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In this note, we provide an adaptation of the Kohler-Jobin rearrangement technique to the setting of the Gauss space. As a result, we prove the Gaussian analogue of the Kohler-Jobin resolution of a conjecture of Pólya-Szegö: when the Gaussian torsional rigidity of a domain is fixed, the Gaussian principal frequency is minimized for the half-space. At the core of this rearrangement technique is the idea of considering a "modified" torsional rigidity, with respect to a given function, and rearranging its layers to half-spaces, in a particular way; the Rayleigh quotient decreases with this procedure. We emphasize that the analogy of the Gaussian case with the Lebesgue case is not to be expected here, as in addition to some soft symmetrization ideas, the argument relies on the properties of some special functions; the fact that this analogy does hold is somewhat of a miracle. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

In this article, we propose, analyze and demonstrate a dynamic momentum method to accelerate power and inverse power iterations with minimal computational overhead. The method can be applied to real diagonalizable matrices, is provably convergent with acceleration in the symmetric case, and does not require a priori spectral knowledge. We review and extend background results on previously developed static momentum accelerations for the power iteration through the connection between the momentum accelerated iteration and the standard power iteration applied to an augmented matrix. We show that the augmented matrix is defective for the optimal parameter choice. We then present our dynamic method which updates the momentum parameter at each iteration based on the Rayleigh quotient and two previous residuals. We present convergence and stability theory for the method by considering a power‐like method consisting of multiplying an initial vector by a sequence of augmented matrices. We demonstrate the developed method on a number of benchmark problems, and see that it outperforms both the power iteration and often the static momentum acceleration with optimal parameter choice. Finally, we present and demonstrate an explicit extension of the algorithm to inverse power iterations. [ABSTRACT FROM AUTHOR]

This work presents the mathematical/theoretical framework of the "nth-Order Feature Adjoint Sensitivity Analysis Methodology for Response-Coupled Forward/Adjoint Linear Systems" (abbreviated as "nth-FASAM-L"), which enables the most efficient computation of exactly obtained mathematical expressions of arbitrarily-high-order (nth-order) sensitivities of a generic system response with respect to all of the parameters (including boundary and initial conditions) underlying the respective forward/adjoint systems. Responses of linear models can depend simultaneously on both the forward and the adjoint state functions. This is in contradistinction to responses for nonlinear systems, which can depend only on the forward state functions since nonlinear operators do not admit bona-fide adjoint operators. Among the best-known model responses that depend simultaneously on both the forward and adjoint state functions are Lagrangians used for system optimization, the Schwinger and Roussopoulos functionals for analyzing reaction rates and ratios thereof, and the Rayleigh quotient for analyzing eigenvalues and/or separation constants. The sensitivity analysis of such responses makes it necessary to treat linear models/systems in their own right, rather than treating them just as particular cases of nonlinear systems. The unparalleled efficiency and accuracy of the nth-FASAM-L methodology stems from the maximal reduction of the number of adjoint computations (which are "large-scale" computations) for computing high-order sensitivities, since the number of large-scale computations when applying the nth-FASAM-N methodology is proportional to the number of model features as opposed to the number of model parameters (which are considerably more than the number of features). The mathematical framework underlying the nth-FASAM-L is developed in linearly increasing higher-dimensional Hilbert spaces, as opposed to the exponentially increasing "parameter-dimensional" spaces in which response sensitivities are computed by other methods (statistical, finite differences, etc.), thus providing the basis for overcoming the curse of dimensionality in sensitivity analysis and all other fields (uncertainty quantification, predictive modeling, etc.) which need such sensitivities. [ABSTRACT FROM AUTHOR]

