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A NUMERICAL STUDY OF THE GAUSSIAN BEAM METHODS FOR SCHR(O|¨)DINGER-POISSON EQUATIONS

Journal of Computational mathematics 2010年第2期28卷 261-272页

作者： Shi Jin Hao Wu Xu Yang Department of Mathematics University of Wisconsin Madison W1 53706 USA Department of Mathematical Sciences Tsinghua University Beijing 100084 China Department of Mathematics University of Wisconsin Madison WI 53706 USA Program in Applied and Computational Mathematics Princeton University Princeton NJ 08544USA

As an important model in quantum semiconductor devices, the SchrSdinger-Poisson equations have generated widespread interests in both analysis and numerical simulations in recent years. In this paper, we present Gaussian beam methods for the numerical simulation of the one-dimensional Schrodinger-Poisson equations. The Gaussian beam methods for high frequency waves outperform the geometrical optics method in that the former are accurate even around caustics. The purposes of the paper are first to develop the Gaussian beam methods, based on our previous methods for the linear SchrSdinger equation, for the Schrodinger-Poisson equations, and then check their validity for this weakly-nonlinear system.

关键词： Schrodinger-Poisson equations Gaussian beam methods Vlasov-Poisson equations

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ANOMALOUS DIMENSIONS IN TURBULENCE

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MODERN PHYSICS LETTERS A 1991年第11期6卷 1023-1043页

作者： MIGDAL, AA Physics Department and Program in Applied and Computational Mathematics Princeton University Princeton NJ 08544 USA

A time-independent field theoretical framework for turbulence is suggested, based upon a variational principle for a stationary solution of the Fokker-Planck equation. We obtain a functional equation for the effective Action of this spatial field theory and investigate its general properties and some numerical solutions. The equation is completely universal, and allows for the scale invariant solutions in the inertial range. The critical indices are not fixed at the kinematical level, but rather should be found from certain eigenvalue conditions, as in the field theory of critical phenomena. Unlike the Wyld field theory, there are no divergences in our Feynman integrals, due to some magic cancellations. The simplest possible Gaussian approximation yields crude but still reasonable results (there are deviations from Kolmogorov scaling in 3 dimensions, but at 2.7544 dimensions it would be exact). Our approach allows us to study some new problems, such as spontaneous parity breaking in 3d turbulence. It turns out that with the appropriate helicity term added to the velocity correlation function, logarithmic infrared divergences arise in our field theory which effectively eliminates these terms. In order to build a quantitative theory of turbulence, one should consider more sophisticated Ansatz for the effective Action, which would require serious numerical work.

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From Dyson models to many-body quantum chaos

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Physical Review B 2025年第3期111卷 035147-035147页

作者： Alexei Andreanov Matteo Carrega Jeff Murugan Jan Olle Dario Rosa Ruth Shir Center for Theoretical Physics of Complex Systems Institute for Basic Science (IBS) Daejeon 34126 Korea Basic Science Program Korea University of Science and Technology (UST) Daejeon 34113 Korea CNR-SPIN Via Dodecaneso 33 16146 Genova Italy The Laboratory for Quantum Gravity & Strings Department of Mathematics and Applied Mathematics University of Cape Town Cape Town 7701 South Africa The National Institute for Theoretical and Computational Sciences Private Bag X1 Matieland 7602 South Africa Max Planck Institute for the Science of Light 91058 Erlangen Germany ICTP South American Institute for Fundamental Research Instituto de Física Teórica UNESP—University of Estadual Paulista Rua Dr. Bento Teobaldo Ferraz 271 01140-070 São Paulo SP Brazil Department of Physics and Materials Science University of Luxembourg L-1511 Luxembourg Luxembourg

A deep understanding of the mechanisms underlying many-body quantum chaos is one of the big challenges in contemporary theoretical physics. We tackle this problem in the context of a set of perturbed quadratic Sachdev-Ye-Kitaev (SYK) Hamiltonians defined on graphs. This allows us to disentangle the geometrical properties of the underlying single-particle problem and the importance of the interaction terms, showing that the former is the dominant feature ensuring the single-particle to many-body chaotic transition. Our results are verified numerically with state-of-the-art numerical techniques, capable of extracting eigenvalues in a desired energy window of very large Hamiltonians. Our approach essentially provides a new way of viewing many-body chaos from a single-particle perspective.

