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arXiv 2017年

作者： Audoux, Benjamin Bobkov, Vladimir Parini, Enea Aix Marseille Univ CNRS Centrale Marseille I2M 39 Rue Frederic Joliot Curie Marseille13453 France University of West Bohemia Faculty of Applied Sciences Department of Mathematics and NTIS Univerzitní 8 Plzen306 14 Czech Republic

We investigate multiplicity and symmetry properties of higher eigenvalues and eigenfunctions of the p-Laplacian under homogeneous Dirichlet boundary conditions on certain symmetric domains RN. By means of topological arguments, we show how symmetries of help to construct subsets of W1;p 0 () with suitably high Krasnosel'skii genus. In particular, if is a ball B RN, we obtain the following chain of inequalities: (formula presented) (p;B): Here i(p;B) are variational eigenvalues of the p-Laplacian on B, and (p;B) is the eigenvalue which has an associated eigenfunction whose nodal set is an equatorial section of B. If 2(p;B) =(p;B), as it holds true for p = 2, the result implies that the multiplicity of the second eigenvalue is at least N. In the case N = 2, we can deduce that any third eigenfunction of the p-Laplacian on a disc is nonradial. The case of other symmetric domains and the limit cases p = 1, p = 1 are also *** Codes 35J92, 35P30, 35A15, 35A16, 55M25, 35B06 Copyright © 2017, The Authors. All rights reserved.

关键词： eigenvalues and eigenfunctions

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ASYMPTOTIC EXPRESSIONS FOR eigenfunctions AND eigenvalues OF A DIELECTRIC OR OPTICAL WAVEGUIDE

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IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES 1969年第12期MT17卷 1130-&页

作者： SNYDER, AW Department of Electronic and Electrical Engineering University College London London UK

An asymptotic technique is presented, resulting in an analytically simple self-consistent description of the modes of a circular dielectric structure. When the dielectric difference between the rod and surrounding medium is small, the asymptotic expressions are valid for all frequencies. Even when the inside dielectric constant is twice the outside, less than a 10 percent error is usually involved. A simple functional expression for the eigenvalues of both the circular rod and the dielectric slab results from the analysis, thus eliminating the need for numerical or graphical methods.

关键词： Optical waveguides eigenvalues and eigenfunctions Optical refraction Frequency Dielectric constant Optical scattering Electromagnetic scattering Equations Electromagnetic waveguides Slabs

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Asymptotic behaviors and numerical computations of the eigenfunctions and eigenvalues associated with the classical and circular prolate spheroidal wave functions

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APPLIED MATHEMATICS AND COMPUTATION 2012年第22期218卷 10871-10888页

作者： Karoui, Abderrazek Mehrzi, Issam Univ Carthage Dept Math Fac Sci Bizerte Jarzouna 7021 Bizerte Tunisia

For a fixed bandwidth c, the classical prolate spheroidal wave functions (PSWFs) psi(n,c), and the circular ones (CPSWFs) phi(nc), form remarkable Hilbertian bases for the spaces of Fourier and Hankel band-limited functions with bandwidth c, respectively. The prolate spheroidal wave functions have already found many applications from various scientific area such as signal processing, numerical analysis, Physics and random matrix theory. These concrete applications of the PSWFs and CPSWFs, require the accurate computation of these special functions. To this end, we develop in this work two approximate procedures. The first procedure is based on pushing forward the WKB method and provide accurate approximations of the PSWFs and the CPSWFs in terms of some classical special functions. The second procedure is based on efficient and accurate quadrature methods for the computation of the values and the eigenvalues associated with PSWFs and the CPSWFs. Both procedures are valid for small as well as for large values of the frequency band-width c. Also, we provide the reader with some numerical examples that illustrate the different results of this work. (C) 2012 Elsevier Inc. All rights reserved.

关键词： eigenvalues and eigenfunctions Quadrature methods Prolate spheroidal wave functions WKB method Asymptotic behavior Bessel functions Orthogonal polynomials

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ASYMPTOTIC eigenfunctions AND eigenvalues OF A HOMOGENEOUS INTEGRAL-EQUATION

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IRE TRANSACTIONS ON INFORMATION THEORY 1962年第1期8卷 2-&页

作者： CAPON, J Federal Scientific Corporation NY USA

The eigenfunctions and eigenvalues of a certain integral equation are of importance in the Karhunen-Loéve expansion of second-order stationary random functions. In this note the asymptotic eigenfunctions and eigenvalues of this integral equation are derived for the case where the kernel is the Fourier transform of a rational function of \omega^2 .

