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arXiv 2024年

作者： Athanasiou, Nikolaos Fournodavlos, Grigorios Department of Mathematics & Applied Mathematics University of Crete Voutes Campus Heraklion70013 Greece Institute of Applied and Computational Mathematics FORTH Heraklion70013 Greece

We construct local, in spacetime, singular solutions to the Einstein vacuum equations that exhibit Kasner-like behavior in their past boundary. Our result can be viewed as a localization (in space) of the construction in [18]. We also prove a refined uniqueness statement and give a simple argument that generates general asymptotic data for Kasner-like singularities, enjoying all expected degrees of freedom, albeit only locally in space. The key difference of the present work with [18] is our use of a first order symmetric hyperbolic formulation of the Einstein vacuum equations, relative to the connection coefficients of a parallelly propagated orthonormal frame which is adapted to the Gaussian time foliation. This makes it easier to localize the construction, since elliptic estimates are no longer required to complete the energy *** Codes 83C75 (Primary) 35A21, 35Q76, 83C05, 83F05 (Secondary) Copyright © 2024, The Authors. All rights reserved.

关键词： Statistical mechanics

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Fully relativistic energies,transition properties,and lifetimes of lithium-like germanium

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Chinese Physics B 2024年第10期33卷 52-61页

作者： Shuang Li Jing Zhou Liu-Hong Zhu Xiu-Fei Mei Jun Yan School of Electrical and Optoelectronic Engineering West Anhui UniversityLu’an 237012China Institute of Applied Physics and Computational Mathematics Beijing 100088China Shanghai EBIT Laboratory Institute of Modern PhysicsDepartment of Nuclear Science and TechnologyFudan UniversityShanghai 200433China

Employing two fully relativistic methods,the multi-reference configuration Dirac-Hartree-Fock(MCDHF)methodand the relativistic many-body perturbation theory(RMBPT)method,we report energies and lifetime values for the lowest35 energy levels of the(1s^(2))nl configurations(where the principal quantum number n=2-6 and the angular quantum numberl=0,...,n-1)of lithium-like germanium(Ge XXX),as well as complete data on the transition wavelengths,radiativerates,absorption oscillator strengths,and line strengths between the *** the allowed(E1)and forbidden(magneticdipole M1,magnetic quadrupole M2,and electric quadrupole E2)ones are *** results from the two methodsare consistent with each other and align well with previous accurate experimental and theoretical *** assess theoverall accuracies of present RMBPT results to be likely the most precise ones to *** present fully relativistic resultsshould be helpful for soft x-ray laser research,spectral line identification,plasma modeling and *** datasetspresented in this paper are openly available at https://***/10.57760/sciencedb.j00113.00135.

关键词： multi-reference Dirac-Hartree-Fock relativistic many-body perturbation radiative rate lifetime

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Continuous dependence on data for linear evolution PDEs on the quarter-plane

arXiv

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arXiv 2024年

作者： Chatziafratis, Andreas Kamvissis, Spyridon Department of Mathematics National and Kapodistrian University of Athens Greece Department of Pure and Applied Mathematics University of Crete Greece Institute of Applied and Computational Mathematics FORTH Crete Greece

In this note, we announce a systematic analysis of "continuous dependence on the data" in classical spaces for the initial-boundary-value problem of the diffusion equation on the half-line, with data that are not necessarily compatible at the quadrant corner. This is based on a recent approach [1-9] to rigorously analyzing integral representations derived via the unified transform method [10-18] of Fokas. No exotic phenomena were discovered in this case, yet our findings appear to be new in the pertinent literature. These results supplement our previous investigations [2,6] on existence and (non)uniqueness within the framework of "well-posedness". The present detailed exposition elucidates the subtleties involved while also demonstrating a generic technique. Applications of the latter to several other IBVPs and PDEs will be reported elsewhere. Copyright © 2024, The Authors. All rights reserved.