关键词： Quantum chaos

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A comparative analysis of optimization and generalization properties of two-layer neural network and random feature models under gradient descent dynamics

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Science China mathematics 2020年第7期63卷 1235-1258页

作者： Weinan E Chao Ma Lei Wu Department of Mathematics Princeton UniversityPrincetonNJ 08544USA The Program in Applied and Computational Mathematics Princeton UniversityNJ 08544USA Beijing Institute of Big Data Research Beijing 100871China

A fairly comprehensive analysis is presented for the gradient descent dynamics for training two-layer neural network models in the situation when the parameters in both layers are *** initialization schemes as well as general regimes for the network width and training data size are *** the overparametrized regime,it is shown that gradient descent dynamics can achieve zero training loss exponentially fast regardless of the quality of the *** addition,it is proved that throughout the training process the functions represented by the neural network model are uniformly close to those of a kernel *** general values of the network width and training data size,sharp estimates of the generalization error are established for target functions in the appropriate reproducing kernel Hilbert space.

关键词： two-layer neural network random feature model Gram matrix generalization error

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Solving algebraic riccati equations via proper orthogonal decomposition 19

Solving algebraic riccati equations via proper orthogonal de...

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19th IFAC World Congress on International Federation of Automatic Control, IFAC 2014

作者： Kramer, Boris Department of Mathematics Interdisciplinary Center for Applied Mathematics Virginia Tech. BlacksburgVA24061 United States

ISBN: (纸本)9783902823625

In this paper we present a method to solve algebraic Riccati equations by employing a projection method based on Proper Orthogonal Decomposition. The method only requires simulations of linear systems to compute the solution of a Lyapunov equation. The leading singular vectors are then used to construct a projector which is employed to produce a reduced order system. We compare this approach to an extended Krylov subspace method and a standard Gramian based method. © IFAC.

关键词： Algebra

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Fréchet sensitivity analysis for partial differential equations with distributed parameters

Fréchet sensitivity analysis for partial differential equat...

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作者： Borggaard, Jeff Nunes, Vitor Leite Interdisciplinary Center for Applied Mathematics Department of Mathematics Virginia Tech. United States

ISBN: (纸本)9781457700804

This paper reviews Fréchet sensitivity analysis for partial differential equations with variations in distributed parameters. The Fréchet derivative provides a linear map between parametric variations and the linearized response of the solution. We propose a methodology based on representations of the Fréchet derivative operator to find those variations that lead to the largest changes to the solution (the most significant variations). This includes an algorithm for computing these variations that only requires the action of the Fréchet operator on a given direction (the Gateaux derivative) and its adjoint. This algorithm is applicable since it does not require an approximation of the entire Fréchet operator, but only typical sensitivity analysis software for partial differential equations. The proposed methodology can be utilized to find worst case distributed disturbances and is thus applicable to uncertainty quantification and the optimal placement of sensors and actuators. © 2011 AACC American Automatic Control Council.

关键词： Partial differential equations

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Bifurcation from the essential spectrum for an elliptic equation with general nonlinearities

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Science China mathematics 2023年第10期66卷 2243-2260页

作者： Jianjun Zhang Xuexiu Zhong Huansong Zhou College of Mathematics and Statistics Chongqing Jiaotong UniversityChongqing 400074China South China Research Center for Applied Mathematics and Interdisciplinary Studies South China Normal UniversityGuangzhou 510631China Center for Mathematical Sciences and Department of Mathematics Wuhan University of TechnologyWuhan 430070China

In this paper,based on some prior estimates,we show that the essential spectrum λ=0 is a bifurcation point for a superlinear elliptic equation with only local conditions,which generalizes a series of earlier results ... 详细信息

关键词： elliptic equation bifurcation point essential spectrum

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A Deep Learning Approach for Nerve Injury Classification in Brachial Plexopathies Using Magnetic Resonance Neurography with Modified Hiking Optimization Algorithm

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Academic Radiology 2025年

作者： Dahou, Abdelghani Elaziz, Mohamed Abd Khattap, Mohamed G. Hassan, Hend Galal Eldeen Mohamed Ali School of Computer Science and Technology Zhejiang Normal University Jinhua 321004 China Mathematics and Computer Science department University of Ahmed DRAIA Adrar 01000 Algeria Department of Mathematics Faculty of Science Zagazig University Zagazig 44519 Egypt Faculty of Computer Science and Engineering Galala University Suez 435611 Egypt Artificial Intelligence Research Center (AIRC) Ajman University Ajman 346 United Arab Emirates Department of Diagnostic Interventional Radiology and Molecular Imaging Faculty of Medicine Ain Shams University Cairo 11591 Egypt Technology of Radiology and Medical Imaging Program Faculty of Applied Health Sciences Technology Galala University Suez 435611 Egypt

