Minimum Eigenvalue Research Articles

Distance-2 MDS Codes and Latin Colorings in the Doob Graphs

The maximum independent sets in the Doob graphs D(m,n) are analogs of the distance-2 MDS codes in Hamming graphs and of the latin hypercubes. We prove the characterization of these sets stating that every such set is semilinear or reducible. As related objects, we study vertex sets with maximum cut (edge boundary) in D(m,n) and prove some facts on their structure. We show that the considered two classes (the maximum independent sets and the maximum-cut sets) can be defined as classes of completely regular sets with specified 2-by-2 quotient matrices. It is notable that for a set from the considered classes, the eigenvalues of the quotient matrix are the maximum and the minimum eigenvalues of the graph. For D(m,0), we show the existence of a third, intermediate, class of completely regular sets with the same property.

Uniform bounds for Weil–Petersson curvatures

We find bounds for Weil–Petersson holomorphic sectional curvature, and the Weil–Petersson curvature operator in several regimes, that do not depend on the topology of the underlying surface. Among other results, we show that the minimal Weil–Petersson holomorphic sectional curvature of a sufficiently thick hyperbolic surface is comparable to − 1 , independently of the genus. This provides a counterexample to some suggestions that the Weil–Petersson metric becomes asymptotically flat, as the genus g goes to infinity, in the thick loci in Teichmüller space. Adopting a different perspective on curvature, we also show that the minimal (most negative) eigenvalue of the curvature operator at any point in the Teichmüller space Teich ( S g ) of a closed surface S g of genus g is uniformly bounded away from zero. Restricting to a thick part of Teich ( S g ) , we show that the minimal eigenvalue is uniformly bounded below by an explicit constant which does not depend on the topology of the surface but only on the given bound on injectivity radius.

Homotopy analysis method for predicting multiple solutions in the channel flow with stability analysis

The present theoretical investigation covers a study of finding the multiple solutions of a Jeffery–Hamel flow and heat transfer under the influence of magnetic field. An analytical technique (Homotopy Analysis Method) is applied to solve the cylindrical form of governing equations after getting non-dimensional system using suitable transformation. It is noticed that dual solutions exist only for convergent channel case. Critical values of channel angle (αc) are obtained which reveals the existence domain (αc < α < 0) for multiple solutions in convergent channel flow. Stability analysis is also performed by constructing eigenvalue problem to predict the physically stable solution. The minimum positive eigenvalue justifies the fact that the growth of disturbances given to the solution decays with time. The effect of various pertinent physical parameters is shown graphically on velocity, temperature, skin friction coefficient and Nusselt number.

Entangling and disentangling many-electron quantum systems with an electric field

We show that the electron correlation of a molecular system can be enhanced or diminished through the application of a homogeneous electric field antiparallel or parallel to the system's intrinsic dipole moment. More generally, we prove that any external stimulus that significantly changes the expectation value of a one-electron operator with nondegenerate minimum and maximum eigenvalues can be used to control the degree of a molecule's electron correlation. Computationally, the effect is demonstrated in ${\mathrm{HeH}}^{+}, {\mathrm{MgH}}^{+}$, BH, HCN, ${\mathrm{H}}_{2}\mathrm{O}$, HF, formaldehyde, and a fluorescent dye. Furthermore, we show in calculations with an array of formaldehyde (${\mathrm{CH}}_{2}\mathrm{O}$) molecules that the field can control not only the electron correlation of a single formaldehyde molecule but also the entanglement among formaldehyde molecules. The quantum control of correlation and entanglement has potential applications in the design of molecules with tunable properties and the stabilization of qubits in quantum computations.

Optimization of some eigenvalue problems with large drift

This paper is concerned with eigenvalue problems for elliptic operators with large drifts in bounded domains under Dirichlet boundary conditions. We consider the minimal principal eigenvalue and the related principal eigenfunction in the class of drifts having a given, but large, pointwise upper bound. We show that, in the asymptotic limit of large drifts, the maximal points of the optimal principal eigenfunctions converge to the set of points maximizing the distance to the boundary of the domain. We also show the uniform asymptotic profile of these principal eigenfunctions and the direction of their gradients in neighborhoods of the boundary.