In this work, we prove existence and multiplicity of solutions to a Kirchhoff elliptic problem in the whole space ℝN. More specifically, we consider the following nonlocal elliptic problem: − m(∥∇u∥22)Δu + V (x)u = λa(x)|u|q−2u − 휃b(x)|u|p−2uin ℝN,u ∈ H1(ℝN), where N ≥ 3, the parameters λ,휃 > 0, 2 < 2(σ + 1) < q < p < 2∗ := 2N/(N − 2), σ ∈ (0, 2/(N − 2)) and a,b ∈ L∞(ℝN) with a(x),b(x) > 0 almost everywhere in ℝN. This type of problem contains the function m : ℝ+ → ℝ+ known as the Kirchhoff function given by m(t) = α1 + α2tσ with α1,α2 > 0 and t ∈ ℝ+. Under our assumptions the potential V : ℝN → ℝ and the nonlinearities can be sign changing functions. Hence, our main objective is to prove that the above problem has at least two distinct nontrivial solutions, one of them being a ground state, whenever λ ∈ (λ∗, +∞) for some suitable λ∗ > 0. The main idea is to use the minimization method in the Nehari manifold together with the nonlinear Rayleigh quotient. In our setting, the main difficulty is ensuring the existence of nontrivial solutions by using the Nehari method considering the Lagrange Multipliers Theorem. In other words, we study the case where the fibering map admits inflections points with λ,휃 > 0. Furthermore, for each λ ∈ (−∞,λ∗], we show a nonexistence result for our main problem. It is important to emphasize that λ∗ > 0 is sharp in order to find the existence and multiplicity of nontrivial solutions for our main problem. [ABSTRACT FROM AUTHOR]

We examine a method for solving an infinite‐dimensional tensor eigenvalue problem Hx=λx$$ Hx=\lambda x $$, where the infinite‐dimensional symmetric matrix H$$ H $$ exhibits a translational invariant structure. We provide a formulation of this type of problem from a numerical linear algebra point of view and describe how a power method applied to e−Ht$$ {e}^{- Ht} $$ is used to obtain an approximation to the desired eigenvector. This infinite‐dimensional eigenvector is represented in a compact way by a translational invariant infinite Tensor Ring (iTR). Low rank approximation is used to keep the cost of subsequent power iterations bounded while preserving the iTR structure of the approximate eigenvector. We show how the averaged Rayleigh quotient of an iTR eigenvector approximation can be efficiently computed and introduce a projected residual to monitor its convergence. In the numerical examples, we illustrate that the norm of this projected iTR residual can also be used to automatically modify the time step t$$ t $$ to ensure accurate and rapid convergence of the power method. [ABSTRACT FROM AUTHOR]

A two dimensional eigenvalue problem (2DEVP) of a Hermitian matrix pair (A,C) is introduced in this paper. The 2DEVP can be regarded as a linear algebra formulation of the well-known eigenvalue optimization problem of the parameter matrix (A-µC). We first present fundamental properties of the 2DEVP, such as the existence and variational characterizations of 2D-eigenvalues, and then devise a Rayleigh quotient iteration (RQI)-like algorithm, 2DRQI in short, for computing a 2D-eigentriplet of the 2DEVP. The efficacy of the 2DRQI is demonstrated by large scale eigenvalue optimization problems arising from the minmax of Rayleigh quotients and the distance to instability of a stable matrix. [ABSTRACT FROM AUTHOR]

Background: The common spatial patterns (CSP) is a widely used EEG feature extractor for motor imagerybased brain-computer interfaces, with the optimal spatial filter formulated as a generalized Rayleigh quotient. However, the traditional CSP uses Euclidean metric, which ignores the specific geometric structure of symmetry positive definite (SPD) matrices, resulting in issues such as swelling effect, noncomplete space, and indefinite matrices. Methods: To address these limitations, this paper introduces three alternative approaches with considering the geometric properties of SPD matrices. The geometry-aware CSP with diagonalization (gaCSPd) replaces the Euclidean means in the joint diagonalization principle of CSP with Riemannian means. The geometry-aware CSP with maximum discriminative information between classes (gaCSPb) aims to find an optimal projection matrix on a Riemannian manifold while maximizing the Riemannian distance between classes. The geometry-aware CSP with maximum within-class variance (gaCSPw) seeks a low-dimensional submanifold with the maximum intra-class variance in the projected data. Results: Experiment results on two BCI competition datasets demonstrate the competitiveness against state-of-the-art methods and confirm the effectiveness of geometry-aware CSP as a feature extractor for motor imagery-based brain-computer interfaces. [ABSTRACT FROM AUTHOR]

