Rayleigh Quotient Research Articles

Polarized Communities Search via Co-guided Random Walk in Attributed Signed Networks

Polarized communities search aims at locating query-dependent communities, in which mostly nodes within each community form intensive positive connections, while mostly nodes across two communities are connected by negative links. Current approaches towards polarized communities search typically model the network topology, while the key factor of node, i.e., the attributes, are largely ignored. Existing studies have shown that community formation is strongly influenced by node attributes and the formation of communities are determined by both network topology and node attributes simultaneously. However, it is nontrivial to incorporate node attributes for polarized communities search. Firstly, it is hard to handle the heterogeneous information from node attributes. Secondly, it is difficult to model the complex relations between network topology and node attributes in identifying polarized communities. To address the above challenges, we propose a novel method Co-guided Random Walk in Attributed signed networks (CoRWA) for polarized communities search by equipping with reasonable attribute setting. For the first challenge, we devise an attribute-based signed network to model the auxiliary relation between nodes and a weight assignment mechanism is designed to measure the reliability of the edges in the signed network. As to the second challenge, a co-guided random walk scheme in two signed networks is designed to explicitly model the relations between topology-based signed network and attribute-based signed network so as to enhance the search result of each other. Finally, we can identify polarized communities by a well-designed Rayleigh quotient in the signed network. Extensive experiments on three real-world datasets demonstrate the effectiveness of the proposed CoRWA. Further analysis reveals the significance of node attributes for polarized communities search.

Nehari manifold method for singular double phase problem with optimal control on parameter

In this paper, we consider the following singular double phase problem −div(|∇u|p−2∇u + a(x)|∇u|q−2∇u) = λf(x)u−γ + g(x)ur−1, u > 0 in Ω and u = 0 on ∂Ω, where Ω⊂RN is an open bounded domain with smooth boundary, dimension N ≥ 2, exponents p < q < r < p* = Np/(N − p) with 1 < p < N, while 0 < γ < 1 and λ > 0 is real parameter. The weight functions f, g are bounded continuous functions which may change sign and the modulating function a is non-negative, continuous and has compact support in Ω. Using fibering map and Nehari manifold method, we show the existence of at least two positive solutions for (0, λ* + ϵ) for some ϵ > 0, where λ* is an extremal parameter, characterized via nonlinear Rayleigh quotient. An estimate on the extremal value λ* is also obtained.

A finding of the maximal saddle-node bifurcation for systems of differential equations

A variational method is presented for directly finding the bifurcation point of nonlinear equations as the saddle-node point of the extended nonlinear Rayleigh quotient. In the main result, this method is justified for finding the maximum saddle-node bifurcation point of the set of stable positive solutions to a system of equations with nonlinearities of convex-concave type.

On the uniqueness of eigenfunctions for the vectorial p-Laplacian

We study a nonlinear eigenvalue problem for vector-valued eigenfunctions and give a succinct uniqueness proof for minimizers of the associated Rayleigh quotient.

Post-Buckling Solutions for the Gao Beam

Summary This article analyses static buckling of the so-called Gao beam nonlinear model. It considers pure buckling problems in which the vertical loads are omitted. The analysis, using minimisation of energy and the concept of a modified Rayleigh quotient, leads to new results regarding the critical load necessary for buckling, and the existence and number of post-buckling solutions. Computational results are provided for cases with fixed axial loading. Furthermore, the authors explore the impact of the system parameters on the solutions, which are summarised in a table. The new findings in this research are unique and help to better understand the behaviour of the static and dynamic Gao beam.

