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1. Decomposition and conformal mapping techniques for the quadrature of nearly singular integrals

   https://doi.org/10.1007/s10543-023-00984-w  

   Mitchell, William; Natkin, Abbie; Robertson, Paige; Sullivan, Marika; Yu, Xuechen; Zhu, Chenxin (July 2023, BIT Numerical Mathematics)

   Abstract Gauss–Legendre quadrature, Clenshaw–Curtis quadrature and the trapezoid rule are powerful tools for numerical integration of analytic functions. For nearly singular problems, however, these standard methods become unacceptably slow. We discuss and generalize some existing methods for improving on these schemes when the location of the nearby singularity is known. We conclude with an application to some nearly singular surface integrals that arise in three-dimensional viscous fluid flow. 

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2. The parallel-transported (quasi)-diabatic basis

   https://doi.org/10.1063/5.0122781  

   Littlejohn, Robert; Rawlinson, Jonathan; Subotnik, Joseph (November 2022, The Journal of Chemical Physics)

   This article concerns the use of parallel transport to create a diabatic basis. The advantages of the parallel-transported basis include the facility with which Taylor series expansions can be carried out in the neighborhood of a point or a manifold such as a seam (the locus of degeneracies of the electronic Hamiltonian), and the close relationship between the derivative couplings and the curvature in this basis. These are important for analytic treatments of the nuclear Schrödinger equation in the neighborhood of degeneracies. The parallel-transported basis bears a close relationship to the singular-value basis; in this article, both are expanded in power series about a reference point and are shown to agree through second order but not beyond. Taylor series expansions are effected through the projection operator, whose expansion does not involve energy denominators or any type of singularity and in terms of which both the singular-value basis and the parallel-transported basis can be expressed. The parallel-transported basis is a version of Poincaré gauge, well known in electromagnetism, which provides a relationship between the derivative couplings and the curvature and which, along with a formula due to Mead, affords an efficient method for calculating Taylor series of the basis states and the derivative couplings. The case in which fine structure effects are included in the electronic Hamiltonian is covered. 

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3. Locally finite two-loop amplitudes for off-shell multi-photon production in electron-positron annihilation

   https://doi.org/10.1007/JHEP04(2021)222  

   Anastasiou, Charalampos; Haindl, Rayan; Sterman, George; Yang, Zhou; Zeng, Mao (April 2021, Journal of High Energy Physics)

   null (Ed.)

   A bstract We study the singularity structure of two-loop QED amplitudes for the production of multiple off-shell photons in massless electron-positron annihilation and develop counterterms that remove their infrared and ultraviolet divergences point by point in the loop integrand. The remainders of the subtraction are integrable in four dimensions and can be computed in the future with numerical integration. The counterterms capture the divergences of the amplitudes and factorize in terms of the Born amplitude and the finite remainder of the one-loop amplitude. They consist of simple one- and two-loop integrals with at most three external momenta and can be integrated analytically in a simple manner with established methods. We uncover novel aspects of fully local IR factorization, where vertex and self energy subdiagrams must be modified by new symmetrizations over loop momenta, in order to expose their tree-like tensor structures and hence factorization of IR singularities prior to loop integration. This work is a first step towards isolating locally the hard contributions of generic gauge theory amplitudes and rendering them integrable in exactly four dimensions with numerical methods. 

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4. Local Rankin–Selberg integrals for Speh representations

   https://doi.org/10.1112/S0010437X2000706X  

   Lapid, Erez M.; Mao, Zhengyu (May 2020, Compositio Mathematica)

   We construct analogues of Rankin–Selberg integrals for Speh representations of the general linear group over a $$p$$ -adic field. The integrals are in terms of the (extended) Shalika model and are expected to be the local counterparts of (suitably regularized) global integrals involving square-integrable automorphic forms and Eisenstein series on the general linear group over a global field. We relate the local integrals to the classical ones studied by Jacquet, Piatetski-Shapiro and Shalika. We also introduce a unitary structure for Speh representation on the Shalika model, as well as various other models including Zelevinsky’s degenerate Whittaker model. 

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5. A multipoint stress mixed finite element method for elasticity on quadrilateral grids

   https://doi.org/10.1002/num.22624  

   Ambartsumyan, Ilona; Khattatov, Eldar; Nordbotten, Jan Martin; Yotov, Ivan (November 2020, Numerical Methods for Partial Differential Equations)

   Abstract We develop a multipoint stress mixed finite element method for linear elasticity with weak stress symmetry on quadrilateral grids, which can be reduced to a symmetric and positive definite cell centered system. The method utilizes the lowest order Brezzi–Douglas–Marini finite element spaces for the stress and the trapezoidal quadrature rule in order to localize the interaction of degrees of freedom, which allows for local stress elimination around each vertex. We develop two variants of the method. The first uses a piecewise constant rotation and results in a cell‐centered system for displacement and rotation. The second uses a continuous piecewise bilinear rotation and trapezoidal quadrature rule for the asymmetry bilinear form. This allows for further elimination of the rotation, resulting in a cell‐centered system for the displacement only. Stability and error analysis is performed for both methods. First‐order convergence is established for all variables in their natural norms. A duality argument is employed to prove second order superconvergence of the displacement at the cell centers. Numerical results are presented in confirmation of the theory. 

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