关键词： Initial value problems

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Trustworthy Actionable Perturbations 41

Trustworthy Actionable Perturbations

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41st International Conference on Machine Learning, ICML 2024

作者： Friedbaum, Jesse Adiga, Sudarshan Tandon, Ravi Program in Applied Mathematics University of Arizona TucsonAZ United States Department of Electrical and Computer Engineering University of Arizona TucsonAZ United States

Counterfactuals, or modified inputs that lead to a different outcome, are an important tool for understanding the logic used by machine learning classifiers and how to change an undesirable classification. Even if a counterfactual changes a classifier's decision, however, it may not affect the true underlying class probabilities, i.e. the counterfactual may act like an adversarial attack and "fool" the classifier. We propose a new framework for creating modified inputs that change the true underlying probabilities in a beneficial way which we call Trustworthy Actionable Perturbations (TAP). This includes a novel verification procedure to ensure that TAP change the true class probabilities instead of acting adversarially. Our framework also includes new cost, reward, and goal definitions that are better suited to effectuating change in the real world. We present PAC-learnability results for our verification procedure and theoretically analyze our new method for measuring reward. We also develop a methodology for creating TAP and compare our results to those achieved by previous counterfactual methods. Copyright 2024 by the author(s)

关键词： Adversarial machine learning

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Comparative Analysis of Radial Basis Functions and Cubic Splines for Data Imputation 10

Comparative Analysis of Radial Basis Functions and Cubic Spl...

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10th International Conference on Fuzzy Systems and Data Mining, FSDM 2024

作者： Kitchainukoon, Wisut Sangsuwan, Amornrat Pongchalee, Pornthip Chanthawara, Krittidej Department of Mathematics Faculty of Science and Technology Loei Rajabhat University Loei42000 Thailand Department of Mathematics Faculty of Education Loei Rajabhat University Loei42000 Thailand Department of Applied Mathematics and Statistics Faculty of Science and Liberal Arts Rajamangala University of Technology Isan Nakhon Ratchasima30000 Thailand Program of Mathematics Faculty of Science Ubon Ratchathani Rajabhat University Ubon Ratchathani34000 Thailand

ISBN: (纸本)9781643685694

This study evaluates the effectiveness of Radial Basis Function (RBF) approaches, specifically Gaussian and Multiquadric RBFs, compared to Cubic and Adaptive Splines for data imputation in time-series datasets. Three datasets-multiseasonal, non-stationary, and spikes and drops-were used, with varying data corruption levels to simulate real-world scenarios. Imputation performance was assessed using Root Mean Square Error (RMSE). The results indicate that spline-based methods, particularly Cubic and Adaptive Splines, consistently deliver lower RMSE values, showcasing superior robustness to data variability and high corruption levels compared to RBF methods. This comparative analysis offers insights into selecting appropriate techniques for effective data imputation in time-series analysis. © 2024 The Authors.

关键词： Splines

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Character Modelling with Sketches and ODE-Based Shape Creation

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Numerical mathematics(Theory,Methods and Applications) 2023年第3期16卷 720-751页

作者： Ouwen Li Haibin Fu Shaojun Bian Xiaosong Yang Xiaogang Jin Andres Iglesias Algirdas Noreika Lihua You Jian Jun Zhang National Centre for Computer Animation Bournemouth UniversityUnited Kingdom State Key Lab of CAD&CG Zhejiang UniversityChina Department of Applied Mathematics and Computational Sciences University of CantabriaSpain Department of Information Science Toho UniversityJapan Indeform Ltd Lithuania

Character models have enormous applications in *** creation of detailed character models is an important *** paper proposes a new and easy-to-use technique to quickly create detailed character models from *** proposed technique consists of two main components:primitive deformer and shape *** this technique,2D silhouette contours of a character model are drawn or extracted from an image or ***,proper geometric primitives are selected and aligned with the corresponding 2D silhouette *** that,a primitive deformer is used to create a base mesh and three shape generators are used to add 3D details to the base *** primitive deformer and three shape generators are developed from ODE-driven *** primitive deformer deforms the aligned geometric primitives to exactly match the 2D silhouette contours in one view plane and obtains a base mesh of a character model consisting of deformed *** shape generators are used to add 3D details to the base mesh by creating local 3D *** experimental results demonstrate that the new technique can quickly create detailed 3D character models from sketches with few manual *** new technique is physics-based and easy to learn and use.