Computational routine for obtaining the Rayleigh quotient for a truss system using its self-weight and corresponding static deformation

When subjected to dynamic loads, structural systems can lose stability or undergo material failure. Furthermore, if the frequency of the loads in the system approaches one of its natural vibration frequencies, resonance can occur, which can potentially increase the failure. In general, the first natural frequency of the structure is the most important for excitation loads with low frequencies. Therefore, a simplified analysis to obtain this frequency and vibration mode can be highly useful. This study aims to assess the application of the deformation of a spatial truss under static loading (self-weight) as an approximation of its first vibration mode. In this way, the Rayleigh Quotient is determined, and consequently, an approximation of the fundamental natural frequency of the system is obtained. A computational routine implemented in FORTRAN© was used for calculating this Quotient, and the routine's validation was performed by comparing its result with that obtained in the SAP2000© program. A good agreement between the results was observed, thus validating the mentioned routine.

Multiplicity of solutions to a Schrödinger problem with square diffusion term

In this paper we show multiplicity of solutions for a parameterized quasilinear Schrödinger equation in the presence of a square diffusion and indefinite superlinear term. Due to the presence of the quasilinear term, we can no longer work on the standard Sobolev spaces to show existence and non-existence of solutions. We overcome these difficulties by using perturbations arguments, Nehari sets and nonlinear Rayleigh quotients. As a byproduct of this approach we show that the associated energy functional has non-zero global minimizers only for small parameters.

The Neumann problem for a class of semilinear fractional equations with critical exponent

We establish the existence of solutions to the following semilinear Neumann problem for fractional Laplacian and critical exponent:{(−Δ)su+λu=|u|p−1uinΩ,Nsu(x)=0inRn∖Ω‾,u≥0inΩ, where λ>0 is a constant and Ω⊂Rn is a bounded domain with smooth boundary. Here, p=n+2sn−2s is a critical exponent, n>max⁡{4s,8s+23}, s∈(0,1). Due to the critical exponent in the problem, the corresponding functional Jλ does not satisfy the Palais-Smale (PS)-condition and therefore one cannot use standard variational methods to find the critical points of Jλ. We overcome such difficulties by establishing a bound for Rayleigh quotient and with the aid of nonlocal version of the Cherrier's optimal Sobolev inequality in bounded domains. We also show the uniqueness of these solutions in small domains.

A singular linear statistic for a perturbed LUE and the Hankel matrices

In this paper, we investigate the Hankel determinant generated by a singular Laguerre weight with two parameters. Using ladder operators adapted to monic orthogonal polynomials associated with the weight, we show that one of the auxiliary quantities is a solution to the Painlevé III′ equation and derive the discrete σ-forms of two logarithmic partial derivatives of the Hankel determinant. We approximate the second-order differential equation satisfied by the monic orthogonal polynomials with respect to the singular Laguerre weight with two parameters to the double confluent Heun equation, leveraging the scaling limit for two parameters and the dimension of the Hankel determinant. In addition, we establish the asymptotic behavior of the smallest eigenvalue of large Hankel matrices associated with the weight with two parameters, using the Coulomb fluid method and the Rayleigh quotient.

A homogeneous Rayleigh quotient with applications in gradient methods

Given an approximate eigenvector, its (standard) Rayleigh quotient and harmonic Rayleigh quotient are two well-known approximations of the corresponding eigenvalue. We propose a new type of Rayleigh quotient, the homogeneous Rayleigh quotient, and analyze its sensitivity with respect to perturbations in the eigenvector. Furthermore, we study the inverse of this homogeneous Rayleigh quotient as stepsize for the gradient method for unconstrained optimization. The notion and basic properties are also extended to the generalized eigenvalue problem.

Superdirective Unidirectional Mixed-Multipole Antennas: Designs, Analysis, and Simulations

