differentiation operators of generalized Jacobi polynomials generate a sparse linear system for discretization of differential operators acting on the gyroscopic bases. This enables efficient simulation of systems with strong gyroscopic alignment. 

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