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With the recent emergence of mixed precision hardware, there has been a renewed interest in its use for solving numerical linear algebra problems fast and accurately. The solution of total least squares problems, i.e., solving min E , r ‖ [ E , r ] ‖ F subject to (A + E) x = b + r , arises in numerous applications. Solving this problem requires finding the smallest singular value and corresponding right singular vector of [ A , b ] , which is challenging when A is large and sparse. An efficient algorithm for this case due to Björck et al. (SIAM J. Matrix Anal. Appl. 22(2), 413–429 2000), called RQI-PCGTLS, is based on Rayleigh quotient iteration coupled with the preconditioned conjugate gradient method. We develop a mixed precision variant of this algorithm, RQI-PCGTLS-MP, in which up to three different precisions can be used. We assume that the lowest precision is used in the computation of the preconditioner and give theoretical constraints on how this precision must be chosen to ensure stability. In contrast to standard least squares, for total least squares, the resulting constraint depends not only on the matrix A , but also on the right-hand side b . We perform a number of numerical experiments on model total least squares problems used in the literature, which demonstrate that our algorithm can attain the same accuracy as RQI-PCGTLS albeit with a potential convergence delay due to the use of low precision. Performance modeling shows that the mixed precision approach can achieve up to a 4 × speedup depending on the size of the matrix and the number of Rayleigh quotient iterations performed. [ABSTRACT FROM AUTHOR]

Quaternions are a non-commutative associative number system that extends complex numbers, first described by Hamilton in 1843. We present algorithms for solving the eigenvalue problem for arrowhead and DPRk (diagonal-plus-rank-k) matrices of quaternions. The algorithms use the Rayleigh Quotient Iteration with double shifts (RQIds), Wielandt's deflation technique and the fact that each eigenvector can be computed in O (n) operations. The algorithms require O (n 2) floating-point operations, n being the order of the matrix. The algorithms are backward stable in the standard sense and compare well to the standard QR method in terms of speed and accuracy. The algorithms are elegantly implemented in Julia, using its polymorphism feature. [ABSTRACT FROM AUTHOR]

It is established existence of weak solutions for nonlocal elliptic problems driven by the fractional Laplacian where the nonlinearity is indefinite in sign. More specifically, we shall consider the following nonlocal elliptic problem: (−Δ)su+V(x)u=μa(x)|u|q−2u−λ|u|p−2uinRN,u∈Hs(RN),$$\begin{equation*} {\left\lbrace \def\eqcellsep{&}\begin{array}{rcl}(-\Delta)^s u +V(x)u & = & \mu a(x)|u|^{q-2}u-\lambda |u|^{p-2}u \mbox{ in }\, \mathbf {R}^N, \\[3pt] u\in H^s(\mathbf {R}^N),&&{} \end{array} \right.} \end{equation*}$$where s∈(0,1),s0$\mu , \lambda >0$. The potentials V,a:RN→R$V, a : \mathbf {R}^N \rightarrow \mathbf {R}$ satisfy some extra assumptions. The main feature is to find sharp parameters λ>0$\lambda > 0$ and μ>0$\mu > 0$ where the Nehari method can be applied. In order to do that, we employ the nonlinear Rayleigh quotient together a fine analysis on the fibering maps associated to the energy functional. It is important to mention also that for each parameters λ>0$\lambda > 0$ and μ>0$\mu > 0$, there exist degenerate points in the Nehari set that give serious difficulties. Furthermore, we consider nonlinearities that are superlinear at the origin and superlinear at infinity. In order to overcome these difficulties, we apply some estimates together with a carefully analysis on the fibering maps. Here, we also consider the asymptotic behavior of the weak solutions for our main problem when λ→0$\lambda \rightarrow 0$ or μ→∞$\mu \rightarrow \infty$. Furthermore, we consider a nonexistence result for our main problem under some appropriate conditions on the parameters λ>0$\lambda >0$ and μ>0$\mu > 0$. [ABSTRACT FROM AUTHOR]

It is established existence and multiplicity of solutions for semilinear elliptic problems defined in the whole space R N considering subcritical nonlinearities with some parameters. Here we emphasize that our nonlinearities can be sign-changing functions. The main difficulty is proving the existence of nontrivial solutions by using the Nehari method, taking into account that the Lagrange multipliers theorem cannot be directly applied in our setting. In fact, we consider the case where the fibering map admits inflection points. In other words, we consider the case where the Nehari set admits degenerate critical points. Hence our main contribution is to consider a huge class of semilinear elliptic problems where the standard Nehari method cannot be applied. Using some fine estimates and recovering some compactness results together with the nonlinear Rayleigh quotient, we prove that our main problem admits at least three nontrivial solutions depending on the parameters. [ABSTRACT FROM AUTHOR]

