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Ground state phases of the two-dimension electron gas with a unified variational approach

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arXiv 2024年

作者： Smith, Conor Chen, Yixiao Levy, Ryan Yang, Yubo Morales, Miguel A. Zhang, Shiwei Center for Computational Quantum Physics Flatiron Institute New YorkNY10010 United States Department of Electrical and Computer Engineering University of New Mexico AlbuquerqueNM87131 United States Program in Applied and Computational Mathematics Princeton University PrincetonNJ08544 United States

The two-dimensional electron gas (2DEG) is a fundamental model, which is drawing increasing interest because of recent advances in experimental and theoretical studies of 2D materials. Current understanding of the ground state of the 2DEG relies on quantum Monte Carlo calculations, based on variational comparisons of different ansatze for different phases. We use a single variational ansatz, a general backflow-type wave function using a message-passing neural quantum state architecture, for a unified description across the entire density range. The variational optimization consistently leads to lower ground-state energies than previous best results. Transition into a Wigner crystal (WC) phase occurs automatically at rs = 37±1, a density lower than currently believed. Between the liquid and WC phases, the same ansatz and variational search strongly suggest the existence of intermediate states in a broad range of densities, with enhanced short-range nematic spin correlations. Copyright © 2024, The Authors. All rights reserved.

关键词： Message passing

Engaging Engineering mathematics students online: Tutoring using MS Teams

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Engaging Engineering Mathematics students online: Tutoring u...

World Engineering Education Forum - Global Engineering Deans Council (WEEF-GEDC)

作者： Pragashni Padayachee Anita Campbell Ebrahim Fredericks Academic Support Program in Engineering University of Cape Town Cape Town South Africa Academic Support Program in Engineering Centre for Research in Engineering Education University of Cape Town Cape Town South Africa Department of Mathematics and Applied Mathematics University of Cape Town Cape Town South Africa

ISBN: (数字)9781665475280

ISBN: (纸本)9781665475297

Uncommon in the South African higher education landscape, online learning came to the fore during the global pandemic. We present an account of the use of Microsoft Teams for hybrid mathematics tutorials in a one-semester Vector Calculus course at a South African university in 2022. Interviews with the lecturer, analysed through a Community of Inquiry lens, showed the lecturer's perspective of the design and experience for tutors and students. Our aim is to improve the design of future hybrid tutorials to ensure engineering students' capacity in mathematics is well developed, their communication skills are improved, and that they experience working in a team. Future research will evaluate students' and tutors' accounts of their experiences. Our findings raise awareness of the possibilities and potential difficulties when using Microsoft Teams for communication, teaching and learning mathematics.

关键词： Visualization Pandemics Navigation Tutorials mathematics Reflection Calculus

Microstructural and Transport Characteristics of Triply Periodic Bicontinuous Materials

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arXiv 2024年

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

Three-dimensional (3D) bicontinuous two-phase materials are increasingly gaining interest because of their unique multifunctional characteristics and advancements in techniques to fabricate them. Because of their complex topological and structural properties, it still has been nontrivial to develop explicit microstructure-dependent formulas to predict accurately their physical properties. A primary goal of the present paper is to ascertain various microstructural and transport characteristics of five different models of triply periodic bicontinuous porous materials at a porosity 1 = 1/2: those in which the two-phase interfaces are the Schwarz P, Schwarz D and Schoen G minimal surfaces as well as two different pore-channel structures. We ascertain their spectral densities, pore-size distribution functions, local volume-fraction variances, and hyperuniformity order metrics and then use this information to estimate certain effective steady-state as well as time-dependent transport properties via closed-form microstructure-property formulas. Specifically, the recently introduced time-dependent diffusion spreadability is determined exactly from the spectral density. Moreover, we accurately estimate the fluid permeability of such porous materials from a closed-form formula that depends on the second moment of the pore-size function and the formation factor, a measure of the tortuosity of the pore space, which is exactly obtained for the three minimal-surface structures. We also rigorously bound the permeability from above using the spectral density. For the five models with identical cubic unit cells, we find that the permeability, inverse of the specific surface, hyperuniformity order metric, pore-size second moment and long-time spreadability behavior are all positively correlated and rank order the structures in exactly the same way. We also conjecture what structures maximize the fluid permeability for arbitrary porosities and show that this conjecture must be true in

关键词： Pore size

Identifying Equivalent Training Dynamics

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arXiv 2023年

作者： Redman, William T. Bello-Rivas, Juan Fonoberova, Maria Mohr, Ryan Kevrekidis, Yannis Mezić, Igor AIMdyn Inc. United States Interdepartmental Program in Dynamical Neuroscience UC Santa Barbara United States Department of Chemical and Biomolecular Engineering Whiting School of Engineering Johns Hopkins University United States Departments of Applied Mathematics and Statistics Johns Hopkins University United States Department of Mechanical Engineering UC Santa Barbara United States Department of Mathematics UC Santa Barbara United States

Study of the nonlinear evolution deep neural network (DNN) parameters undergo during training has uncovered regimes of distinct dynamical behavior. While a detailed understanding of these phenomena has the potential to advance improvements in training efficiency and robustness, the lack of methods for identifying when DNN models have equivalent dynamics limits the insight that can be gained from prior work. Topological conjugacy, a notion from dynamical systems theory, provides a precise definition of dynamical equivalence, offering a possible route to address this need. However, topological conjugacies have historically been challenging to compute. By leveraging advances in Koopman operator theory, we develop a framework for identifying conjugate and non-conjugate training dynamics. To validate our approach, we demonstrate that comparing Koopman eigenvalues can correctly identify a known equivalence between online mirror descent and online gradient descent. We then utilize our approach to: (a) identify non-conjugate training dynamics between shallow and wide fully connected neural networks;(b) characterize the early phase of training dynamics in convolutional neural networks;(c) uncover non-conjugate training dynamics in Transformers that do and do not undergo grokking. Our results, across a range of DNN architectures, illustrate the flexibility of our framework and highlight its potential for shedding new light on training dynamics. © 2023, CC BY.