Highly directive antenna systems that will address the needs of space-limited Internet-of-Things devices and their NextG applications are in demand. Compact high-frequency high-directivity alternatives to complex, power hungry phased arrays are desired. Several compact, unidirectional mixed-multipole antennas (UMMAs) based upon mixtures of primarily dipoles and quadrupoles are reported. The simulated performance characteristics of these 28 GHz near-field resonant parasitic (NFRP) antennas, notably their large peak directivities and front-to-back ratios, are presented; they are all demonstrated to be superdirective while having attractive radiation efficiencies. Uniform endfire arrays of active closely-spaced equal-length idealized wire dipoles are exemplified as reference cases. The Rayleigh quotient (RQ) method typically used to determine their amplitude coefficients is extended for the first time to achieve unidirectional versions. Comparisons between these unconstrained and constrained RQ-designed arrays are made. While the performance characteristics of the single-source broadside-radiating UMMAs approach those of the reference arrays, a direct comparison of a realizable UMMA with the analogous unequal-length, non-uniformly-spaced idealized dipole array demonstrates its superiority. A two-element UMMA array that is scalable to a larger number of elements is also presented and shown to be superdirective, illustrating that the attractive properties of its individual elements are maintained even when mutual coupling occurs.

Variational principles for a double Rayleigh beam system undergoing vibrations and connected by a nonlinear Winkler–Pasternak elastic layer

Abstract Variational principles and variationally consistent boundary conditions are derived for a system of double Rayleigh beams undergoing vibrations and subject to axial loads. The elastic layer connecting the beams are modelled as a three-parameter nonlinear Winkler–Pasternak layer with the Winkler layer having linear and nonlinear components and Pasternak layer having only a linear component. Variational principles are derived for the forced and freely vibrating double beam system using a semi-inverse approach. Hamilton’s principle for the system is given and the Rayleigh quotients are derived for the vibration frequency of the freely vibrating system and for the buckling load. Natural and geometric variationally consistent boundary conditions are derived which leads to a set of coupled boundary conditions due to the presence of Pasternak layer connecting the beams.

Multi-fidelity information fusion with hierarchical surrogate guided by feature mapping

Multi-fidelity information fusion has attracted increasing attention in the recent for its promising in engineering design and optimization. However, most of the existing fusion methods are sensitive to the correlation between low-fidelity (LF) and high-fidelity (HF) information, which could be further improved in terms of approximation performance. To address this problem, we develop a feature mapping-based hierarchical surrogate model for the fusion of LF and HF information. Compared to most current models, the presented model has a distinct hierarchical structure, where the LF model is first built through radial basis function. Then, a mapping relationship is established to describe the correlation between the transitional prediction and the HF prediction, and the mapping relationship is optimized using the Rayleigh quotient. Finally, a surrogate model is developed that can fuse information with different fidelities. To verify the effectiveness of the developed model, widely used numerical problems are analyzed, and five state-of-the-art methods are selected as comparative surrogates. Moreover, one engineering case is investigated to illustrate the capability of the proposed model in support of complex engineering design. Experiments demonstrate that our method outperforms other methods in 97.436% of the thirteen test problems under three different quantitative evaluations.

Free vibration disturbance and local mesh refinement induced by microcrack damage in circularly curved beams

