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Rigorous justification of the hydrostatic approximation limit of viscous compressible flows

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arXiv 2023年

作者： Liu, Xin Titi, Edriss S. Department of Mathematics Texas A&M University College StationTX77843 United States Department of Applied Mathematics and Theoretical Physics University of Cambridge CambridgeCB3 0WA United Kingdom Department of Computer Science and Applied Mathematics Weizmann Institute of Science Rehovot76100 Israel

This paper considers the asymptotic limit of small aspect ratio between vertical and horizontal spatial scales for viscous isothermal compressible flows. In particular, it is observed that fast vertical acoustic waves arise and induce an averaging mechanism of the density in the vertical variable, which at the limit leads to the hydrostatic approximation of compressible flows, i.e., the compressible primitive equations of atmospheric dynamics. We justify the hydrostatic approximation for general as well as "well-prepared" initial data. The initial data is called well-prepared when it is close to the hydrostatic balance in a strong topology. Moreover, the convergence rate is calculated in the well-prepared initial data case in terms of the aspect ratio, as the latter goes to *** Codes 35B25, 35B40, 76N10, 76N30 © 2023, CC BY-NC-ND.

关键词： Compressible flow

On hitting-set generators for polynomials that vanish rarely 23

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On hitting-set generators for polynomials that vanish rarely

23rd International Conference on Approximation Algorithms for Combinatorial Optimization Problems and 24th International Conference on Randomization and Computation, APPROX/RANDOM 2020

作者： Doron, Dean Ta-Shma, Amnon Tell, Roei Department of Computer Science Stanford University CA United States Blavatnik School of Computer Science Tel-Aviv University Israel Department of Computer Science and Applied Mathematics Weizmann Institute of Science Rehovot Israel

ISBN: (纸本)9783959771641

The problem of constructing hitting-set generators for polynomials of low degree is fundamental in complexity theory and has numerous well-known applications. We study the following question, which is a relaxation of this problem: Is it easier to construct a hitting-set generator for polynomials p: Fn → F of degree d if we are guaranteed that the polynomial vanishes on at most an Ε > 0 fraction of its inputs? We will specifically be interested in tiny values of Ε d/|F|. This question was first considered by Goldreich and Wigderson (STOC 2014), who studied a specific setting geared for a particular application, and another specific setting was later studied by the third author (CCC 2017). In this work our main interest is a systematic study of the relaxed problem, in its general form, and we prove results that significantly improve and extend the two previously-known results. Our contributions are of two types: Over fields of size 2 ≤ |F| ≤ poly(n), we show that the seed length of any hitting-set generator for polynomials of degree d ≤ n.49 that vanish on at most Ε = |F|−t of their inputs is at least Ω ((d/t) · log(n)). Over F2, we show that there exists a (non-explicit) hitting-set generator for polynomials of degree d ≤ n.99 that vanish on at most Ε = |F|−t of their inputs with seed length O ((d − t) · log(n)). We also show a polynomial-time computable hitting-set generator with seed length O ((d − t) · (2d−t+ log(n))). In addition, we prove that the problem we study is closely related to the following question: "Does there exist a small set S ⊆ Fn whose degree-d closure is very large?", where the degree-d closure of S is the variety induced by the set of degree-d polynomials that vanish on S. © 2020 Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing. All rights reserved.

关键词： Polynomial approximation

W-schemes in spectral methods for reaction-diffusion equations

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PAMM 2024年第3期24卷

作者： Roland Pulch Institute of Mathematics and Computer Science University of Greifswald Greifswald Germany

We consider reaction-diffusion equations, which are parabolic systems of partial differential equations. Assuming periodic boundary conditions in space, a multidimensional Fourier transform can be used. A spectral method yields an initial value problem of a stiff system of ordinary differential equations for time-dependent Fourier coefficients. We investigate the application of Rosenbrock-Wanner W-schemes in the time integration of the ordinary differential equations including step size selection. Results of numerical computations are presented using a Turing system in two space dimensions.

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When do graph covers preserve the clique dynamics of infinite graphs?

