Smallest Positive Eigenvalue Research Articles

Bvp4c approach and duality of hybrid nanofluid over extending and contracting sheet with chemical reaction and cross-diffusion effects

Heat transfer has a major effect on material selection and mechanical efficiency. The importance of peristaltic fluid motion in transmitting heat is obvious in the domains of biomedical research through the processes of metabolic heat generation, blood transportation, capillary dehydration, thermal control, and bio-heat exchange pathways. The present work aims to explore the computational and theoretical evaluation of hybrid nanofluid (HNF)-suspended Cu and Al2O3 nanoparticles in water in the presence of chemical reaction, Lorentz force and ross-diffusion. The fundamental flow system based on Navier-Stokes in terms of partial differential equations is converted to dimensionless nonlinear ordinary differential equations via similarity transformations. The transformed ODEs are computationally solved by bvp4c approach. Dual solutions have been observed for emerging factors, so stability examinations are implemented to find the stable solution. Based on eigenvalues, it is witnessed that smallest positive eigenvalues indicates the stable while negative depicts the unstable solutions. The influences of involved factors on the flow characteristics are depicted through graphs and tables. The confirmation of the present analysis is done with the published study. Decreasing behavior is investigated for both branches of velocity as the M is improved while f′(η) is an increasing function of K. From this analysis it is reported that θ(η) is directly proportional to R, Q and Q1. Increasing phenomena is reported for θη and ϕη profiles as the quantities of Ec are augmented from 0.1 to 1. The conclusions of this theoretical analysis have impending applications in numerous fields such as technological devices, solar cooling and heating systems for cars, surfactants, lubricating qualities, hybrid generators, metallic welding process, and the production of medical equipment.

Stability analysis of radiative-magnetic hybrid nanofluid slip flow due to an exponentially stretching/shrinking permeable sheet with heat generation

The present article aims to find the duality of solutions and investigate the magneto-radiative hybrid nanofluid flow over an exponentially stretching/shrinking permeable sheet with heat generation and multiple slip boundary conditions. The governing nonlinear partial differential equations are transformed into dimensionless nonlinear ordinary differential equations which are computed by bvp4c in Matlab platform. The hybridisation of alumina and copper nanoparticles enhances the system's heat transfer performance. The dual nature solution is obtained only if the suction is applied to the stretching/shrinking surface. From stability analysis, we can conclude that the first solution is stable, but the other is unstable. It is realistic to have the solution stable with the smallest positive eigenvalues, while the lower solution branch is unstable with the smallest negative eigenvalues. It is found that the upsurges of solid volume fraction of the copper nanoparticle enhanced the skin friction coefficient and the local Nusselt number.

Regularity-based spectral clustering and mapping the Fiedler-carpet

Abstract We discuss spectral clustering from a variety of perspectives that include extending techniques to rectangular arrays, considering the problem of discrepancy minimization, and applying the methods to directed graphs. Near-optimal clusters can be obtained by singular value decomposition together with the weighted k k -means algorithm. In the case of rectangular arrays, this means enhancing the method of correspondence analysis with clustering, while in the case of edge-weighted graphs, a normalized Laplacian-based clustering. In the latter case, it is proved that a spectral gap between the ( k − 1 ) \left(k-1) st and k k th smallest positive eigenvalues of the normalized Laplacian matrix gives rise to a sudden decrease of the inner cluster variances when the number of clusters of the vertex representatives is 2 k − 1 {2}^{k-1} , but only the first k − 1 k-1 eigenvectors are used in the representation. The ensemble of these eigenvectors constitute the so-called Fiedler-carpet.

A conjecture on the eigenvalues of threshold graphs

Let An be the connected anti-regular graph of order n. It was conjectured that among all threshold graphs on n vertices, An has the smallest positive eigenvalue and the largest eigenvalue less than −1. Recently, in [1] was given partial results for this conjecture and identified the critical cases where a more refined method is needed. In this paper, we deal with these cases and confirm that conjecture holds.