The Spectral Gaps of Generalized Flag Complexes and a Geometric Hall-type Theorem

Abstract Let $X$ be a simplicial complex on $n$ vertices without missing faces of dimension larger than $d$. Let $L_{k}$ denote the $k$-Laplacian acting on real $k$-cochains of $X$ and let $\mu _{k}(X)$ denote its minimal eigenvalue. We study the connection between the spectral gaps $\mu _{k}(X)$ for $k\geq d$ and $\mu _{d-1}(X)$. In particular, we establish the following vanishing result: if $\mu _{d-1}(X)&amp;gt;\big(1-\binom{k+1}{d}^{-1}\big)n$, then $\tilde{H}^{j}\left (X;{\mathbb{R}}\right )=0$ for all $d-1\leq j \leq k$. As an application we prove a fractional extension of a Hall-type theorem of Holmsen, Martínez-Sandoval, and Montejano for general position sets in matroids.

The method of perpendiculars of finding estimates from below for minimal singular eigenvalues of random matrices

Abstract The lower bounds for the minimal singular eigenvalue of the matrix are obtained under the G-Lindeberg condition and the G-double stochastic condition for the variances of the matrix entries. The new method is based on the G-method of perpendiculars, the REFORM method, the martingale method, and the theory of canonical spectral equations.

The elastic wave propagation through the finite and infinite periodic laminated structure of micropolar elasticity

The reflection and transmission coefficients through multilayered structure of finite thickness and the dispersive relations of Bloch waves in one-dimension infinite periodic structure (phononic crystals) consisting of two repetitive different micropolar slabs are studied. Due to the additional microrotational degree of freedom, the modes of elastic waves in a micropolar solid are different from that in the traditional elastic medium. The transfer matrix and the stiffness matrix of each slab and further one single cell are obtained for both in-plane and anti-plane situations. The dispersive equations are derived by using Bloch theorem. The relation between the minimum eigenvalue of transfer matrix of single cell and the stop bands are presented. The reflection and transmission coefficients through finite multilayered structure are also calculated based on the stiffness matrix method to avoid the numerical instability of the transfer matrix method. The energy fluxes carried by the reflection and transmission waves are estimated and the energy conservation is checked to validate the numerical results. Based on the numerical results, the influence of micropolar parameters on the dispersive feature of Bloch waves are discussed. It is found that the dispersive feature and the bandgap of Bloch waves in the micropolar phononic crystal are evidently different from that in the classic phononic crystal. The dispersive feature and bandgap of Bloch wave at high frequency are more sensitive to the micropolar constants than at low frequency range.

Holomorphic Hermite Functions in Segal–Bargmann Spaces

We study systems of holomorphic Hermite functions in the Segal–Bargmann spaces, which are Hilbert spaces of entire functions on the complex Euclidean space, and are determined by the Bargmann-type integral transform on the real Euclidean space. We prove that for any positive parameter which is strictly smaller than the minimum eigenvalue of the positive Hermitian matrix associated with the transform, one can find a generator of holomorphic Hermite functions whose annihilation and creation operators satisfy canonical commutation relations. In other words, we find the necessary and sufficient conditions so that some kinds of entire functions can be such generators. Moreover, we also study the complete orthogonality, the eigenvalue problems and the Rodrigues formulas.

The Least Eigenvalue of the Graphs Whose Complements Are Connected and Have Pendent Paths

AbstractThe adjacency matrix of a graph is a matrix which represents adjacent relation between the vertices of the graph. Its minimum eigenvalue is defined as the least eigenvalue of the graph. Let Gnbe the set of the graphs of order n, whose complements are connected and have pendent paths. This paper investigates the least eigenvalue of the graphs and characterizes the unique graph which has the minimum least eigenvalue in Gn.

Adaptive multi objective differential evolution with fuzzy decision making in preventive and corrective control approaches for voltage security enhancement