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arXiv 2025年

作者： Limbach, Anna M. Winter, Martin Institute of Computer Science Czech Academy of Sciences Pod Vodárenskou věží 271/2 Praha 8182 00 Czech Republic Mathematics Institute Technische Universität Berlin Straße des 17. Juni 135 Berlin10623 Germany

We investigate for which classes of (potentially infinite) graphs the clique dynamics is cover stable, i. e., when clique convergence/divergence is preserved under triangular covering maps. We first present an instructive counterexample: a clique convergent graph which covers a clique divergent graph and which is covered by a clique divergent graph. Based on this we then focus on local conditions (i. e., conditions on the neighbourhoods of vertices) and show that the following are sufficient to imply cover stability: local girth ≥ 7 and local minimum degree ≥ 2;being locally cyclic and of minimum degree ≥ 6. © 2025, CC BY.

关键词： Graph theory

Neural Networks Efficiently Learn Low-Dimensional Representations with SGD

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arXiv 2022年

作者： Mousavi-Hosseini, Alireza Park, Sejun Girotti, Manuela Mitliagkas, Ioannis Erdogdu, Murat A. Department of Computer Science University of Toronto Vector Institute Canada Department of Artificial Intelligence Korea University Korea Republic of Department of Mathematics and Computing Science Saint Mary's University Canada Department of Computer Science Université de Montréal Mila Institute Canada

We study the problem of training a two-layer neural network (NN) of arbitrary width using stochastic gradient descent (SGD) where the input x ∈ Rd is Gaussian and the target y ∈ R follows a multiple-index model, i.e., y = g(〈u1, x〉,..., 〈uk, x〉) with a noisy link function g. We prove that the first-layer weights of the NN converge to the k-dimensional principal subspace spanned by the vectors u1,..., uk of the true model, when online SGD with weight decay is used for training. This phenomenon has several important consequences when k d. First, by employing uniform convergence on this smaller subspace, we establish a generalization error bound of O(√kd/T) after T iterations of SGD, which is independent of the width of the NN. We further demonstrate that, SGD-trained ReLU NNs can learn a single-index target of the form y = f(〈u, x〉) + ∊ by recovering the principal direction, with a sample complexity linear in d (up to log factors), where f is a monotonic function with at most polynomial growth, and ∊ is the noise. This is in contrast to the known dΩ(p) sample requirement to learn any degree p polynomial in the kernel regime, and it shows that NNs trained with SGD can outperform the neural tangent kernel at initialization. Finally, we also provide compressibility guarantees for NNs using the approximate low-rank structure produced by SGD. Copyright © 2022, The Authors. All rights reserved.

关键词： Stochastic systems

FedVC: Virtual Clients for Federated Autonomous Driving With Imbalanced Label Distribution

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IEEE Transactions on Consumer Electronics 2025年

作者： Zhang, Yongquan Chen, Hongda Wang, Zhihui Kumar, Rahul Wang, Changmiao Zhejiang University of Finance and Economics School of Data Science Hangzhou310018 China Xi'an Jiaotong University School of Software Engineering Xi'an710049 China Shandong University of Science and Technology College of Computer Science and Engineering Qingdao266510 China Ssv Pg College Department Of Mathematics Hapur245101 India Shenzhen Research Institute of Big Data Shenzhen518172 China

Federated learning in autonomous driving safeguards the privacy of individual vehicles during collaborative training by avoiding the exchange of raw data. These vehicles often suffer from imbalanced label distribution, making the federated learning model developed from them critical for safety. Most current solutions addressing label imbalance in federated learning are oriented towards classification tasks. However, there is limited research focused on prediction tasks involving imbalanced datasets relevant to autonomous driving. To address the identified gap, we introduce an innovative peer-to-peer federated learning framework called FedVC. This framework includes the use of virtual clients to enhance the ability of the collaborative training process to recognize the global data distribution effectively. It strategically samples relevant segments of local data from various clients for each training round and controls the execution time for backpropagation associated with each virtual client. Importantly, FedVC constructs a global perspective by utilizing metadata from each client, thereby maintaining data privacy while overcoming the challenge of label imbalance in autonomous driving prediction tasks, all without the need to share any raw datasets. Experimental results demonstrate that FedVC outperforms classical FedAvg and the most recent methods at three steering angle prediction datasets with different levels of imbalanced label distribution. The source code is publicly available at https://***/aikedaerC/FedVC. © 1975-2011 IEEE.

