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1. Some matrix properties preserved by generalized matrix functions

   https://doi.org/10.1515/spma-2019-0003  

   Benzi, Michele; Huang, Ru (January 2019, Special Matrices)

   null (Ed.)

   Abstract Generalized matrix functions were first introduced in [J. B. Hawkins and A. Ben-Israel, Linear and Multilinear Algebra, 1(2), 1973, pp. 163-171]. Recently, it has been recognized that these matrix functions arise in a number of applications, and various numerical methods have been proposed for their computation. The exploitation of structural properties, when present, can lead to more efficient and accurate algorithms. The main goal of this paper is to identify structural properties of matrices which are preserved by generalized matrix functions. In cases where a given property is not preserved in general, we provide conditions on the underlying scalar function under which the property of interest will be preserved by the corresponding generalized matrix function. 

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2. Enhanced alternating energy minimization methods for stochastic galerkin matrix equations

   https://doi.org/10.1007/s10543-021-00903-x  

   Lee, Kookjin; Elman, Howard C.; Powell, Catherine E.; Lee, Dongeun (January 2022, BIT numerical mathematics)

   In uncertainty quantification, it is commonly required to solve a forward model consisting of a partial differential equation (PDE) with a spatially varying uncertain coefficient that is represented as an affine function of a set of random variables, or parameters. Discretizing such models using stochastic Galerkin finite element methods (SGFEMs) leads to very high-dimensional discrete problems that can be cast as linear multi-term matrix equations (LMTMEs). We develop efficient computational methods for approximating solutions of such matrix equations in low rank. To do this, we follow an alternating energy minimization (AEM) framework, wherein the solution is represented as a product of two matrices, and approximations to each component are sought by solving certain minimization problems repeatedly. Inspired by proper generalized decomposition methods, the iterative solution algorithms we present are based on a rank-adaptive variant of AEM methods that successively computes a rank-one solution component at each step. We introduce and evaluate new enhancement procedures to improve the accuracy of the approximations these algorithms deliver. The efficiency and accuracy of the enhanced AEM methods is demonstrated through numerical experiments with LMTMEs associated with SGFEM discretizations of parameterized linear elliptic PDEs. 

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3. Lump solutions to a (2+1)-dimensional fifth-order KdV-like equation

   Batwa, Sumayah; Ma, Wen-Xiu (July 2018, Advances in Mathematical Physics (Print))

   A (2+1)-dimensional fifth-order KdV-like equation is introduced through a generalized bilinear equation with the prime number . The new equation possesses the same bilinear form as the standard (2+1)-dimensional fifth-order KdV equation. By Maple symbolic computation, classes of lump solutions are constructed from a search for quadratic function solutions to the corresponding generalized bilinear equation. We get a set of free parameters in the resulting lump solutions, of which we can get a nonzero determinant condition ensuring analyticity and rational localization of the solutions. Particular classes of lump solutions with special choices of the free parameters are generated and plotted as illustrative examples. 

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4. Spin-density calculation via the graphical unitary group approach

   https://doi.org/10.1080/00268976.2022.2091049  

   Spada, Rene F.; Franco, Maurício P.; Nieman, Reed; Aquino, Adelia J.; Shepard, Ron; Plasser, Felix; Lischka, Hans (June 2022, Molecular Physics)

   In this work we discuss the calculation of the spin-density matrix from fundamental spin principles as implemented in the Columbus Program System employing the graphical unitary group approach (GUGA). First, a general equation for the spin-density matrix is derived in terms of the one- and two-particle reduced density matrices, quantities that are spin-independent and readily available within the GUGA formalism. Next, the evaluation of this equation using the Shavitt loop values is discussed. Finally, the spatially resolved counterpart of the spin-density matrix, the spin distribution, is calculated for the phenalenyl radical and structures produced by heteroatoms with mono- and di-substitutions. The physical meaning of the spin-density along with its computational description using various methods is discussed putting special emphasis on negative contributions to the spin-density and their quantification via a spin-promotion index. 

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5. Log-canonical coordinates for symplectic groupoid and cluster algebras

   https://doi.org/10.1093/imrn/rnac101  

   Chekhov, L.; Shapiro, M. (January 2022, International mathematics research notices)

   Using Fock--Goncharov higher Teichmüller space variables we derive Darboux coordinate representation for entries of general symplectic leaves of the A_n groupoid of upper-triangular matrices and, in a more general setting, of higher-dimensional symplectic leaves for algebras governed by the reflection equation with the trigonometric R-matrix. The obtained results are in a perfect agreement with the previously obtained Poisson and quantum representations of groupoid variables for A_3 and A_4 in terms of geodesic functions for Riemann surfaces with holes. We represent braid-group transformations for A_n via sequences of cluster mutations in the special A_n-quiver. We prove the groupoid relations for quantum transport matrices and, as a byproduct, obtain the Goldman bracket in the semiclassical limit. 

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