We discuss the existence and non-existence of periodic in one variable and compactly supported in the other variables least energy solutions for equations with non-Lipschitz nonlinearity of the form: − Δ u = λ u p − u q in R N + 1 , where 0 < q < p < 1 and λ ∈ R. The approach is based on the Nehari manifold method supplemented by a one-sided constraint given through the functional of the suitable Pohozaev identity. The limit value of the parameter λ, where the approach is applicable, corresponds to the existence of periodic in one variable and compactly supported in the other variables least energy solutions. This value is found through the extrem values of nonlinear generalized Rayleigh quotients and the so-called curve of the critical exponents of p, q. Important properties of the solutions are derived for suitable ranges of the parameters, such as that they are not trivial with respect to the periodic variable and do not coincide with compactly supported solutions on the entire space R N + 1 . [ABSTRACT FROM AUTHOR]

For the adjacency matrix A(G) and diagonal matrix of the vertex degrees D(G) of a simple graph G, the A(G) matrix is the convex combinations of D(G) and A(G), and is defined as A(G) = D(G)+(1)A(G), for 0 n be the eigenvalues of A(G) (which we call -adjacency eigenvalues of the graph G). The generalized adjacency energy also called -adjacency energy of the graph G is defined as EA (G) = is the average vertex degree, m is the size and n is the order of G. The -adjacency energy of a graph G merges the theory of energy (adjacency energy) and the signless Laplacian energy, as EA0 (G) = E (G) and 2E A 12 (G) = QE(G), where E (G) is the energy and QE(G) is the signless Laplacian energy of G. In this paper, we obtain some new upper and lower bounds for the generalized adjacency energy of a graph, in terms of different graph parameters like the vertex covering number, the Zagreb index, the number of edges, the number of vertices, etc. We characterize the extremal graphs attained these bounds. [ABSTRACT FROM AUTHOR]

We prove that the biharmonic NLS equation Δ 2 u + 2 Δ u + (1 + ε) u = | u | p - 2 u i n R d has at least k + 1 geometrically distinct solutions if ε > 0 is small enough and 2 < p < 2 ⋆ k , where 2 ⋆ k is an explicit critical exponent arising from the Fourier restriction theory of O (d - k) × O (k) -symmetric functions. This extends the recent symmetry breaking result of Lenzmann–Weth (Symmetry breaking for ground states of biharmonic NLS via Fourier extension estimates, 2023) and relies on a chain of strict inequalities for the corresponding Rayleigh quotients associated with distinct values of k. We further prove that, as ε → 0 + , the Fourier transform of each ground state concentrates near the unit sphere and becomes rough in the scale of Sobolev spaces. [ABSTRACT FROM AUTHOR]

Due to its improved spectral efficiency, non-orthogonal multiple access (NOMA) is regarded as a promising multiple access technology for beyond fifth-generation (5G) networks. In this paper, we examine the performance of a cell-edge user and suggest a beamforming scheme in a cooperative two-user multiple-input multiple-output (MIMO)-NOMA system with Rayleigh fading channels. In the envisaged scenario, a cell-center user with better channel gain harvests energy and assists a cell-edge user with poorer channel conditions by using a simultaneous wireless information and power transfer mechanism. We first obtain the outage probability expressions in closed form for the cell-edge user for the Kronecker structured channel model in the covariance shaping and indefinite quadratic form. Next, the beamformers at the transmitter and receiver are introduced to reduce outage probability, with transmit beamformers maximizing the signal-to-leakage-plus-noise ratio and receive beamformers minimizing cross-covariance across all users. Furthermore, beamformers are adopted in the two-user network to adjust the power ratio and power allocation coefficients for better performance of the cell-edge user. Moreover, our scheme is also compared with a transmit antenna selection baseline scheme. Simulation results demonstrate that our approach enhances the performance of the two-user cooperative MIMO-NOMA system’s performance, validating the theoretical analysis.

