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Norm estimates of the partial derivatives for harmonic and harmonic elliptic mappings

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arXiv 2020年

作者： Chen, Shaolin Ponnusamy, Saminathan Wang, Xiantao College of Mathematics and Statistics Hengyang Normal University HengyangHunan421008 China Department of Mathematics Indian Institute of Technology Madras Chennai600 036 India Key Laboratory of High Performance Computing and Stochastic Information Processing College of Mathematics and Statistics Hunan Normal University ChangshaHunan410081 China

Let f = P[F] denote the Poisson integral of F in the unit disk D with F being absolutely continuous in the unit circle T and F∈ Lp(0, 2π), where F (eit) = dtd F (eit) and p ≥ 1. Recently, the author in [12] proved that (1) if f is a harmonic mapping and 1 ≤ p z and fz ∈ Bp(D), the classical Bergman spaces of D [12,_ Theorem 1.2];(2) if f is a harmonic quasiregular mapping and 1 ≤ p ≤ ∞, then fz, fz ∈ Hp(D), the classical Hardy spaces of D [12, Theorem 1.3]. These are the main results in [12]. The purpose of this paper is to generalize these two results. First, we prove that, under the same assumptions, [12, Theorem 1.2] is true when 1 ≤ p MSC Codes Primary: 30C62, 31A05, Secondary: 30H10, 30H20 Copyright © 2020, The Authors. All rights reserved.

关键词： Harmonic analysis

An application of nakayama functor in representation stability theory

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arXiv 2017年

作者： Gan, Wee Liang Li, Liping Xi, Changchang Department of Mathematics University of California RiversideCA92521 United States Key Laboratory of High Performance Computing and Stochastic Information Processing Ministry of Education College of Mathematics and Computer Science Hunan Normal University Changsha Hunan410081 China School of Mathematical Sciences Capital Normal University Beijing100048 China

Using the Nakayama functor, we construct an equivalence from a Serre quotient category of a category of finitely generated modules to a category of finite-dimensional modules. We then apply this result to the categories FIG and VIq, and answer positively an open question of Nagpal on representation stability theory. Copyright © 2017, The Authors. All rights reserved.

关键词： Algebra

A fourth-order maximum principle preserving operator splitting scheme for three-dimensional fractional allen-cahn equations

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arXiv 2018年

作者： He, Dongdong Pan, Kejia Hu, Hongling School of Science and Engineering Chinese University of Hong Kong Shenzhen Guangdong518172 China School of Mathematics and Statistics Central South University Changsha Hunan410083 China School of Mathematics and Statistics Key Laboratory of High Performance Computing and Stochastic Information Processing Hunan Normal University Changsha Hunan410081 China

In this paper, by using Strang's second-order splitting method, the numerical procedure for the three-dimensional (3D) space fractional Allen-Cahn equation can be divided into three steps. The first and third steps involve an ordinary differential equation, which can be solved analytically. The intermediate step involves a 3D linear fractional diffusion equation, which is solved by the Crank-Nicolson alternating directional implicit (ADI) method. The ADI technique can convert the multidimensional problem into a series of one-dimensional problems, which greatly reduces the computational cost. A fourth-order difference scheme is adopted for discretization of the space fractional derivatives. Finally, Richardson extrapolation is exploited to increase the temporal accuracy. The proposed method is shown to be unconditionally stable by Fourier analysis. Another contribution of this paper is to show that the numerical solutions satisfy the discrete maximum principle under reasonable time step constraint. For fabricated smooth solutions, numerical results show that the proposed method is unconditionally stable and fourth-order accurate in both time and space variables. In addition, the discrete maximum principle is also numerically *** Codes 65M06, 65M12 Copyright © 2018, The Authors. All rights reserved.