关键词： Topology

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

PERMUTATION INVARIANT REPRESENTATIONS with APPLICATIONS to GRAPH DEEP LEARNING

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arXiv 2022年

作者： Balan, Radu Haghani, Naveed Singh, Maneesh Department of Mathematics University of Maryland College ParkMD20742 United States Applied Mathematics and Statistics and Scientific Computation Program University of Maryland College ParkMD20742 United States Verisk Analytics Jersey CityNJ07310 United States

This paper presents primarily two Euclidean embeddings of the quotient space generated by matrices that are identified modulo arbitrary row permutations. The original application is in deep learning on graphs where the learning task is invariant to node relabeling. Two embedding schemes are introduced, one based on sorting and the other based on algebras of multivariate polynomials. While both embeddings exhibit a computational complexity exponential in problem size, the sorting based embedding is globally bi-Lipschitz and admits a low dimensional target space. Additionally, an almost everywhere injective scheme can be implemented with minimal redundancy and low computational cost. In turn, this proves that almost any classifier can be implemented with an arbitrary small loss of performance. Numerical experiments are carried out on two data sets, a chemical compound data set (QM9) and a proteins data set (PROTEINS FULL).MSC Codes 05C62, 20C32, 68R12 © 2022, CC BY.

关键词： Embeddings

Consensus dynamics and coherence in hierarchical small-world networks

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arXiv 2023年

作者： Liao, Yunhua Maama, Mohamed Aziz-Alaoui, M.A. Department of Mathematics Hunan University of Technology and Business Hunan Changsha410205 China Key Laboratory of Hunan Province for Statistical Learning and Intelligent Computation Hunan Changsha410205 China Applied Mathematics and Computational Science Program KAUST Thuwal23955-6900 Saudi Arabia Normandie Univ UNIHAVRE LMAH FR-CNRS-3335 ISCN Le Havre76600 France

The hierarchical small-world network is a real-world network. It models well the benefit transmission web of the pyramid selling in China and many other countries. In this paper, by applying the spectral graph theory, we study three important aspects of the consensus problem in the hierarchical small-world network: convergence speed, communication time-delay robustness, and network coherence. Firstly, we explicitly determine the Laplacian eigenvalues of the hierarchical small-world network by making use of its treelike structure. Secondly, we find that the consensus algorithm on the hierarchical small-world network converges faster than that on some well-studied sparse networks, but is less robust to time delay. The closed-form of the first-order and the second-order network coherence are also derived. Our result shows that the hierarchical small-world network has an optimal structure of noisy consensus dynamics. Therefore, we provide a positive answer to two open questions of Yi et al. Finally, we argue that some network structure characteristics, such as large maximum degree, small average path length, and large vertex and edge connectivity, are responsible for the strong robustness with respect to external perturbations. Copyright © 2023, The Authors. All rights reserved.

关键词： Eigenvalues and eigenfunctions

Thermodynamics of the Solid-Liquid Phase Equilibrium of a Binary System: Effect of a Chemical Reaction in the Liquid and Epitaxial Strain in the Solid

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SSRN

SSRN 2023年

作者： Deymier, Alix C. Deymier, Pierre Alix Muralidharan, Krishna Latypov, Marat I. Department of Biomedical Engineering UConn Health FarmingtonCT06030 United States Department of Materials Science and Engineering University of Arizona TucsonAZ85721 United States Graduate Interdisciplinary Program in Applied Mathematics University of Arizona TucsonAZ85721 United States

We introduce two thermodynamics models of solid-liquid phase equilibrium of a binary system in the presence of (a) a chemical reaction in the liquid involving only one of the binary components and, (b) strain-inducing epitaxial conditions on the solid in addition to the chemical reaction in the liquid. These models show the existence of the counter intuitive behavior that epitaxy may promote recrystallization of the solid in spite of the reaction driving dissolution. The conditions for this recrystallization behavior are determined by the value of the epitaxial lattice matching composition, the temperature and the overall composition of the binary solid-liquid system. © 2023, The Authors. All rights reserved.

关键词： Chemical reactions

Alignment of density maps in Wasserstein distance

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Biological imaging 2024年 4卷 e5页

作者： Amit Singer Ruiyi Yang Department of Mathematics Princeton University Princeton NJ USA. Program in Applied and Computational Mathematics Princeton University Princeton NJ USA.

In this article, we propose an algorithm for aligning three-dimensional objects when represented as density maps, motivated by applications in cryogenic electron microscopy. The algorithm is based on minimizing the 1-Wasserstein distance between the density maps after a rigid transformation. The induced loss function enjoys a more benign landscape than its Euclidean counterpart and Bayesian optimization is employed for computation. Numerical experiments show improved accuracy and efficiency over existing algorithms on the alignment of real protein molecules. In the context of aligning heterogeneous pairs, we illustrate a potential need for new distance functions.

关键词： Bayesian optimization Wasserstein distance cryogenic electron microscopy heterogeneity analysis volume alignment