This paper presents a multi objective differential evolution (MODE) based voltage security enhancement through combined preventive-corrective control strategy. Load shedding, generation rescheduling and optimal utilization of FACTS devices are considered for security enhancement. Maximum l-index of load buses is taken as the indicator of voltage stability. Minimization of cost of FACTS devices, minimization of amount of load shedding along with improvement in voltage stability are the objectives of this multi objective optimization problem. The optimal location of FACTS devices are selected using modal analysis technique. The buses for load shedding are selected based on the minimum eigen value of load flow Jacobian. The proposed MODE algorithm employs DE/randSF/1/bin strategy scheme with self tuned parameter which employs binomial crossover and difference vector based mutation. A fuzzy based decision making algorithm is employed to get the best compromise solution from the non dominated solutions. The proposed MODE is also tested with statistical performance metrices. The proposed methodology is implemented on IEEE 30 bus and IEEE 57 bus test systems. The proposed MODE method provides better solutions in the pareto optimal front than the other optimization techniques such as MOGA and NSGA II under combined preventive–corrective control approach. In IEEE 30 bus system, the amount of load shedding is reduced by 40% and voltage stability is improved by 15% and in IEEE 57 bus system, the amount of load shedding is reduced by 15.4% and voltage stability is improved by 13% by the proposed approach. Hence the simulation results show that the proposed approach provides considerable reduction in the amount of load shedding and enhancement of voltage stability by including generation rescheduling and utilization of FACTS devices.

Local buckling of composite channel columns

The investigation concerns local buckling of compressed flanges of axially compressed composite channel columns. Cooperation of the member flange and web is taken into account here. The buckling mode of the member flange is defined by rotation angle a flange about the line of its connection with the web. The channel column under investigation is made of unidirectional fibre-reinforced laminate. Two approaches to member orthotropic material modelling are performed: the homogenization with the aid of theory of mixture and periodicity cell or homogenization upon the Voigt–Reuss hypothesis. The fundamental differential equation of local buckling is derived with the aid of the stationary total potential energy principle. The critical buckling stress corresponding to a number of buckling half-waves is assumed to be a minimum eigenvalue of the equation. Some numerical examples dealing with columns are given here. The analytical results are compared with the finite element stability analysis carried out by means of ABAQUS software. The paper is focused on a close analytical solution of the critical buckling stress and the associated buckling mode while the web–flange cooperation is assumed.

Evaluation of Local Matching Methods in Image Analysis for Mineral Grain Tracking in Microscope Images of Rock Sections

Modern geological techniques have resulted in vast and growing databases of digital images and video sequences of rocks, which are available for the use of researchers. The number of database images continues to increase exponentially, creating a need for techniques that will enable the automation of data set management. Desired techniques include query by image, a topic that has been extensively elaborated on in the literature recently. Unfortunately, using such techniques in the geological sciences has been very sporadic and insufficient. This paper presents the evaluation of characteristic local features within rock images for tracking objects on images or video sequences. It also discusses the possibilities for using selected local feature descriptors for content-based image retrieval (CBIR) in the area of geological sciences. The evaluation was performed for the Speeded Up Robust Features (SURF), Binary Robust Invariant Scalable Keypoints (BRISK), Harris–Stephens Algorithm (HSA), Minimum Eigenvalue Algorithm (MEA), and Features from Accelerated Segment Test algorithm (FAST) methods, which are widely known and appreciated in the computer vision field. These methods were analysed for their application to microscopic images of rocks. Five functional cases of geological grain tracking were investigated, based on a selected non-transformed query image, as well as a computer-rotated, acquisitive-rotated, computer-magnified, and an acquisitive-magnified query image. The results demonstrated that these methods can be successfully used for geological applications.

Structural physical approximation for the realization of the optimal singlet fraction with two measurements

Structural physical approximation (SPA) has been exploited to approximate non-physical operation such as partial transpose. It has already been studied in the context of detection of entanglement and found that if the minimum eigenvalue of SPA to partial transpose is less than $\frac{2}{9}$ then the two-qubit state is entangled. We find application of SPA to partial transpose in the estimation of optimal singlet fraction. We show that optimal singlet fraction can be expressed in terms of minimum eigenvalue of SPA to partial transpose. We also show that optimal singlet fraction can be realized using Hong-Ou-Mandel interferometry with only two detectors. Further we have shown that the generated hybrid entangled state between a qubit and a binary coherent state can be used as a resource state in quantum teleportation.

Minimum norm partial quadratic eigenvalue assignment for vibrating structures using receptances and system matrices

Abstract This paper is concerned with the minimum norm partial quadratic eigenvalue assignment problem (MNPQEAP) for vibrating structures via active feedback control. We propose a gradient-based optimization method for solving the MNPQEAP using the receptance measurements, system matrices and a few undesired open-loop eigenvalues with associated eigenvectors. The real form of our method is also derived. Numerical tests show that the proposed methods are effective for solving the MNPQEAP with multiple-input vibration controls.