关键词： Maximum principle

Practical security of continuous-variable quantum key distribution with reduced optical attenuation

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Physical Review A 2019年第1期100卷 012313-012313页

作者： Yi Zheng Peng Huang Anqi Huang Jinye Peng Guihua Zeng College of Information Science and Technology Northwest University Xi'an 710127 Shaanxi China State Key Laboratory of Advanced Optical Communication Systems and Networks and Center of Quantum Information Sensing and Processing Shanghai Jiao Tong University Shanghai 200240 China Institute for Quantum Information & State Key Laboratory of High Performance Computing College of Computer National University of Defense Technology Changsha 410073 China Greatwall Quantum Laboratory China Greatwall Technology Changsha 410205 China

In a practical continuous-variable quantum key distribution (CVQKD) system, the optical attenuator can adjust the Gaussian-modulated coherent states and the local oscillator signal to an optimal value for guaranteeing the security of the system and optimizing the performance of the system. However, the performance of the optical attenuator may deteriorate due to the intentional and unintentional damage of the device. In this paper, we investigate the practical security of a CVQKD system with reduced optical attenuation. We find that the secret key rate of the system may be overestimated based on the investigation of parameter estimation under the effects of reduced optical attenuation. This opens a security loophole for Eve to successfully perform an intercept-resend attack in a practical CVQKD system. To close this loophole, we add an optical fuse at Alice's output port and design a scheme to monitor the level of optical attenuation in real time, which can make the secret key rate of the system evaluated precisely. The analysis shows that these countermeasures can effectively resist this potential attack.

关键词： Quantum communication Quantum cryptography Quantum information processing with continuous variables

Vortex patterns and the critical rotational frequency in rotating dipolar Bose-Einstein condensates

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Physical Review A 2018年第2期98卷 023610-023610页

作者： Yongyong Cai Yongjun Yuan Matthias Rosenkranz Han Pu Weizhu Bao Beijing Computational Science Research Center Haidian District Beijing 100193 People's Republic of China Key Laboratory of High Performance Computing and Stochastic Information Processing (Ministry of Education of China) College of Mathematics and Statistics Hunan Normal University Changsha Hunan 410081 People's Republic of China Department of Mathematics National University of Singapore 119076 Singapore Department of Physics and Astronomy and Rice Quantum Institute Rice University Houston Texas 77251 USA

Based on the two-dimensional mean-field equations for pancake-shaped dipolar Bose-Einstein condensates in a rotating frame, for both attractive and repulsive dipole-dipole interaction (DDI) as well as arbitrary polarization direction, we study the profiles of the single vortex state and show how the critical rotational frequency changes with the s-wave contact interaction strength, DDI strength, and polarization angle. In addition, we find numerically that at the “magic angle” ϑ=arccos(3/3), the critical rotational frequency is almost independent of the DDI strength. By numerically solving the dipolar Gross-Pitaevskii equation at high rotation speed, we identify different patterns of vortex lattices which strongly depend on the polarization direction. As a result, we undergo a study of vortex lattice structures for the whole regime of polarization direction and find evidence that the vortex lattice orientation tends to be aligned to the dipole polarization axis for positive DDI strength and to the perpendicular direction of the dipole axis for negative DDI strength.

关键词： Bose-Einstein condensates Dipolar gases

Correction to: Bifurcations in a Hamiltonian system with two degrees of freedom associated to the reversible hyperbolic umbilic

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Nonlinear Dynamics 2021年第3期105卷 2849-2849页

作者： Zhou, Xing Li, Xuemei Key Laboratory of High Performance Computing and Stochastic Information Processing College of Mathematics and Statistics Hunan Normal University Changsha China

Distributed Online Generalized Nash Equilibrium Tracking for Prosumer Energy Trading Games

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Distributed Online Generalized Nash Equilibrium Tracking for...