关键词： eigenvalues and eigenfunctions Integral equations Kernel Fourier transforms Information theory Personal communication networks Laplace equations Polynomials

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Efficient Switched-Beam Detection for Dynamic Spectrum Sharing in 6G Wireless Networks With Full Duplex Technology at the THz Band

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IEEE ACCESS 2025年 13卷 57662-57675页

作者： Tani, Andrea Marabissi, Dania Univ Florence Dept Informat Engn I-50139 Florence Italy

The demand for higher data rates and bandwidth-intensive applications in the digital age is driving the need for cutting-edge technologies in future 6G systems, such as terahertz (THz) and full duplex (FD) communications. These technologies promise exceptional performance enhancement when combined with a dynamic spectrum management approach based on Cognitive Radio (CR). This paper explores spectrum sensing (SS) methods that do not require any prior information suitable for FD-THz communications, addressing challenges posed by residual self-interference (RSI) and molecular noise, which characterize THz communications. Both molecular re-radiation and RSI are sources of colored noise that dominate additive white Gaussian noise, complicating the operations of SS algorithms that do not require prior information based on the eigenvalues of the sample covariance matrix. To address this issue, we propose a low-complexity solution using a switched beam technique, along with a whitening matrix derived from a dataset containing RSI, molecular noise, and two largest eigenvalue (LE) detectors. The LE detector is introduced as an upper bound applicable only when perfect knowledge of the variances of all the noise components is available, which is unlikely in most practical scenarios. Therefore, we propose a novel, low-complexity detector not relying on prior information, based on scaled LE (SLE), which estimates the molecular noise and the AWGN variance online. The algorithm also includes a heuristic to adjust the contribution of the RSI and molecular noise components, ensuring that the detector retains an approximately constant false alarm rate (CFAR) property under typical operating conditions. Numerical simulations show that the proposed SS method is feasible and robust. Compared to the recursive least squares (RLS) adaptive filtering benchmark, it offers a more robust false alarm probability, particularly in the low-AWGN regime, with detection performance similar to or sligh

关键词： Colored noise Detectors Terahertz communications Delays Terahertz radiation Absorption Training Switches Sensors eigenvalues and eigenfunctions Cognitive radio spectrum sensing full duplex molecular noise terahertz

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Rotation Switching for Generating Multiplex Multiwing Hyperchaotic Hidden Attractors

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IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS 2025年第4期72卷 4147-4156页

作者： Wang, Ning Cui, Mengkai Yang, Yan Parastesh, Fatemeh Iu, Herbert Ho-Ching Xu, Quan Changzhou Univ Sch Microelect & Control Engn Changzhou 213159 Peoples R China Harbin Engn Univ Coll Informat & Commun Engn Harbin 150001 Peoples R China Chennai Inst Technol Ctr Nonlinear Syst Chennai 600069 India Univ Western Australia Sch Elect Elect & Comp Engn Perth WA 6009 Australia

The generation of multiplex-structure attractor mainly profits from the control of phase orbits in terms of location and orientation. It is an attractive but challenging task to implement trajectory control of stable-equilibrium-type hyperchaotic system to generate multiplex hidden attractors. This article presents a rotation switching scheme for generating multiplex multiwing hyperchaotic hidden attractors. First, a rotation transformation is applied to a single-stable-equilibrium hyperchaotic system to generate separate hidden attractors with arc-shaped, ring-shaped, and double-ring-shaped distributions. The rotation is linear operation and does not change the number and stability conditions of the equilibrium, which provides foundation for stringing together multiple separate units via multiangle switching. Consequently, multiplex multiwing hyperchaotic hidden attractors are generated under multilevel pulse excitation. A significant feature is that this method has certain flexibility in position and direction control of a seed attractor. Finally, hardware implementation using microcontroller-based platform verifies the feasibility of the designed system.

关键词： Switches Chaotic communication Autonomous systems Orbits Frequency division multiplexing eigenvalues and eigenfunctions Delay effects Hyperchaotic system hidden attractor multiplex attractor rotation switching single stable equilibrium

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Synchronous Condenser Placement for Multiple HVDC Power Systems Considering Short-Circuit Ratio Requirements

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IEEE TRANSACTIONS ON POWER SYSTEMS 2025年第1期40卷 765-779页

作者： Wang, Jiaxin Zhang, Jiawei Hou, Qingchun Zhang, Ning Tsinghua Univ Dept Elect Engn State Key Lab Power Syst Beijing 100084 Peoples R China Alibaba Grp Hangzhou 310000 Peoples R China

High voltage direct current (HVDC) power injections decrease power systems' short-circuit ratio (SCR) and the grid strength. They may cause voltage stability problems, especially in systems with a high share of renewable. The synchronous condenser (SynCon) is a promising solution for increasing the SCR. However, planning SynCons to meet the SCR requirements is challenging since the SCR requirement is a set of nonlinear eigenvalue inequality constraints that are not friendly to solvers. Based on generalized spectral decomposition, this paper proposes a Rayleigh cut method to equivalently transform eigenvalue inequality constraints into a set of linear constraints. Then, a mixed-integer linear programming model is built to solve the SCR-constrained SynCon placement problem. The proposed approach's convergence, complexity, and error analysis are all analyzed theoretically. Also, numerical comparisons with the other seven models suggest the proposed model is effective and efficient, outperforming the other models in terms of solution quality and computation time.