关键词： Autonomous Driving Data Imbalance Federated Learning Virtual Clients

ASYMPTOTICS FOR d-FOLD PARTITION DIAMONDS AND RELATED INFINITE PRODUCTS

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arXiv 2023年

作者： Bringmann, Kathrin Craig, William Males, Joshua Department of Mathematics and Computer Science Division of Mathematics University of Cologne Weyertal 86-90 Cologne50931 Germany School of Mathematics University of Bristol BristolBS8 1TW United Kingdom The Heilbronn Institute for Mathematical Research Bristol United Kingdom

We prove an asymptotic formula for the number of d-fold partition diamonds of n and their Schmidt-type counterparts. In order to do so, we study the asymptotic behavior of certain infinite products. We also remark on interesting potential connections with mathematical physics and Bloch groups. Copyright © 2023, The Authors. All rights reserved.

关键词： Diamonds

Investigation of Automatic Part-of-Speech Tagging using CRF, HMM and LSTM on Misspelled and Edited Texts 5

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Investigation of Automatic Part-of-Speech Tagging using CRF,...

5th Artificial Intelligence and Cloud Computing Conference, AICCC 2022

作者： Aydinov, Farhad Huseynov, Igbal Sayadzada, Sofiya Rustamov, Samir Data Science University of Glasgow United Kingdom Computational Linguistics and Language Technologies Charles University Prague Czech Republic Department of Applied Mathematics and Computer Science Technical University of Denmark Copenhagen Denmark School of Information Technology and Engineering ADA University Baku Azerbaijan Institute of Control Systems BakuVarna Bulgaria Azerbaijan

ISBN: (纸本)9781450398749

Part-of-speech tagging is the process of assigning words in a given text to appropriate parts-of speech in order to reduce the disambiguation which may arise depending on the contextual usage of the words. In this paper, the problem of word sense disambiguation in Azerbaijani language is addressed by applying part of speech tagging on two varying data corpora, misspelled, and edited (clean) text using 3 different machine learning algorithms: Hidden Markov Model, Long Short-Term Memory, and Conditional Random Fields. The comparative analysis on the outcomes of the mentioned algorithms and their accuracy scores were analysed in the paper. The misspelled dataset for the experiments is provided by Unibank from their chatbot dialogues while the clean textual data was retrieved from the books and newspapers in Azerbaijani. The experiments showed that the Bidirectional LSTM has the highest accuracy scores for both edited (98.2%) and noisy (96.2%) data corpora. Suggested models can be used in the application of algorithms focuses on part of speech tags and syntactic structure of Azerbaijani language which is an agglutinative language belonging to Turkic languages family, thus enabling the research to be further investigated in other agglutinative languages with similar grammatical structure. © 2022 ACM.

关键词： Hidden Markov models

Quantum and classical low-degree learning via a dimension-free Remez inequality

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arXiv 2023年

作者： Klein, Ohad Slote, Joseph Volberg, Alexander Zhang, Haonan School of Computer Science and Engineering Hebrew University Israel Department of Computing and Mathematical Sciences California Institute of Technology United States Department of Mathematics Michigan State University Hausdorff Center for Mathematics University of Bonn Germany Department of Mathematics University of South Carolina United States

Recent efforts in Analysis of Boolean Functions aim to extend core results to new spaces, including to the slice ([nk]), the hypergrid [K]n, and noncommutative spaces (matrix algebras). We present here a new way to relate functions on the hypergrid (or products of cyclic groups) to their harmonic extensions over the polytorus. We show the supremum of a function f over products of the cyclic group {exp(2πik/K)}Kk=1 controls the supremum of f over the entire polytorus ({z ∈ C: |z| = 1}n), with multiplicative constant C depending on K and deg(f) only. This Remez-type inequality appears to be the first such estimate that is dimension-free (i.e., C does not depend on n). This dimension-free Remez-type inequality removes the main technical barrier to giving O(log n) sample complexity, polytime algorithms for learning low-degree polynomials on the hypergrid and low-degree observables on level-K qudit systems. In particular, our dimension-free Remez inequality implies new Bohnenblust–Hille-type estimates which are central to the learning algorithms and appear unobtainable via standard techniques. Thus we extend to new spaces a recent line of work [10, 13, 22] that gave similarly efficient methods for learning low-degree polynomials on the hypercube and observables on qubits. An additional product of these efforts is a new class of distributions over which arbitrary quantum observables are well-approximated by their low-degree truncations—a phenomenon that greatly extends the reach of low-degree learning in quantum science [13].MSC Codes 46B10, 46B09, 46B07, 60E15 Copyright © 2023, The Authors. All rights reserved.

关键词： Learning algorithms