In this paper, buckling and nonlinear post-buckling behaviors of a bidirectional porous (BDP) beam are investigated under bidirectional hygrothermal environment. Euler–Bernoulli beam theory with the von Kármán nonlinearity is employed to derive the nonlinear variable coefficient governing differential equations based on Rayleigh quotient method. Analytical solutions of critical buckling load and load–deflection equilibrium path in post-buckling are deduced for the single directional varying (SDV) porous beam. The general numerical solutions for bidirectional varying (BDV) porous beam are obtained by differential quadrature finite element method (DQFEM) with Newton–Raphson iteration method based on the variation principle. The high accuracy of the present numerical method with higher computing efficiency is verified by comparison with published reports and the analytical results in this work. Parametric analysis on effects of the porosity bidirectional distributions, porosity coefficients, distributions of hygrothermal environment and boundary conditions on buckling load and post-buckling response is carried out to enhance the buckling and deformation resistances in design, manufacture and usage of porous structures. The results show that the bidirectional porosity pattern, linear and nonlinear hygrothermal distribution and boundary conditions play a significant role on buckling critical external load and critical hygrothermal increments, buckling form and post-buckling path. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

We propose a supervised principal component regression method for relating functional responses with high-dimensional predictors. Unlike the conventional principal component analysis, the proposed method builds on a newly defined expected integrated residual sum of squares, which directly makes use of the association between the functional response and the predictors. Minimizing the integrated residual sum of squares gives the supervised principal components, which is equivalent to solving a sequence of nonconvex generalized Rayleigh quotient optimization problems. We reformulate the nonconvex optimization problems into a simultaneous linear regression with a sparse penalty to deal with high dimensional predictors. Theoretically, we show that the reformulated regression problem can recover the same supervised principal subspace under certain conditions. Statistically, we establish nonasymptotic error bounds for the proposed estimators when the covariate covariance is bandable. We demonstrate the advantages of the proposed method through numerical experiments and an application to the Human Connectome Project fMRI data. for this article are available online. [ABSTRACT FROM AUTHOR]

Bent functions are Boolean functions in even number of variables that have maximal nonlinearity. They have flat Walsh–Hadamard spectrum and are of interest for their applications in algebra, coding theory and cryptography. A bent function is called self-dual if it coincides with its dual bent function. In current work we study the decomposition of the form f 0 , f 1 , ... , f 2 k - 1 of the vector of values of a self-dual bent function, formed by the concatenation of 2 k Boolean functions f j in n - k variables. We treat the cases k = 1 , 2 . Based on a spectral characterization, we introduce a notion of self-dual near-bent function in odd number of variables and prove that there exists a one-to-one correspondence between the notions of self-duality for even and odd number of variables. As a result the characterization for the decomposition f 0 , f 1 is obtained. For the decomposition f = f 0 , f 1 , f 2 , f 3 we prove that if sign vectors of subfunctions f j are linearly dependent, then all these subfunctions are bent. We prove that for n ⩾ 6 the converse does not hold, that is the obtained condition is sufficient only. These results are also generalized for the case of an arbitrary bent function. Three new iterative constructions of self-dual bent functions are proposed. One of them allows to build a class of self-dual bent functions which cannot be decomposed into the concatenation of four bent functions. Based on the constructions a new iterative lower bound on the cardinality of the set of self-dual bent functions is obtained. [ABSTRACT FROM AUTHOR]

Within planetary gear transmissions (PGTs), mode shapes and eigenfrequencies hold a crucial significance in operational reliability and efficacy. Mode shapes explain the unique motion patterns inherent in PGT systems. Conversely, eigenfrequencies describe the inherent frequencies at which PGT systems undergo vibration or oscillation upon exposure to external forces or disruptions. This research paper presents a comprehensive investigation into the dynamic behavior of a three-stage PGT utilized in medium and heavy trucks. This study introduces the Rayleigh energy method to assess system dynamics, revealing a bounded Rayleigh quotient related to the highest related eigenvalue. Then, this study delves into eigenfrequencies and the mode shape behavior of the adopted PGT model. The eigenfrequencies are identified as encompassing diverse vibrational modes of central components and planet gears. Moreover, a multi-scale analysis of the adopted PGT model is presented by deriving matrices for mass, bearing stiffness, and mesh stiffness. Comparisons with the Rayleigh energy method demonstrate the new approach's efficiency, exhibiting a low margin of error in the determination of eigenfrequencies. This investigation also highlights the alignment of identified mode shapes with the established literature, detailing the multi-scale approach's minor deviation in mode shape determination compared to the Rayleigh energy method. [ABSTRACT FROM AUTHOR]