American Control Conference (ACC)

作者： Yongkai Xie Zhaojian Wang John Z.F. Pang Bo Yang Xinping Guan Department of Automation Key Laboratory of System Control and Information Processing Ministry of Education of China Shanghai Jiao Tong University Shanghai China Agency for Science Technology and Research (A*STAR) Institute of High Performance Computing (IHPC) Singapore Republic of Singapore

With the proliferation of distributed generations, traditional passive consumers in distribution networks are evolving into "prosumers", which can both produce and consume energy. Energy trading with the main grid or between prosumers is inevitable if the energy surplus and shortage exist. To this end, this paper investigates the peer-to-peer (P2P) energy trading market, which is formulated as a generalized Nash game. We first prove the existence and uniqueness of the generalized Nash equilibrium (GNE). Then, an distributed online algorithm is proposed to track the GNE in the time-varying environment. Its regret is proved to be bounded by a sublinear function of learning time, which indicates that the online algorithm has an acceptable accuracy in practice. Finally, numerical results with six microgrids validate the performance of the algorithm.

关键词：

-SPECTRUM OF SELF-SIMILAR MEASURES WITH OVERLAPS IN THE ABSENCE OF SECOND-ORDER IDENTITIES

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Journal of the Australian Mathematical Society 2018年第1期106卷 56-103页

作者： SZE-MAN NGAI YUANYUAN XIE College of Mathematics and Computer Science Hunan Normal University Changsha Hunan 410081 China Department of Mathematical Sciences Georgia Southern University Statesboro GA 30460-8093 USA email smngai@georgiasouthern.edu Key Laboratory of High Performance Computing and Stochastic Information Processing (HPCSIP) (Ministry of Education of China) College of Mathematics and Computer Science Hunan Normal University Changsha Hunan 410081 PR China email xieyuanyuan198767@***

For the class of self-similar measures in $\mathbb{R}^{d}$ with overlaps that are essentially of finite type, we set up a framework for deriving a closed formula for the $L^{q}$-spectrum of the measure for $q\geq 0$. This framework allows us to include iterated function systems that have different contraction ratios and those in higher dimension. For self-similar measures with overlaps, closed formulas for the $L^{q}$-spectrum have only been obtained earlier for measures satisfying Strichartz’s second-order identities. We illustrate how to use our results to prove the differentiability of the $L^{q}$-spectrum, obtain the multifractal dimension spectrum, and compute the Hausdorff dimension of the measure.

关键词： fractal Lq-spectrum multifractal formalism self-similar measure essentially of finite type

ECML: An Ensemble Cascade Metric Learning Mechanism towards Face Verification

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arXiv 2020年

作者： Xiong, Fu Xiao, Yang Cao, Zhiguo Wang, Yancheng Zhou, Joey Tianyi Wu, Jianxin National Key Laboratory of Science and Technology on Multispectral Information Processing School of Artificial Intelligence and Automation Huazhong University of Science and Technology Wuhan430074 China Institute of High Performance Computing A*STAR Singapore Singapore National Key Laboratory for Novel Software Technology Nanjing University Nanjing210023 China

Face verification can be regarded as a 2-class fine-grained visual recognition problem. Enhancing the feature’s discriminative power is one of the key problems to improve its performance. Metric learning technology is often applied to address this need, while achieving a good tradeoff between underfitting and overfitting plays the vital role in metric learning. Hence, we propose a novel ensemble cascade metric learning (ECML) mechanism. In particular, hierarchical metric learning is executed in the cascade way to alleviate underfitting. Meanwhile, at each learning level, the features are split into non-overlapping groups. Then, metric learning is executed among the feature groups in the ensemble manner to resist overfitting. Considering the feature distribution characteristics of faces, a robust Mahalanobis metric learning method (RMML) with closed-form solution is additionally proposed. It can avoid the computation failure issue on inverse matrix faced by some well-known metric learning approaches (e.g., KISSME). Embedding RMML into the proposed ECML mechanism, our metric learning paradigm (EC-RMML) can run in the one-pass learning manner. Experimental results demonstrate that EC-RMML is superior to state-of-the-art metric learning methods for face verification. And, the proposed ensemble cascade metric learning mechanism is also applicable to other metric learning approaches. Copyright © 2020, The Authors. All rights reserved.

关键词： Inverse problems