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TRACE FORMS ON THE CYCLOTOMIC HECKE ALGEBRAS AND COCENTERS OF THE CYCLOTMIC SCHUR ALGEBRAS

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arXiv 2022年

作者： He, Zhekun Hu, Jun Lin, Huang School of Mathematical and Statistics Beijing Institute of Technology Beijing100081 China MIIT Key Laboratory of Mathematical Theory and Computation in Information Security School of Mathematical and Statistics Beijing Institute of Technology Beijing100081 China

We define a unified trace form τ on the cyclotomic Hecke algebras Hn,K of type A, which generalize both Malle-Mathas’ trace form on the non-degenerate version (with Hecke parameter ξ 6= 1) and Brundan-Kleshchev’s trace form on the degenerate version. We use seminormal basis theory to construct an explicit pair of dual bases for Hn,K with respect to the form. We also construct an explicit basis for the cocenter (i.e., the 0th Hochschild homology) of the corresponding cyclotomic Schur algebra, which shows that the cocenter has dimension independent of the ground field K, the Hecke parameter ξ and the cyclotomic parameters Q1, · · · , Q Copyright © 2022, The Authors. All rights reserved.

关键词： Algebra

On the seminormal bases and dual seminormal bases of the cyclotomic Hecke algebras of type G(, 1, n)

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arXiv 2022年

作者： Hu, Jun Wang, Shixuan MIIT Key Laboratory of Mathematical Theory & Computation In Information Security School of Mathematical & Statistics Beijing Institute of Technology Beijing100081 China School of Mathematical & Statistics Beijing Institute of Technology Beijing100081 China

This paper studies the seminormal bases {fst}, {fst}, and the dual seminormal bases {gst}, {gst} of the non-degenerate and the degenerate cyclotomic Hecke algebras H,nof type G(, 1, n). We present some explicit formulae for the constants αst:= gst/fst∈ K×, ast:= gst/fst∈ K×in terms of the γ-coefficients {γu, γ′u} and the r-coefficients {ru, r′u} of H,n. In particular, we answer a question [21, Remark 3.6] of Mathas on the rationality of square roots of some quotients of products of γ-coefficients. We obtain some explicit formulae for the expansion of each seminormal bases of H,n-1as a linear combination of the seminormal bases of H,nunder the natural inclusion H,n-1→ H,n Copyright © 2022, The Authors. All rights reserved.

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THE SPIDERS S(4m + 2, 2m, 1) ARE e-POSITIVE

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arXiv 2024年

作者： Tang, Davion Q.B. Wang, David G.L. Wang, Monica M.Y. School of Mathematics and Statistics Beijing Institute of Technology Beijing102400 China School of Mathematics and Statistics Beijing Key Laboratory on MCAACI Beijing Institute of Technology MIIT Key Laboratory of Mathematical Theory and Computation in Information Security Beijing102400 China School of Basic Education Beijing Institute of Graphic Communication Beijing102600 China School of Mathematical Sciences Capital Normal University Beijing100089 China

By using the composition method, we establish the e-positivity of spider graphs of the form S(4m + 2, 2m, 1), which was conjectured by Aliniaeifard, Wang and van Willigenburg. We expand chromatic symmetric functions in terms of eI for compositions I. Following the divide-and-conquer strategy, we group one or two eJ-terms that have positive coefficients with each eI-term that has a negative coefficient, where the compositions J are selected to be distinct compositions obtained by rearranging the parts of I, and show the positivity of the sum of those *** Codes 05E05 05C15 05A20 © 2024, CC BY-NC-ND.

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A NOTE ON DECAY PROPERTY OF NONLINEAR SCHRÖDINGER EQUATIONS

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arXiv 2022年

作者： Fan, Chenjie Zhao, Zehua Academy of Mathematics and Systems Science CAS China Department of Mathematics and Statistics Beijing Institute of Technology MIIT Key Laboratory of Mathematical Theory and Computation in Information Security Beijing China

In this note, we show the existence of a special solution u to defocusing cubic NLS in 3d, which lives in Hs for all s > 0, but scatters to a linear solution in a very slow way. We prove for this u, for all ǫ > 0, one has (Equation presented). Note that such a slow asymptotic convergence is impossible if one further pose the initial data of u(0) be in L1. We expect that similar construction hold the for other NLS models. It can been seen the slow convergence is caused by the fact that there are delayed backward scattering profile in the initial data, we also illustrate why L1 condition of initial data will get rid of this phenomena. Copyright © 2022, The Authors. All rights reserved.