关键词： HVDC transmission eigenvalues and eigenfunctions Admittance Linear matrix inequalities Computational modeling Programming Vectors Synchronous condenser placement short-circuit ratio eigenvalue inequality constraint Rayleigh cut constraint

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A Robust VMD-Based Modified Multivariable Output Error Statespace Method for Detection of Low-Frequency Oscillations From Noisy Signals of WAMS

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IEEE ACCESS 2025年 13卷 2152-2172页

作者： Shetty, Dinesh Prabhu, Nagesh NITTE NMAM Inst Technol Dept Elect & Elect Engn Nitte 574110 Karnataka India

A poorly damped set of power oscillations interconnected power systems may lead to system-wide breakups or considerably reduce the power transfer over critical channels and corridors. As a result, identifying and suppressing these oscillatory modes is a critical challenge in modern power systems. Advances in wide area monitoring systems in modern power system facilitates the numerous data for the analysis of the system characteristics. Wide Area Monitoring Systems (WAMS) have enabled the collection of large amounts of data, aiding in system analysis. Phasor measurement units may experience noisy signals during measurement which may lead to poor identification of low frequency oscillatory modes or introduce extraneous modes during analysis using state of the art methods. This research examines the performance of the Robust Variable Mode Decomposition (VMD) technique for analyzing signals contaminated with noise. The study employs particle swarm optimization to determine the appropriate quantity of Intrinsic Mode Functions (IMFs). Additionally, it applies Anderson-Darling (AD) and Kolmogorov-Smirnov (KD) statistical distance measures to identify the effective number of IMFs, enabling the differentiation between signal IMFs and noise IMFs. Wavelet de-noising technique is employed for the de-noising the sum of chosen IMF signals. To validate the selection of adequate IMFs a heuristic approach is used to monitor the Signal-to-Noise Ratio (SNR) and Mean Square Error (MSE) for effectively seperating noisy contents from actual content of the signal. The proposed system incorporates the wavelet based filter in conventional VMD which not only helps in fetching the frequency but also helps in filtering the noise with appropriate selection of the mother wavelet, i.e. it utilizes the fusion of VMD and Wavelet Transform. The approach leverages the adaptability of VMD's decomposition alongside the precise denoising capabilities of WT, resulting in high effectiveness for signals th

关键词： Oscillators Power system stability Signal processing algorithms Noise measurement Transforms Noise Wavelet transforms Damping Noise reduction eigenvalues and eigenfunctions Anderson Darlington intrinsic mode functions low frequency oscillations variable mode decomposition

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Average Controllability of Complex Networks With Exponential Degree Distribution

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IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS I-REGULAR PAPERS 2025年第4期72卷 1813-1821页

作者： Xiang, Linying Zhu, Zeya Zhu, Jiawei Yao, Shuwei Chen, Fei Tiangong Univ Sch Artificial Intelligence Tianjin 300387 Peoples R China Tiangong Univ Tianjin Key Lab Intelligent Control Elect Equipmen Tianjin 300387 Peoples R China Shanghai Jiao Tong Univ Sch Elect Informat & Elect Engn Shanghai 200240 Peoples R China Nankai Univ Coll Artificial Intelligence Tianjin 300350 Peoples R China

This paper investigates the average controllability of Laplacian dynamical networks characterized by an exponential degree distribution. We introduce a novel configuration network model with an exponential degree distribution, incorporating a degree distribution parameter to adjust the heterogeneity of the distribution. We thoroughly examine the impact of degree distribution and degree correlation on average controllability. Our results reveal that increased heterogeneity in degree distribution tends to enhance average controllability, particularly when edges connect nodes with higher degrees. Moreover, networks with high assortativity exhibit improved average controllability. Building on these findings, we propose two effective strategies for optimizing average controllability. These strategies offer valuable guidance for the design of optimal complex networks in practical settings.

关键词： Controllability Laplace equations eigenvalues and eigenfunctions Complex networks Correlation Vectors Dynamical systems Approximation algorithms Power system dynamics Circuits Complex network average controllability Laplacian dynamics exponential degree distribution degree correlation

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Rate-Coded Secure Control for Multiagent Systems

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IEEE TRANSACTIONS ON AUTOMATIC CONTROL 2025年第3期70卷 1957-1964页

作者： Ji, Ruihang Ge, Shuzhi Sam Natl Univ Singapore Elect & Comp Engn Singapore 117583 Singapore

In this article, a rate-coded secure control is investigated for the consensus of multiagent systems (MASs). The challenge lies in how to design a broadcasting protocol when the communication bit is critical, and to strengthen its security at the same time. A novel rate-coded algorithm is designed for MASs from an encoding-decoding viewpoint, where only an encrypted L-length codeword, instead of the sensitive system information, is transmitted over the network between each agent and its neighbors whenever the triggering condition is satisfied. Unlike most existing results, our algorithm's triggering condition is correlated with the changing rate of the transmitted signal, effectively alleviating the signal distortion problem when its changing rate is too small or too large. As a result, the proposed rate-coded secure control not only ensures uniformly bounded consensus outcomes without continuous monitoring of neighbors' states but also reduces bandwidth requirements and enhances communication security. Finally, simulation results are given to illustrate the effectiveness of our theoretical finding.

关键词： Encoding Security Decoding Protocols Directed graphs Multi-agent systems eigenvalues and eigenfunctions Distortion Cryptography Vectors Consensus control rate-coded algorithm security communication

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