The smallest positive eigenvalue of graphs under perturbation

The effect on the smallest positive eigenvalue of a bipartite graph is studied when the graph is perturbed by attaching a pendant vertex at one of its vertices. Let $${\widehat{T}}(v)$$ be the graph obtained by attaching a pendant at vertex v of T. We characterize the vertices v such that the smallest positive eigenvalue of $${\widehat{T}}(v)$$ is equal or greater than that of T. As an application, we obtain the pairs of nonisomorphic noncospectral trees having the same smallest positive eigenvalue.

Rayleigh-Ritz Majorization Error Bounds for the Linear Response Eigenvalue Problem

Abstract In the linear response eigenvalue problem arising from computational quantum chemistry and physics, one needs to compute a few of smallest positive eigenvalues together with the corresponding eigenvectors. For such a task, most of efficient algorithms are based on an important notion that is the so-called pair of deflating subspaces. If a pair of deflating subspaces is at hand, the computed approximated eigenvalues are partial eigenvalues of the linear response eigenvalue problem. In the case the pair of deflating subspaces is not available, only approximate one, in a recent paper [SIAM J. Matrix Anal. Appl., 35(2), pp.765-782, 2014], Zhang, Xue and Li obtained the relationships between the accuracy in eigenvalue approximations and the distances from the exact deflating subspaces to their approximate ones. In this paper, we establish majorization type results for these relationships. From our majorization results, various bounds are readily available to estimate how accurate the approximate eigenvalues based on information on the approximate accuracy of a pair of approximate deflating subspaces. These results will provide theoretical foundations for assessing the relative performance of certain iterative methods in the linear response eigenvalue problem.

Uncertainty principle and geometry of the infinite Grassmann manifold

We study the pairs of projections $$ P_If=\chi_If Q_Jf= \left(\chi_J \hat{f}\right)\check{ } , f\in L^2(\mathbb{R}^n), $$ where $I, J\subset \mathbb{R}^n$ are sets of finite Lebesgue measure, $\chi_I, \chi_J$ denote the corresponding characteristic functions and $\hat{ } , \check{ }$ denote the Fourier-Plancherel transformation $L^2(\mathbb{R}^n)\to L^2(\mathbb{R}^n)$ and its inverse. These pairs of projections have been widely studied by several authors in connection with the mathematical formulation of Heisenberg's uncertainty principle. Our study is done from a differential geometric point of view. We apply known results on the Finsler geometry of the Grassmann manifold ${\cal P}({\cal H})$ of a Hilbert space ${\cal H}$ to establish that there exists a unique minimal geodesic of ${\cal P}({\cal H})$, which is a curve of the form $$ \delta(t)=e^{itX_{I,J}}P_Ie^{-itX_{I,J}} $$ which joins $P_I$ and $Q_J$ and has length $\pi/2$. As a consequence we obtain that if $H$ is the logarithm of the Fourier-Plancherel map, then $$ \|[H,P_I]\|\ge \pi/2. $$ The spectrum of $X_{I,J}$ is denumerable and symmetric with respect to the origin, it has a smallest positive eigenvalue $\gamma(X_{I,J})$ which satisfies $$ \cos(\gamma(X_{I,J}))=\|P_IQ_J\|. $$

Correction to: Finite Element Approximation of the Spectrum of the Curl Operator in a Multiply Connected Domain

In the published article, Figure 5 corresponds to an eigenfunction associated not with the first smallest positive eigenvalue.

MARKOV SPECTRA OF SELF-SIMILAR NETWORKS BY SUBSTITUTION RULE

For a class of self-similar networks generated by substitution rules, we investigate them in terms of normalized Laplacian spectra. Accordingly, we obtain the recurrent structure of Markov spectra for these self-similar networks, and also estimate the smallest positive eigenvalue for Laplace operator.

A projected preconditioned conjugate gradient method for the linear response eigenvalue problem

The linear response eigenvalue problem aims at computing a few smallest positive eigenvalues together with the associated eigenvectors of a special Hamiltonian matrix and plays an important role for estimating the excited states of physical systems. A subspace version of the Thouless minimization principle was established by Bai and Li (SIAM J. Matrix Anal. Appl., 33:1075-1100, 2012) which characterizes the desired eigenpairs as its solution. In this paper, we propose a Projected Preconditioned Conjugate Gradient ( \begin{document}$\texttt{PPCG_lrep}$\end{document} ) method to solve this subspace version of Thouless's minimization directly. We show that \begin{document}$\texttt{PPCG_lrep}$\end{document} is an efficient implementation of the inverse power iteration and can be performed in parallel. It also enjoys several properties including the monotonicity and constraint preservation in the Thouless minimization principle. Convergence of both eigenvalues and eigenvectors are established and numerical experiences on various problems are reported.