关键词： Sketch-based character modelling shape generators ordinary differential equations finite difference solution

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Inverse Volume Scaling of Finite-Size Error in Periodic Coupled Cluster Theory

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Physical Review X 2024年第1期14卷 011059-011059页

作者： Xin Xing Lin Lin Department of Mathematics University of California Berkeley California 94720 USA Applied Mathematics and Computational Research Division Lawrence Berkeley National Laboratory Berkeley California 94720 USA

Coupled cluster theory is one of the most popular post-Hartree-Fock methods for ab initio molecular quantum chemistry. The finite-size error of the correlation energy in periodic coupled cluster calculations for three-dimensional insulating systems has been observed to satisfy the inverse volume scaling, even in the absence of any correction schemes. This is surprising, as simpler theories that utilize only a subset of the coupled cluster diagrams exhibit much slower decay of the finite-size error, which scales inversely with the length of the system. In this study, we review the current understanding of finite-size error in quantum chemistry methods for periodic systems. We introduce new tools that elucidate the mechanisms behind this phenomenon in the context of coupled cluster doubles calculations. This reconciles some seemingly paradoxical statements related to finite-size scaling. Our findings also show that singularity subtraction can be a powerful method to effectively reduce finite-size errors in practical quantum chemistry calculations for periodic systems.

关键词： Crystalline systems Coupled cluster Finite-size scaling

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Point forces in elasticity equation and their alternatives in multi dimensions

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mathematics AND COMPUTERS IN SIMULATION 2022年第0期199卷 182-201页

作者： Peng, Q. Vermolen, F. J. [a]Mathematical Institute Leiden University Niels Bohrweg 1 2333 CA Leiden The Netherlands [b]Delft Institute of Applied Mathematics Delft University of Technology Mekelweg 4 2628 CD Delft The Netherlands [c]Computational Mathematics Group Department of Statistics and Mathematics Hasselt University Agoralaan Gebouw D 3590 Diepenbeek Belgium

Deep dermal wounds induce skin contraction as a result of the traction forcing exerted by (myo)fibroblasts on their immediate environment. These (myo)fibroblasts are skin cells that are responsible for the regeneration of collagen that is necessary for the integrity of skin We consider several mathematical issues regarding models that simulate traction forces exerted by (myo)fibroblasts. Since the size of cells (e.g. (myo)fibroblasts) is much smaller than the size of the domain of computation, one often considers point forces, modelled by Dirac Delta distributions on boundary segments of cells to simulate the traction forces exerted by the skin cells. In the current paper, we treat the forces that are directed normal to the cell boundary and toward the cell centre. Since it can be shown that there exists no smooth solution, at least not in H1 for solutions to the governing momentum balance equation, we analyse the convergence and quality of approximation. Furthermore, the expected finite element problems that we get necessitate to scrutinize alternative model formulations, such as the use of smoothed Dirac Delta distributions, or the so-called smoothed particle approach as well as the so-called 'hole' approach where cellular forces are modelled through the use of (natural) boundary conditions. In this paper, we investigate and attempt to quantify the conditions for consistency between the various approaches. This has resulted into error analyses in the L2-norm of the numerical solution based on Galerkin principles that entail Lagrangian basis functions. The paper also addresses well-posedness in terms of existence and uniqueness. The current analysis has been performed for the linear steady-state (hence neglecting inertia and damping) momentum equations under the assumption of Hooke's law. (C) 2022 The Author(s). Published by Elsevier B.V. on behalf of International Association for mathematics and Computers in Simulation (IMACS).

