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WEIGHTED CYLINDRIC PARTITIONS

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arXiv 2022年

作者： Bridges, Walter Uncu, Ali K. University of Cologne Department of Mathematics and Computer Science Weyertal 86-90 Cologne50931 Germany Johann Radon Institute for Computational and Applied Mathematics Austrian Academy of Science Altenbergerstraße 69 LinzA-4040 Austria University of Bath Faculty of Science Department of Computer Science BathBA2 7AY United Kingdom

Recently Corteel and Welsh outlined a technique for finding new sum-product identities by using functional relations between generating functions for cylindric partitions and a theorem of Borodin. Here, we extend this framework to include very general product-sides coming from work of Han and Xiong. In doing so, we are led to consider structures such as weighted cylindric partitions, symmetric cylindric partitions and weighted skew double shifted plane partitions. We prove some new identities and obtain new proofs of known identities, including the Göllnitz-Gordon and Little Göllnitz identities as well as some beautiful Schmidt-type identities of Andrews and *** Codes Primary 05A15, Secondary 05A17, 11B65, 11P81, 11P84 © 2022, CC0.

关键词： Algebra

Highly excited quantum state-selective capture dynamics in slow Ar8+−H2 collisions

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Physical Review A 2024年第3期109卷 032819-032819页

作者： Md Abul Kalam Azad Siddiki G. P. Zhao L. Liu Deepankar Misra Department of Nuclear and Atomic Physics Tata Institute of Fundamental Research Homi Bhabha Road Colaba Mumbai 400005 India College of Data Science Jiaxing University Jiaxing 314001 China Data Center for High Energy Density Physics Institute of Applied Physics and Computational Mathematics Beijing 100088 China

We report on the combined experimental and theoretical studies of the single-electron capture collisions of Ar8+ projectiles with the H2 molecules at 1, 2, and 4 keV/u collision energies. The nondissociative recoil H2+ molecular ions are measured in coincidence with the charge-changing Ar7+ projectiles. The relative cross sections of the different state-selective capture channels are obtained from the experimentally measured Q-value spectra. The 1s electron transfers to the highly excited 4d+4f, 5s, 5p, 5d+5f+5g, 6s+6p, and 6d+6f+6g+6h states are resolved experimentally. The differential scattering angle distributions for the dominant 1s to 5s, 5p, and 5d+5f+5g transitions are compared with the two-center atomic orbital close-coupling methods. A collision energy-dependent dynamical coupling effect is also observed for the 1s→5p0 and 1s→5p±1 transitions. The dominant oscillatory structures in the scattering-angle distributions are attributed to Stueckelberg-type oscillations. In contrast, the less visible undulations in the smaller scattering angles imprint the signature of the quantum matter-wave scattering of the projectiles. The quantum nature of the oscillations in the angular distributions is further validated by classical calculations. Our study thus illustrates the highly excited quantum state-selective electron capture process and sheds light on the scattering-angle-range-dependent collision dynamics for highly charged ion-molecule collisions in the highly perturbative regime.

关键词： Atomic & molecular collisions Scattering of atoms, molecules, clusters & ions Scattering theory

Reflecting (ON) the modulo 9 kanade-russell (conjectural) identities

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arXiv 2021年

作者： Uncu, Ali Zudilin, Wadim University of Bath Faculty of Science Department of Computer Science BathBA2 7AY United Kingdom Johann Radon Institute for Computational and Applied Mathematics Austrian Academy of Science Altenbergerstraße 69 LinzA-4040 Austria Department of Mathematics IMAPP Radboud University PO Box 9010 Nijmegen 6500 GL Netherlands

We examine complexity and versatility of five modulo 9 Kanade- Russell identities through their finite (aka polynomial) versions and images under the q → 1/q *** Codes Primary 11P84, Secondary 05A15, 05A17, 11B65 Cop... 详细信息

关键词： Functions

ERROR ESTIMATES OF FINITE ELEMENT METHODS FOR NONLOCAL PROBLEMS USING EXACT OR APPROXIMATED INTERACTION NEIGHBORHOODS

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arXiv 2024年

作者： Du, Qiang Xie, Hehu Yin, Xiaobo Zhang, Jiwei Department of Applied Physics and Applied Mathematics Data Science Institute Columbia University NY10027 United States LSEC NCMIS Academy of Mathematics and Systems Science Chinese Academy of Sciences Beijing100190 China School of Mathematical Sciences University of Chinese Academy of Sciences Beijing100049 China Central China Normal University Wuhan430079 China School of Mathematics and Statistics Hubei Key Laboratory of Computational Science Wuhan University Wuhan430072 China