An integro-differential inequality related to the smallest positive eigenvalue of p(x)-Laplacian Dirichlet problem

AbstractWe consider the eigenvalue problem for the p(x)-Laplace-Beltrami operator on the unit sphere. We prove same integro-differential inequalities related to the smallest positive eigenvalue of this problem.

Eigenvalue estimates for the Laplacian on a metric tree

We provide explicit upper bounds for the eigenvalues of the Laplacian on a finite metric tree subject to standard vertex conditions. The results include estimates depending on the average length of the edges or the diameter. In particular, we establish a sharp upper bound for the spectral gap, i.e. the smallest positive eigenvalue, and show that equilateral star graphs are the unique maximizers of the spectral gap among all trees of a given average length.

A block Chebyshev-Davidson method for linear response eigenvalue problems

We present a Chebyshev-Davidson method to compute a few smallest positive eigenvalues and corresponding eigenvectors of linear response eigenvalue problems. The method is applicable to more general linear response eigenvalue problems where some purely imaginary eigenvalues may exist. For the Chebyshev filter, a tight upper bound is obtained by a computable bound estimator that is provably correct under a reasonable condition. When the condition fails, the estimated upper bound may not be a true one. To overcome that, we develop an adaptive strategy for updating the estimated upper bound to guarantee the effectiveness of our new Chebyshev-Davidson method. We also obtain an estimate of the rate of convergence for the Ritz values by our algorithm. Finally, we present numerical results to demonstrate the performance of the proposed Chebyshev-Davidson method.

The eigenvalue problem and Saint-Venant decay rate for a nonhomogeneous semi-infinite strip

The eigenvalue problem about a nonhomogeneous semi-infinite strip is investigated using the methodology proposed by Papkovich and Fadle for homogeneous plane problems. Two types of nonhomogeneity are considered: (i) the elastic modulus varying with the thickness coordinate x exponentially, (ii) it varying with the length coordinate y exponentially. The eigenvalues for the two cases are obtained numerically in plane strain and plane stress states, respectively. By considering the smallest positive eigenvalue, the Saint-Venant Decay rates are estimated, which indicates material nonhomogeneity has a significant influence on the Saint-Venant end effect.

A Novel Symmetric Skew-Hamiltonian Isotropic Lanczos Algorithm for Spectral Conformal Parameterizations

In the past decades, many methods for computing conformal mesh parameterizations have been developed in response to demand of numerous applications in the field of geometry processing. Spectral conformal parameterization (SCP) (Mullen et al. in Proceedings of the symposium on geometry processing, SGP '08. Eurographics Association, Aire-la-Ville, Switzerland, pp 1487---1494, 2008) is one of these methods used to compute a quality conformal parameterization based on the spectral techniques. SCP focuses on a generalized eigenvalue problem (GEP) $$L_{C}{\mathbf {f}} = \lambda B{\mathbf {f}}$$ L C f = ? B f whose eigenvector(s) associated with the smallest positive eigenvalue(s) provide the conformal parameterization result. This paper is devoted to studying a novel eigensolver for this GEP. Based on structures of the matrix pair $$(L_{C},B)$$ ( L C , B ) , we show that this GEP can be transformed into a small-scale compressed and deflated standard eigenvalue problem with a symmetric positive definite skew-Hamiltonian operator. We then propose a symmetric skew-Hamiltonian isotropic Lanczos algorithm ( $${\mathbb {S}}$$ S HILA) to solve the reduced problem. Numerical experiments show that our compressed deflating technique can exclude the impact of convergence from the kernel of $$L_{C}$$ L C and transform the original problem to a more robust system. The novel $${\mathbb {S}}$$ S HILA method can effectively avoid the disturbance of duplicate eigenvalues. As a result, based on the spectral model of SCP, our numerical eigensolver can compute the conformal parameterization accurately and efficiently.