In this paper, a modified Dai-Liao type four-term conjugate gradient method is proposed. Based on the estimation of the Rayleigh quotient of the iteration matrix of search direction, choices for the parameter mentioned above self-adjust the weight between the sufficient descent condition and reduction in the condition number of iteration matrix of the search direction, which can be regarded as inheritance and development of the originally optimal choices for a set of parameters $ \overline{t} _{\theta_k }^{\ast } $ (JIMO,13(2):649-658,2017). Meanwhile, sufficient descent condition of the modified search direction is satisfied at each iteration. Under mild conditions, global convergence of the resulting method are established even if the objective function is nonconvex. Some comparative numerical results illustrate the efficiency of the proposed method. [ABSTRACT FROM AUTHOR]

Purpose: Eigenvalues are the natural frequencies of system squared. When designing a system it is important to know the natural frequencies, because if the system is forced near one of these natural frequencies the magnitude of vibration becomes very large. The eigenvalues are typically determined by solving an eigenvalue problem, which is an iterative produce that is expensive for larger systems. If multiple perturbations to the system are made or tested re-solving an eigenvalue problem every time becomes prohibitive. Perturbation methods exist to predict perturbed eigenvalues more quickly. However, these methods typically require matrix–vector products and do not quantify what is considered a small enough perturbation to use these methods. Methods: This paper looks to address these issues using a scalar perturbed eigenvalue expression that avoids calculating matrix–vector products for every perturbation and developing reference plots that can be used to predict the associated error. The reference plots may be used to predict errors in the approximated natural frequencies from nominal modal parameters. The scalar perturbed eigenvalue expression and reference plots for errors were tested using numerical examples. Results: In every case tested the plots were able to accurately predict the expected errors, to be within a predicted range. Conclusion: The proposed method allows one to use the developed scalar expression to predict perturbed eigenvalues, and the developed reference plots may be used to predict the errors associated with using the proposed expression. [ABSTRACT FROM AUTHOR]

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 ⊂ R N 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. [ABSTRACT FROM AUTHOR]

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 RandomWalk in Attributed signed networks (CoRWA) for polarized communities search by equipping with reasonable attribute setting. For the first challenge, we devise an attributebased 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 topologybased 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. [ABSTRACT FROM AUTHOR]

Gradient‐type iterative methods for solving Hermitian eigenvalue problems can be accelerated by using preconditioning and deflation techniques. A preconditioned steepest descent iteration with implicit deflation (PSD‐id) is one of such methods. The convergence behavior of the PSD‐id is recently investigated based on the pioneering work of Samokish on the preconditioned steepest descent method (PSD). The resulting non‐asymptotic estimates indicate a superlinear convergence of the PSD‐id under strong assumptions on the initial guess. The present paper utilizes an alternative convergence analysis of the PSD by Neymeyr under much weaker assumptions. We embed Neymeyr's approach into the analysis of the PSD‐id using a restricted formulation of the PSD‐id. More importantly, we extend the new convergence analysis of the PSD‐id to a practically preferred block version of the PSD‐id, or BPSD‐id, and show the cluster robustness of the BPSD‐id. Numerical examples are provided to validate the theoretical estimates. [ABSTRACT FROM AUTHOR]

We introduce the two‐sided Rayleigh quotient shift to the QR algorithm for non‐Hermitian matrices to achieve a cubic local convergence rate. For the singly shifted case, the two‐sided Rayleigh quotient iteration is incorporated into the QR iteration. A modified version of the method and its truncated version are developed to improve the efficiency. Based on the observation that the Francis double‐shift QR iteration is related to a 2D Grassmann–Rayleigh quotient iteration, A doubly shifted QR algorithm with the two‐sided 2D Grassmann–Rayleigh quotient double‐shift is proposed. A modified version of the method and its truncated version are also developed. Numerical examples are presented to show the convergence behavior of the proposed algorithms. Numerical examples also show that the truncated versions of the modified methods outperform their counterparts including the standard Rayleigh quotient single‐shift and the Francis double‐shift. [ABSTRACT FROM AUTHOR]

In our study we construct a boundary value problem in elasticity of porous piezoelectric bodies with a dipolar structure To construct an eigenvalue problem in this context, we consider two operators defined on adequate Hilbert spaces. We prove that the two operators are positive and self adjoint, which allowed us to show that any eigenvalue is a real number and two eigenfunctions which correspond to two distinct eigenvalues are orthogonal. With the help of a Rayleigh quotient type functional, a variational formulation for the eigenvalue problem is given. Finally, we consider a disturbation analysis in a particular case. It must be emphasized that the porous piezoelectric bodies with dipolar structure addressed in this study are considered in their general form, i.e.,inhomogeneous and anisotropic. [ABSTRACT FROM AUTHOR]