关键词： Nonlinear equations

Event-Triggered Stabilization of Parabolic PDEs by Switching

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Event-Triggered Stabilization of Parabolic PDEs by Switching

IEEE Conference on Decision and Control

作者： Wen Kang Emilia Fridman Jing Zhang Chuan-Xin Liu School of Mathematics and Statistics MIIT Key Laboratory of Mathematical Theory and Computation in Information Security Beijing Institute of Technology China School of Electrical Engineering Tel Aviv University Israel School of Automation and Electrical Engineering University of Science and Technology Beijing China

Although switching-based stabilization of 1D parabolic systems was investigated by employing one actuator moving in spatial domain in our recent paper [18], this method increases the system cost since actuator and sensor switching happens at fixed time regardless of whether the switching is necessary or not. To further reduce operating and production cost, in the present paper, switching-based dynamic event-triggered control law is studied to stabilize the parabolic PDE systems via output-dependent switching law. Constructive exponential stability conditions are established by using Lyapunov method. A numerical example shows the effectiveness of the proposed methods.

关键词：

CLOCKS ARE e-POSITIVE

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arXiv 2024年

作者： Chen, L. He, Y.T. Wang, David G.L. Department of Mathematics Southern University of Science and Technology Guangdong Shenzhen518055 China School of Mathematics and Statistics Xi’an Jiaotong University Shaanxi Xi’an710049 China School of Mathematics and Statistics MIIT Key Laboratory of Mathematical Theory and Computation in Information Security Beijing Institute of Technology Beijing102400 China

Along with his confirmation of the e-positivity of all cycle-chord graphs θab1, the third author conjectured the e-positivity of all theta graphs θabc. In this paper, we establish the e-positivity of all clock graphs θab2 by using the composition method. The key idea is to investigate the fibers of certain partial reversal transformation on compositions with all parts at least *** Codes 05E05 © 2024, CC BY-NC-ND.

关键词： Atomic clocks

An Efficient Method for Sample Adversarial Perturbations against Nonlinear Support Vector Machines

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arXiv 2022年

作者： Su, Wen Li, Qingna School of Mathematics and Statistics Beijing Institute of Technology Beijing China School of Mathematics and Statistics Beijing Key Laboratory on MCAACI Key Laboratory of Mathematical Theory and Computation in Information Security Beijing Institute of Technology Beijing China

Adversarial perturbations have drawn great attentions in various machine learning models. In this paper, we investigate the sample adversarial perturbations for nonlinear support vector machines (SVMs). Due to the implicit form of the nonlinear functions mapping data to the feature space, it is difficult to obtain the explicit form of the adversarial perturbations. By exploring the special property of nonlinear SVMs, we transform the optimization problem of attacking nonlinear SVMs into a nonlinear KKT system. Such a system can be solved by various numerical methods. Numerical results show that our method is efficient in computing adversarial perturbations. Copyright © 2022, The Authors. All rights reserved.

关键词： Numerical methods

Boundary Observer for an unstable cascaded heat-heat system with different reaction coefficients

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Boundary Observer for an unstable cascaded heat-heat system ...

Chinese Control Conference (CCC)

作者： Jia-Qi Tang Jun-Min Wang Wen Kang School of Mathematics and Statistics MIIT Key Laboratory of Mathematical Theory and Computation in Information Security Beijing Institute of Technology Beijing P. R. China Shunde Innovation School University of Science and Technology Beijing Foshan P. R. China

The present paper constructs a finite-dimensional observer subject to boundary measurement for a cascaded heat-heat system with different reaction coefficients. The observer design is proposed via the modal decomposition approach. Based on Lyapunov method, the observer exponential convergence is formulated. Numerical simulation illustrates the performance of observer.

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QCSAS: A Quasi-Constant and Synchronized Aggregate Signature Scheme for AMI Networks

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IEEE Internet of Things Journal 2025年

作者： Sun, Yujiao Sui, Zhiyuan Wang, Huaqun de Meer, Hermann Beijing Institute of Technology MIIT Key Laboratory of Mathematical Theory and Computation in Information Security School of Mathematics and Statistics Beijing China Central University of Finance and Economics Information School Beijing China Nanjing University of Posts and Telecommunications School of Computer Science Nanjing China University of Passau Faculty of Computer Science and Mathematics Passau Germany

Smart grids must safeguard their critical infrastructures against both physical and cyber attacks. Therefore, real-time operation and cybersecurity are two fundamental requirements for smart grid systems. AMI (Advanced Metering Infrastructure) is a critical part of the smart grid that specifically deals with the measurement, collection, and analysis of electricity usage data. Digital signatures help prevent data tampering and ensure cybersecurity during AMI communications. However, as the number of smart meters and transactions in an AMI network increases, deploying digital signatures can lead to communication delays. This paper designs a quasi-constant and synchronized aggregate signature scheme (QCSAS), which can protect usage data without hash functions. The QCSAS scheme simplifies the synchronized aggregate signatures, and is proved secure against chosen message attacks in the random oracle model. The QCSAS scheme incorporates the features of synchronized aggregate signatures, such as the constant aggregate signature size, thereby minimizing the data management system’s storage space. Furthermore, the length of usage data does not exceed 12-bit, in compliance with existing governmental standards. The computational cost of multiplying a 12-bit integer is negligible compared to other cryptographic operations. Thus, the verification cost of the QCSAS scheme remains quasi-constant. © 2014 IEEE.

关键词： Advanced metering infrastructures