PurposeThis study aimed to solve the engineering problem of free vibration disturbance and local mesh refinement induced by microcrack damage in circularly curved beams. The accurate identification of the crack damage depth, number and location depends on high-precision frequency and vibration mode solutions; therefore, it is critical to obtain these reliable solutions. The high-precision finite element method for the free vibration of cracked beams needs to be developed to grasp and control error information in the conventional solutions and the non-uniform mesh generation near the cracks. Moreover, the influence of multi-crack damage on the natural frequency and vibration mode of a circularly curved beam needs to be detected.Design/methodology/approachA scheme for cross-sectional damage defects in a circularly curved beam was established to simulate the depth, location and the number of multiple cracks by implementing cross-section reduction induced by microcrack damage. In addition, the h-version finite element mesh adaptive analysis method of the Timoshenko beam was developed. The superconvergent solution of the vibration mode of the cracked curved beam was obtained using the superconvergent patch recovery displacement method to determine the finite element solution. The superconvergent solution of the frequency was obtained by computing the Rayleigh quotient. The superconvergent solution of the eigenfunction was used to estimate the error of the finite element solution in the energy norm. The mesh was then subdivided to generate an improved mesh based on the error. Accordingly, the final optimised meshes and high-precision solution of natural frequency and mode shape satisfying the preset error tolerance can be obtained. Lastly, the disturbance behaviour of multi-crack damage on the vibration mode of a circularly curved beam was also studied.FindingsNumerical results of the free vibration and damage disturbance of cracked curved beams with cracks were obtained. The influences of crack damage depth, crack damage number and crack damage distribution on the natural frequency and mode of vibration of a circularly curved beam were quantitatively analysed. Numerical examples indicate that the vibration mode and frequency of the beam would be disturbed in the region close to the crack damage, and a greater crack depth translates to a larger frequency change. For multi-crack beams, the number and distribution of cracks also affect the vibration mode and natural frequency. The adaptive method can use a relatively dense mesh near the crack to adapt to the change in the vibration mode near the crack, thus verifying the efficacy, accuracy and reliability of the method.Originality/valueThe proposed combination of methodologies provides an extremely robust approach for free vibration of beams with cracks. The non-uniform mesh refinement in the adaptive method can adapt to changes in the vibration mode caused by crack damage. Moreover, the proposed method can adaptively divide a relatively fine mesh at the crack, which is applied to investigating free vibration under various curved beam angles and crack damage distribution conditions. The proposed method can be extended to crack damage detection of 2D plate and shell structures and three-dimensional structures with cracks.

Theoretical Investigation and Optimization of Power Amplifier Nonlinearity for In-Band Full-Duplex Radios

Although in-band full-duplex (IBFD) communication can theoretically double the spectral efficiency compared with conventional wireless communication technologies, it cannot be realized without reducing a high-power self-interference (SI) signal. Since the SI signal is affected by nonlinear distortion of radio-frequency (RF) circuits of its own terminal, cancellation systems capable of removing the nonlinear distorted SI have been actively developed. Detailed theoretical analysis based on the authors’ past literature shows that for IBFD using a nonlinear self-interference canceller, the symbol error rate is improved when the power amplifier has certain distortion characteristics rather than being ideally linearized. However, it is not yet known what amplifier characteristics are best for the IBFD system. In this study, we theoretically analyze the self-interference signal to clarify the optimum amplifier characteristics for the IBFD system. An optimization problem to obtain an optimum transfer function from theoretical analysis is formulated as an Rayleigh quotient with constraints, and we introduce a numerical technique to solve the optimization problem. By numerical experiments, we show that the SI cancellation performance, signal-to-interference-distortion-plus-noise ratio (SIDNR), and amplifier’s power efficiency are improved by the optimization.

Fractional p-Laplacian elliptic problems with sign changing nonlinearities via the nonlinear Rayleigh quotient

It is established existence and multiplicity of solutions to the fractional p-Laplacian problem in the whole space RN. More precisely, we consider the nonlocal elliptic problem with sign changing nonlinearities in the following form:{(−Δ)psu+V(x)|u|p−2u=λf(x)|u|q−2u+g(x)|u|r−2uinRN,u∈Ws,p(RN), where λ∈(γ⁎,0)∪(0,λ⁎),γ⁎<0, λ⁎>0 and N>ps with s∈(0,1) fixed. Furthermore, we assume that 1<q<p<r<ps⁎=Np/(N−ps). The potential V is a continuous function which is bounded from below by a positive constant. The main objective here is to consider nonlinearities f and g that can be sign changing functions. In this case, by using the nonlinear Rayleigh quotient, we prove that our main problem has at least two nontrivial solutions for each λ∈(γ⁎,0)∪(0,λ⁎). More specifically, the numbers λ⁎>0 and γ⁎<0 are sharp in order to consider the Nehari method, that is, the number λ⁎ is the largest positive number such that the Nehari method can be applied for each λ∈(0,λ⁎). The same assertion is verified for γ⁎, that is, the number γ⁎<0 is the smallest negative number such that the Nehari method can be employed for each λ∈(γ⁎,0).

Higher-order Rayleigh-quotient gradient effect on electron correlations.