We study the Rayleigh quotient of a Hermitian matrix with quaternionic coefficients and prove its main properties. As an application, we give some relationships between left and right eigenvalues of Hermitian and symplectic matrices. [ABSTRACT FROM AUTHOR]

We generalize the recently proposed two‐sided Rayleigh quotient single‐shift and the two‐sided Grassmann–Rayleigh quotient double‐shift used in the QR algorithm and apply the generalized versions to the QZ algorithm. With such shift strategies the QZ algorithm normally has a cubic local convergence rate. Our main focus is on the modified shift strategies and their corresponding truncated versions. Numerical examples are provided to demonstrate the convergence properties and the efficiency of the QZ algorithm equipped with the proposed shifts. For the truncated versions, local convergence analysis is not provided. Numerical examples show they outperform the modified shifts and the standard Rayleigh quotient single‐shift and Francis double‐shift. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

For the symmetric Pareto Eigenvalue Complementarity Problem (EiCP), by reformulating it as a constrained optimization problem on a differentiable Rayleigh quotient function, we present a class of descent methods and prove their convergence. The main features include: using nonlinear complementarity functions (NCP functions) and Rayleigh quotient gradient as the descent direction, and determining the step size with exact linear search. In addition, these algorithms are further extended to solve the Generalized Eigenvalue Complementarity Problem (GEiCP) derived from unilateral friction elastic systems. Numerical experiments show the efficiency of the proposed methods compared to the projected steepest descent method with less CPU time. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

A variational formulation is presented for a system consisting of two coupled singlewalled carbon nanotubes subjected to forced vibrations, longitudinal magnetic field, and axial compression, based on the nonlocal elasticity theory. The variational principle for the double nanotube system is derived, followed by the application of Hamilton's principle to express kinetic and potential energies. Subsequently, the time-independent scenario is investigated, and governing equations for the freely vibrating system are provided. The variational formulation for this case is established, and expressions for Rayleigh quotients concerning the vibration frequency and buckling load are derived. The Rayleigh quotient for the frequency demonstrates that the magnetic field increases the vibration frequency of the coupled nanotube system. Nonlocal effects appear in both the numerator and the denominator of the Rayleigh quotient, influencing the frequency increase or decrease depending on the relative values of various problem parameters. In contrast, the magnetic field reduces the buckling load, as evidenced by its negative contribution to the numerator of the Rayleigh quotient for buckling. The effect of the nonlocal parameter on buckling, however, cannot be inferred directly from the Rayleigh quotient. In this study, involving a system of two coupled partial differential equations, it is crucial to derive variationally consistent boundary conditions. Utilizing the formulated variational principle, variationally consistent natural boundary conditions are established in terms of moment and shear force expressions. It is revealed that the Pasternak interlayer between the nanotubes results in coupled boundary conditions when a shear force and/or a moment is specified at the boundaries of the nanotube system. [ABSTRACT FROM AUTHOR]

A minimax variational method for finding mountain pass-type solutions with prescribed energy levels is introduced. The method is based on application of the Linking Theorem to the energy-level nonlinear Rayleigh quotients which critical points correspond to the solutions of the equation with prescribed energy. An application of the method to nonlinear indefinite elliptic problems with nonlinearities that does not satisfy the Ambrosetti-Rabinowitz growth conditions is also presented. [ABSTRACT FROM AUTHOR]

The accuracy of the optimally blended spectral-element method for wave propagation in a homogeneous transversely isotropic elastic medium was investigated. A nonstandard quadrature rule obtained by combining Gauss quadrature rule and Gauss–Lobatto–Legendre quadrature rule was used to compute elementary matrixes. The mass and stiffness matrixes were represented as the triple tensor-product of elementary matrixes as second-order tensors. The solution of the eigenvalue problem representing the semidiscretized version of elastic wave equation for plane harmonic wave propagation was obtained using the Rayleigh quotient approximation technique. The resulting eigenvalues subsequently were used to estimate the phase and group velocity of three bulk waves. The variations of the errors in these velocities of the bulk waves with the number of grid points per wavelength were depicted graphically. These variations were shown for different polynomial orders and various angles of deviation from the symmetry axis. The effect of a change in the order of time discretization on error variation also was shown. The solution obtained by the optimally blended spectral-element method was found to be more accurate than that from the classical spectral-element method for low-order polynomials. The improvement in solution accuracy was demonstrated through dispersion analysis. [ABSTRACT FROM AUTHOR]