Rayleigh--Ritz Approximation For the Linear Response Eigenvalue Problem

Large scale eigenvalue computation is about approximating certain invariant subspaces associated with the interesting part of the spectrum, and the interesting eigenvalues are then extracted through projecting the problem through approximate invariant subspaces into a much smaller eigenvalue problem. In the case of the linear response eigenvalue problem (aka the random phase eigenvalue problem), it is the pair of deflating subspaces associated with the first few smallest positive eigenvalues that needs to be computed. This paper is concerned with approximation accuracy relationships between a pair of approximate deflating subspaces and approximate eigenvalues extracted by the pair. Lower and upper bounds on eigenvalue approximation errors are obtained in terms of canonical angles between an exact and computed pair of deflating subspaces. These bounds can also be interpreted as lower/upper bounds on the canonical angles in terms of eigenvalue approximation errors. They are useful in analyzing numerical sol...

The Eigenvalue Problem and Saint-Venant Decay Rate for a Nonhomogeneous Semi-Infinite Strip in Plane Strain

The eigenvalue problem referring to a nonhomogeneous semi-infinite strip in plane strain is investigated here, by using the analogous methodology proposed by Papkovich and Fadle in homogeneous plane problem. Two types of nonhomogeneity are considered: (i) the modulus varies with the thickness coordinate exponentially, (ii) it varies with the length coordinate exponentially. The eigenvalues for these two cases are obtained by numerical calculation. By considering the smallest positive eigenvalue, the Saint-Venant Decay rate of the problem is estimated, which indicates material nonhomogeneity can have a significant influence on the Saint-Venant decay rate.

Stability of Impulsive Cohen-Grossberg Neural Networks with Time-Varying Delays and Reaction-Diffusion Terms

This work concerns the stability of impulsive Cohen-Grossberg neural networks with time-varying delays and reaction-diffusion terms as well as Dirichlet boundary condition. By means of Poincaré inequality and Gronwall-Bellman-type impulsive integral inequality, we summarize some new and concise sufficient conditions ensuring the global exponential stability of equilibrium point. The proposed criteria are relevant to the diffusion coefficients and the smallest positive eigenvalue of corresponding Dirichlet Laplacian. In conclusion, two examples are illustrated to demonstrate the effectiveness of our obtained results.

Multiple positive solutions for discrete p-Laplacian equations with potential term

We study the existence of solutions to nonlinear discrete boundary value problems with the discrete p-Laplacian, potential, and nonlinear source terms. Using variational methods, we demonstrate that there exist at least two positive solutions. The existence strongly depends on the smallest positive eigenvalue of Dirichlet eigenvalue problems and the growth conditions of the source terms.

Computing Extremal Eigenvalues for Three-Dimensional Photonic Crystals with Wave Vectors Near the Brillouin Zone Center

The band structures of three-dimensional photonic crystals can be determined numerically by solving a sequence of generalized eigenvalue problems. However, not all of the spectral structures of these eigenvalue problems are well-understood, and not all of these eigenvalue problems can be solved efficiently. This article focuses on the eigenvalue problems corresponding to wave vectors that are close to the center of the Brillouin zone of a three dimensional, simple cubic photonic crystal. For these eigenvalue problems, there are (i) many zero eigenvalues, (ii) a couple of near-zero eigenvalues, and (iii) several larger eigenvalues. As the desired eigenvalues are the smallest positive eigenvalues, these particular spectral structures prevent regular eigenvalue solvers from efficiently computing the desired eigenvalues. We study these eigenvalue problems from the perspective of both theory and computation. On the theoretical side, the structures of the null spaces are analyzed to explicitly determine the number of zero eigenvalues of the target eigenvalue problems. On the computational side, the Krylov-Schur and Jacobi-Davidson methods are used to compute the smallest, positive, interior eigenvalues that are of interest. Intensive numerical experiments disclose how the shift values, conditioning numbers, and initial vectors affect the performance of the tested eigenvalue solvers and suggest the most efficient eigenvalue solvers.