The incomplete understanding of electron correlation is still profound due to the lack of exact solutions of the Schrödinger equation of many electron systems. In this work, we present the correlation-induced changes in the calculated many-electron systems beyond the standard residual. To locate the minimum of the Rayleigh quotient, each iteration is to seek the lowest eigenpairs in a subspace spanned by the current wave function and its gradient of the Rayleigh-quotient as well as the upcoming higher-order residual. Consequently, as the upcoming errors can be introduced and circumvented with the search in the higher-order residual, a concomitant improved performance in terms of number of iterations, convergence rate, and total elapsed time is very significant. The correlation energy components obtained with the original residual are corrected with the higher-order residual application, satisfying the correlation virial theorem with much improved accuracy. The comparison with the original residual, the higher-order residual significantly improves the electron binding, favoring the localization of electrons' distribution, revealed with the increasing peak of the distribution and correlation function and the reduced interelectron distance and its angle.

Design and Performance Analysis of Wireless Legitimate Surveillance Systems With Radar Function

Integrated sensing and communication (ISAC) has recently been considered as a promising approach to save spectrum resources and reduce hardware cost. Meanwhile, as information security becomes increasingly more critical issue, government agencies urgently need to legitimately monitor suspicious communications via proactive eavesdropping. Thus, in this paper, we investigate a wireless legitimate surveillance system with radar function. We seek to jointly optimize the receive and transmit beamforming vectors to maximize the eavesdropping success probability which is transformed into the difference of signal-to-interference-plus-noise ratios (SINRs) subject to the performance requirements of radar and surveillance. The formulated problem is challenging to solve. By employing the Rayleigh quotient and fully exploiting the structure of the problem, we apply the divide-and-conquer principle to divide the formulated problem into two subproblems for two different cases. For the first case, we aim at minimizing the total transmit power, and for the second case we focus on maximizing the jamming power. For both subproblems, with the aid of orthogonal decomposition, we obtain the optimal solution of the receive and transmit beamforming vectors in closed-form. Performance analysis and discussion of some insightful results are also carried out. Finally, extensive simulation results demonstrate the effectiveness of our proposed algorithm in terms of eavesdropping success probability.

An application of Rayleigh quotient for crack detection in simply supported beam

An explicit expression of natural frequencies through crack parameters is derived for multiple cracked beams with simply supported boundaries using the Rayleigh quotient. The obtained expression provides not only a simple tool for calculating natural frequencies of multiple cracked beams, but also allows employing the so-called crack scanning method for detecting multiple cracks in simply supported beams from measured natural frequencies. A numerical example demonstrates that the crack scanning method, in combination with the Rayleigh quotient, enables consistent identification of cracks with 1% relative depth.

Solving Sturm–Liouville inverse problems by an orthogonalized enhanced boundary function method and a product formula for symmetric potential

We develop a fast convergence iterative method after transforming the Sturm–Liouville problem into the weak-form integral equations, and using sinusoidal functions as the testers as well as the bases of eigenfunction. Owing to the orthogonal property of these bases, we can obtain the expansion coefficients in closed-form. A new idea of orthogonalized and enhanced boundary function (OEBF) is introduced to compute eigenvalues. In the Sobolev space, we prove the closeness of the OEBF to the eigenfunction with their distance being reduced when the eigenvalue increases. Moreover, upon expressing the Rayleigh quotient in terms of OEBF, the unknown functions in the Sturm–Liouville operator can be recovered quickly, merely two boundary data of unknown functions and the first eigenvalue are sufficing. The OEBF is a promising method with a few iterations to obtain quite accurate estimations of the unknown potential and weight functions. Thanks to the symmetry property a symmetric matrix eigenvalue problem is derived to recover a symmetric potential. The characteristic equation endowing with a special coefficient matrix is decomposed into a product of two characteristic equations. The Newton iterative method is thus developed by relying on the product formula to reconstruct the unknown symmetric potential function with the aid of a few lower orders eigenvalues.