This study investigated the buckling behavior of Porous Functionally Graded Material (PFGM) plates. The present model assumes unevenly distributed porosity along the plate thickness and the use of the novel hyperbolic shear deformation functions and hyperbolic tangent and secant thickness stretching functions. In the present work, a porous Functionally Graded (FG) plate was analyzed by the principle of virtual work in order to understand the buckling behavior under uniaxial and biaxial compressive loading. The Rayleigh quotient method was applied to find the critical buckling load. The mesh convergence was investigated on a Finite Element (FE) model, and the accuracy of the results was compared with the prior research. The results of the proposed model match reasonably well with the ones of the published literature. Thorough parametric studies were performed to investigate the effect of porosity on the critical buckling load of the PFGM plate. [ABSTRACT FROM AUTHOR]

We study the use of inverse harmonic Rayleigh quotients with target for the stepsize selection in gradient methods for nonlinear unconstrained optimization problems. This not only provides an elegant and flexible framework to parametrize and reinterpret existing stepsize schemes, but it also gives inspiration for new flexible and tunable families of steplengths. In particular, we analyze and extend the adaptive Barzilai–Borwein method to a new family of stepsizes. While this family exploits negative values for the target, we also consider positive targets. We present a convergence analysis for quadratic problems extending results by Dai and Liao (IMA J Numer Anal 22(1):1–10, 2002), and carry out experiments outlining the potential of the approaches. [ABSTRACT FROM AUTHOR]

Purpose: This 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/approach: A 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. Findings: Numerical 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/value: The 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. [ABSTRACT FROM AUTHOR]

Data-driven soft sensor technology has been widely used in process monitoring, quality prediction, etc. However, there are dynamic time-varying and concept drift problems in industrial processes, making the accurate modeling of the soft sensor is still a challenging task. To solve the above problems, this paper proposes a concept drift adaptive dynamic partial least squares method. The method maps high-dimensional process data into a low-dimensional latent variable subspace. Dynamic information is introduced by establishing the dynamic regression relationship between process latent variables and quality latent variables. At the same time, transfer learning is used to align second-order statistics between latent variables of data from different distributions to solve the concept drift problem. The experiments on multiple industrial datasets show that the method proposed in this paper can effectively reduce prediction errors and improve the generalization ability of the model. [Display omitted] • Distribution alignment and dynamic system modeling are realized under NIPALS. • The objective function is organized into the form of Rayleigh quotient. • There are few hyperparameters and they have better interpretability. • Experiments show that our method can adapt to the change of data distribution well. [ABSTRACT FROM AUTHOR]

The local stability of thin‐walled fibre‐reinforced plastic composite beams can be reduced to individual laminates using discrete plate theory. These individual plates receive a supporting effect from their surrounding structure, which is modelled with rotational restraints. In the present investigation, this buckling problem is described by a closed‐form solution. The energy‐based method works with the Rayleigh quotient and the principle of the stationary value of the elastic potential energy. For the analysis of unsymmetrically laminated plates, the classical laminated plate theory (CLPT) considers both the plate deflection and the in‐plane displacements. The first‐order shear deformation theory (FSDT) and third‐order shear deformation theory (TSDT) additionally describe the cross‐sectional rotations and thus take transverse shear deformations into account. In addition to the direct consideration of the bending‐extension couplings, these have also been investigated using the reduced bending stiffness (RBS) method. The investigation shows the influence of bending‐extension coupling on the stability of compressively loaded unsymmetrically laminated plates. Moreover, it is found that the transverse shear stiffness reduces the critical load at relatively high plate thicknesses. The closed‐form analytical solution and the RBS method show good agreement with finite element analyses. The presented closed‐form analytical methods provide explicit solutions for the critical compressive load of unsymmetric laminates under different boundary conditions. Due to the explicit solution, this method is significantly more computationally efficient than numerical, semi‐analytical or exact methods. The present methods are characterised by a simple applicability as well as a very high computational efficiency and are very suitable for preliminary design as well as optimisation of laminated structures. [ABSTRACT FROM AUTHOR]