We study the asymptotic error between the finite element solutions of nonlocal models with a bounded interaction neighborhood and the exact solution of the limiting local model. The limit corresponds to the case when the horizon parameter, the radius of the spherical nonlocal interaction neighborhood of the nonlocal model, and the mesh size simultaneously approach zero. Two important cases are discussed: one involving the original nonlocal models and the other for nonlocal models with polygonal approximations of the nonlocal interaction neighborhood. Results of numerical experiments are also reported to substantiate the theoretical *** Codes 65R20, 74S05, 46N20, 46N40, 45A05 Copyright © 2024, The Authors. All rights reserved.

关键词： Galerkin methods

Numerical approximation of the stochastic Cahn-Hilliard equation near the sharp interface limit

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arXiv 2019年

作者： Antonopoulou, Dimitra Baňas, Ľubomír Nürnberg, Robert Prohl, Andreas Department of Mathematics University of Chester Thornton Science Park CH2 4NU United Kingdom Institute of Applied and Computational Mathematics FO.R.T.H. HeraklionGR-711 10 Greece Department of Mathematics Bielefeld University Bielefeld33501 Germany Department of Mathematics Imperial College London LondonSW7 2AZ United Kingdom Mathematisches Institut der Universität Tübingen Auf der Morgenstelle 10 TübingenD-72076 Germany

We consider the stochastic Cahn-Hilliard equation with additive noise term Εγg Ẇ (γ > 0) that scales with the interfacial width parameter Ε. We verify strong error estimates for a gradient flow structure-inheriting time-implicit discretization, where Ε−1 only enters polynomially;the proof is based on higher-moment estimates for iterates, and a (discrete) spectral estimate for its deterministic counterpart. For γ sufficiently large, convergence in probability of iterates towards the deterministic Hele-Shaw/Mullins-Sekerka problem in the sharp-interface limit Ε → 0 is shown. These convergence results are partly generalized to a fully discrete finite element based discretization. We complement the theoretical results by computational studies to provide practical evidence concerning the effect of noise (depending on its'strength' γ) on the geometric evolution in the sharp-interface limit. For this purpose we compare the simulations with those from a fully discrete finite element numerical scheme for the (stochastic) Mullins-Sekerka problem. The computational results indicate that the limit for γ ≥ 1 is the deterministic problem, and for γ = 0 we obtain agreement with a (new) stochastic version of the Mullins-Sekerka problem. Copyright © 2019, The Authors. All rights reserved.

关键词： Stochastic systems

Quantifying Phase Mixing and Separation Behaviors Across Length and Time Scales

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SSRN

SSRN 2024年

作者： Skolnick, Murray Torquato, Salvatore Department of Chemistry Princeton University PrincetonNJ08544 United States Department of Physics Princeton University PrincetonNJ08544 United States Princeton Institute of Materials Princeton University PrincetonNJ08544 United States Program in Applied and Computational Mathematics Princeton University PrincetonNJ08544 United States

Phase mixing and separation phenomena abound in the formation of natural and synthetic material systems, including alloys, composites, granular media, geological media, complex fluids, and biological media. While characterizing phase mixing is critical to understanding material microstructure formation and physical properties, previous metrics that measure the degree of mixing in condensed phases often rely on empirical observation, are applicable only to certain microstructures, and/or fail to detect nontrivial changes in mixing with respect to length scale. To overcome these shortcomings, we introduce and study a mixing metric that is applicable to a broad spectrum of complex microstructures and leverages the hyperuniformity concept to provide a unifying framework to rank and classify mixing in materials across both length and time scales. Specifically, we employ the local phase volume-fraction/field-intensity variance [[EQUATION]] associated with a spherical sampling window of radius [[EQUATION]] at time [[EQUATION]] as our metric. We demonstrate that of [[EQUATION]] provides a sensitive measure of mixing across both length and time scales of a structurally diverse set of real and simulated material microstructures: exotic disordered multihyperuniform avian retina photoreceptor cell patterns, time-evolving spinodal decomposition patterns, and a model material mixed via the baker's transformation. We also describe how this metric can be used to guide future work on fabricating materials whose bulk properties can be controlled by tuning their mixing characteristics across length and time scales. © 2024, The Authors. All rights reserved.