Rationale and Objectives: Brachial plexopathies (BPs) encompass a complex spectrum of nerve injuries affecting motor and sensory function in the upper extremities. Diagnosis is challenging due to the intricate anatomy and symptom overlap with other neuropathies. Magnetic Resonance Neurography (MRN) provides advanced imaging but requires specialized interpretation. This study proposes an AI-based framework that combines deep learning (DL) with the modified Hiking Optimization Algorithm (MHOA) enhanced by a Comprehensive Learning (CL) technique to improve the classification of nerve injuries (neuropraxia, axonotmesis, neurotmesis) using MRN data. Materials and Methods: The framework utilizes MobileNetV4 for feature extraction and MHOA for optimized feature selection across different MRI sequences (STIR, T2, T1, and DWI). A dataset of 39 patients diagnosed with BP was used. The framework classifies injuries based on Seddon's criteria, distinguishing between normal and abnormal conditions as well as injury severity. Results: The model achieved excellent performance, with 1.0000 accuracy in distinguishing normal from abnormal conditions using STIR and T2 sequences. For injury severity classification, accuracy was 0.9820 in STIR, outperforming the original HOA and other metaheuristic algorithms. Additionally, high classification accuracy (0.9667) was observed in DWI. The proposed framework outperformed traditional methods and demonstrated high sensitivity and specificity. Conclusion: The proposed AI-based framework significantly improves the diagnosis of BP by accurately classifying nerve injury types. By integrating DL and optimization techniques, it reduces diagnostic variability, making it a valuable tool for clinical settings with limited specialized neuroimaging expertise. This framework has the potential to enhance clinical decision-making and optimize patient outcomes through precise and timely diagnoses. © 2025 The Association of University Radiologists

关键词： Brachial Plexopathy (BP) Comprehensive Learning (CL) Deep Learning (DL) Feature Selection (FS) Hiking Optimization Algorithm (HOA) Magnetic Resonance Neurography (MRN) MobileNetV4

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System-Wide Emergency Policy for Transitioning from Main to Secondary Fuel 63

System-Wide Emergency Policy for Transitioning from Main to ...

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63rd IEEE Conference on Decision and Control, CDC 2024

作者： Pagnier, Laurent Hyett, Criston Ferrando, Robert Goldshtein, Igal Alisse, Jean Saban, Lilah Chertkov, Michael University of Arizona Graduate Interdisciplinary Program in Applied Mathematics and Department of Mathematics TucsonAZ85721 United States Noga the Israel Independent System Operator Haifa Israel

ISBN: (纸本)9798350316339

Faced with the complexities of managing natural gas-dependent power system amid the surge of renewable integration and load unpredictability, this study explores strategies for navigating emergency transitions to costlier secondary fuels. Our aim is to develop decision-support tools for operators during such exigencies. We approach the problem through a Markov Decision Process (MDP) framework, accounting for multiple uncertainties. These include the potential for dual-fuel generator failures and operator response during high-pressure situations. Additionally, we consider the finite reserves of primary fuel, governed by gas-flow partial differential equations (PDEs) and constrained by nodal pressure. Other factors include the variability in power forecasts due to renewable generation and the economic impact of compulsory load shedding. For tractability, we address the MDP in a simplified context, replacing it by Markov Processes evaluated against a selection of policies and scenarios for comparison. Our study considers two models for the natural gas system: an oversimplified model tracking linepack and a more nuanced model that accounts for gas flow network heterogeneity. The efficacy of our methods is demonstrated using a realistic model replicating Israel's power-gas infrastructure. © 2024 IEEE.

关键词： Markov processes

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Laboratory modeling of moist convection using a reactive fluid

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Physical Review Fluids 2025年第5期10卷 053505-053505页

作者： Valentin Dorel Daniel Lecoanet Michael Le Bars CNRS Aix Marseille Université Centrale Marseille IRPHE Marseille France Department of Engineering Sciences and Applied Mathematics and Center for Interdisciplinary Exploration and Research in Astrophysics Northwestern University Evanston Illinois USA

We present a laboratory experiment to model convection in the atmosphere driven by latent heat of condensation (“moist convection”). The system consists of a tank filled with an aqueous solution containing a color indicator that renders the fluid yellow and transparent when acidic and blue and opaque when basic. A sodium lamp positioned above the tank serves as a heat source. Initially, the solution is entirely acidic and stably stratified in temperature. A basic layer is progressively generated at the tank's bottom through water electrolysis, forming an initially stable, blue region. This layer is internally heated by absorbing the heat from the sodium lamp, modeling the latent heat of condensation in the atmosphere. Over time, instabilities develop at the interface between the yellow and blue layers. We investigate the variation of the instability wavelength and complement our observations with a linear stability analysis to elucidate the underlying mechanisms and the process of wavelength selection. Direct numerical simulations are then employed to explore the first steps of the nonlinear regime in connection with experimental observations.

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