关键词： Point forces Dirac delta distribution Singular solution Immersed boundary approach "Hole" approach Smoothed particle approach

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Splitting algorithms for paraxial and Itô-Schrödinger models of wave propagation in random media

arXiv

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arXiv 2025年

作者： Bal, Guillaume Nair, Anjali Departments of Statistics and Mathematics Committee on Computational and Applied Mathematics University of Chicago ChicagoIL60637 United States Department of Statistics Committee on Computational and Applied Mathematics University of Chicago ChicagoIL60637 United States

This paper introduces a full discretization procedure to solve wave beam propagation in random media modeled by a paraxial wave equation or an Itô-Schrödinger stochastic partial differential equation. This method bears similarities with the phase screen method used routinely to solve such problems. The main axis of propagation is discretized by a centered splitting scheme with step ∆z while the transverse variables are treated by a spectral method after appropriate spatial truncation. The originality of our approach is its theoretical validity even when the typical wavelength θ of the propagating signal satisfies θ ∆z. More precisely, we obtain a convergence of order ∆z in mean-square sense while the errors on statistical moments are of order (∆z)2 as expected for standard centered splitting schemes. This is a surprising result as splitting schemes typically do not converge when ∆z is not the smallest scale of the problem. The analysis is based on equations satisfied by statistical moments in the Itô-Schrödinger case and on integral (Duhamel) expansions for the paraxial model. Several numerical simulations illustrate and confirm the theoretical findings. Copyright © 2025, The Authors. All rights reserved.

关键词： Schrodinger equation

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On Error-Based Step Size Control for Discontinuous Galerkin Methods for Compressible Fluid Dynamics

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应用数学与计算数学学报 2025年第1期7卷 3-39页

作者： Hendrik Ranocha Andrew R.Winters Hugo Guillermo Castro Lisandro Dalcin Michael Schlottke-Lakemper Gregor J.Gassner Matteo Parsani Applied Mathematics University of HamburgBundesstr.5520146 HamburgGermany Department of Mathematics Applied Mathematics Linköping UniversityLinköpingSweden Extreme Computing Research Center(ECRC) Computer Electrical and Mathematical Science and Engineering Division(CEMSE)King Abdullah University of Science and Technology(KAUST)Thuwal 23955-6900Saudi Arabia Applied and Computational Mathematics RWTH Aachen UniversityAachenGermany High-Performance Computing Center Stuttgart University of StuttgartStuttgartGermany Department of Mathematics and Computer Science University of CologneCologneGermany Center for Data and Simulation Science University of CologneCologneGermany Extreme Computing Research Center(ECRC) Computer Electrical and Mathematical Science and Engineering Division(CEMSE)King Abdullah University of Science and Technology(KAUST)Thuwal 23955-6900Saudi Arabia Physical Science and Engineering Division(PSE) King Abdullah University of Science and Technology(KAUST)Thuwal 23955-6900Saudi Arabia

We study a temporal step size control of explicit Runge-Kutta(RK)methods for com-pressible computational fluid dynamics(CFD),including the Navier-Stokes equations and hyperbolic systems of conservation laws such as the Euler *** demonstrate that error-based approaches are convenient in a wide range of applications and compare them to more classical step size control based on a Courant-Friedrichs-Lewy(CFL)*** numerical examples show that the error-based step size control is easy to use,robust,and efficient,e.g.,for(initial)transient periods,complex geometries,nonlinear shock captur-ing approaches,and schemes that use nonlinear entropy *** demonstrate these properties for problems ranging from well-understood academic test cases to industrially relevant large-scale computations with two disjoint code bases,the open source Julia pack-ages *** with *** and the C/Fortran code SSDC based on PETSc.

关键词： Explicit Runge-Kutta(RK)methods Step size control Compressible fluid dynamics Adaptivity in space and time Shock capturing

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