关键词： Blending

Transductive and inductive outlier detection with robust autoencoders 24

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Transductive and inductive outlier detection with robust aut...

Proceedings of the Fortieth Conference on Uncertainty in Artificial Intelligence

作者： Ofir Lindenbaum Yariv Aizenbud Yuval Kluger Faculty of Engineering Bar-Ilan University Ramat Gan Israel School of Mathematical Sciences Tel Aviv University Tel Aviv Israel Program in Applied Mathematics Yale University New Haven CT and Department of Pathology Yale University School of Medicine New Haven CT and Computational Biology & Bioinformatics Program Yale University New Haven CT

Accurate detection of outliers is crucial for the success of numerous data analysis tasks. In this context, we propose the Probabilistic Robust AutoEncoder (PRAE) that can simultaneously remove outliers during training (transductive) and learn a mapping that can be used to detect outliers in new data (inductive). We first present the Robust AutoEncoder (RAE) objective that excludes outliers while including a subset of samples (inliers) that can be effectively reconstructed using an AutoEncoder (AE). RAE minimizes the autoencoder's reconstruction error while incorporating as many samples as possible. This could be formulated via regularization by subtracting an ℓ0 norm, counting the number of selected samples from the reconstruction term. As this leads to an intractable combinatorial problem, we propose two probabilistic relaxations of RAE, which are differentiable and alleviate the need for a combinatorial search. We prove that the solution to the PRAE problem is equivalent to the solution of RAE. We then use synthetic data to demonstrate that PRAE can accurately remove outliers in various contamination levels. Finally, we show that using PRAE for outlier detection leads to state-of-the-art results for inductive and transductive outlier detection.

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Vector tomography for reconstructing electric fields with non-zero divergence in bounded domains

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arXiv 2024年

作者： Koulouri, Alexandra Brookes, Mike Rimpiläinen, Ville Institute for Computational and Applied Mathematics University of Münster Einsteinstrasse 62 Münster48149 Germany Department of Electrical and Electronic Engineering Imperial College London Exhibition Road LondonSW7 2BT United Kingdom Department of Mathematics University of Auckland Private Bag 92019 Auckland1142 New Zealand

In vector tomography (VT), the aim is to reconstruct an unknown multi-dimensional vector field using line integral data. In the case of a 2-dimensional VT, two types of line integral data are usually required. These data correspond to integration of the parallel and perpendicular projection of the vector field along integration lines. VT methods are non-invasive, non-intrusive and offer more information on the field than classical point measurements;they are typically used to reconstruct divergence-free (or source-free) velocity and flow fields. In this paper, we show that VT can also be used for the reconstruction of fields with non-zero divergence. In particular, we study electric fields generated by dipole sources in bounded domains which arise, for example, in electroencephalography (EEG) source imaging. To the best of our knowledge, VT has not previously been used to reconstruct such fields. We explain in detail the theoretical background, the derivation of the electric field inverse problem and the numerical approximation of the line integrals. We show that fields with non-zero divergence can be reconstructed from the longitudinal measurements with the help of two sparsity constraints that are constructed from the transverse measurements and the vector Laplace operator. As a comparison to EEG source imaging, we note that VT does not require mathematical modelling of the sources. By numerical simulations, we show that the pattern of the electric field can be correctly estimated using VT and the location of the source activity can be determined accurately from the reconstructed magnitudes of the field. © 2024, CC BY.

关键词： Electric fields

Pump-probe scheme for optical coherence tomography using indocyanine green mixed with albumin or human plasma

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Pump-probe scheme for optical coherence tomography using ind...

Conference on Lasers and Electro-Optics (CLEO)

作者： Z. Yaqoob J.P. Fingler S. Lintner B.E. Applegate Changhuei Yang Electrical Engineering California Institute of Technology Pasadena CA USA Applied Physics California Institute of Technology Pasadena CA USA Applied & Computational Mathematics California Institute of Technology Pasadena CA USA Department of Biomedical Engineering Duke University Durham NC USA

Use of indocyanine green (ICG) in a pump-probe scheme for OCT is proposed. The study illustrates that ICG in protein solution shows unusual pump-probe imaging potential, indicating its usefulness as a contrast agent f... 详细信息

ISBN: (纸本)1557527954

关键词： Optical pumping Tomography Humans Plasmas Biomedical optical imaging High-resolution imaging Absorption Optical imaging Laser beams Pump lasers