Stabilization of Kelvin–Voigt viscoelastic fluid flow model

ABSTRACTIn this article, stabilization result for the viscoelastic fluid flow problem is governed by Kelvin–Voigt model, that is, convergence of the unsteady solution to a steady state solution is proved under the assumption that linearized self-adjoint steady state eigenvalue problem has a minimal positive eigenvalue. Both power and exponential convergence results are derived under various conditions on the forcing function. It is shown that results are valid uniformly in the time relaxation or some times called regularization parameter as , which in turn, establishes results for the Navier–Stokes system.

The least eigenvalue of graphs whose complements have only two pendent vertices

Let G be a simple graph and A(G) be the adjacency matrix of G. The eigenvalues of A(G) are referred to as the eigenvalues of G. In this paper, we characterize the graphs with the minimal least eigenvalue among all graphs whose complements are connected and have only two pendent vertices.

Tension cable distribution of a membrane antenna frame based on stiffness analysis of the equivalent 4-SPS-S parallel mechanism

Stiffness has significant influence on the surface accuracy of reflector of membrane antennas. This paper proposes a method to analyze the stiffness of a membrane antenna frame supported by compliant booms and tensioned with cables. An equivalent 4-SPS-S parallel mechanism of the membrane antenna frame is established. Velocity Jacobian matrix is obtained through kinematic analysis. Stiffness of the actuating SPS branches and the constrained central strut is constructed. Based on the force-equilibrium equation and the principle of virtual work, total stiffness matrix of the mechanism, which has been decomposed into the SPS branches stiffness matrix and the central strut stiffness matrix, is derived. Influences of cable distribution and platform pose on stiffness are identified by using the stiffness index – the reflector surface accuracy and the minimal eigenvalue. Accuracy of the proposed approach is evaluated. On account of stiffness, numerical simulations have been performed for acquiring the optimal distribution of tension cables and pose of the platform in the design of antennas.

The conditioning of least‐squares problems in variational data assimilation

SummaryIn variational data assimilation a least‐squares objective function is minimised to obtain the most likely state of a dynamical system. This objective function combines observation and prior (or background) data weighted by their respective error statistics. In numerical weather prediction, data assimilation is used to estimate the current atmospheric state, which then serves as an initial condition for a forecast. New developments in the treatment of observation uncertainties have recently been shown to cause convergence problems for this least‐squares minimisation. This is important for operational numerical weather prediction centres due to the time constraints of producing regular forecasts. The condition number of the Hessian of the objective function can be used as a proxy to investigate the speed of convergence of the least‐squares minimisation. In this paper we develop novel theoretical bounds on the condition number of the Hessian. These new bounds depend on the minimum eigenvalue of the observation error covariance matrix and the ratio of background error variance to observation error variance. Numerical tests in a linear setting show that the location of observation measurements has an important effect on the condition number of the Hessian. We identify that the conditioning of the problem is related to the complex interactions between observation error covariance and background error covariance matrices. Increased understanding of the role of each constituent matrix in the conditioning of the Hessian will prove useful for informing the choice of correlated observation error covariance matrix and observation location, particularly for practical applications.

Genetic and hybrid algorithms for optimization of non-singular 3PRR planar parallel kinematics mechanism for machining application

SUMMARYThis paper proposes a special non-symmetric topology of a 3PRR planar parallel kinematics mechanism, which naturally avoids singularity within the workspace and can be utilized for hybrid kinematics machine tools. Subsequently, single-objective and multi-objective optimizations are conducted to improve the performance. The workspace area and minimum eigenvalue, as well as the condition number of the homogenized Cartesian stiffness matrix across the workspace, have been chosen as the objectives in the optimization based on their relevance to the machining application. The single-objective optimization is conducted by using a single-objective genetic algorithm and a hybrid algorithm, whereas the multi-objective optimization is conducted by using a multi-objective genetic algorithm, a weighted sum single-objective genetic algorithm, and a weighted sum hybrid algorithm. It is shown that the single-objective optimization gives superior value in the optimized objective, while sacrificing the other objectives, whereas the multi-objective optimization compromises the improvement of all objectives by providing non-dominated values. In terms of the algorithms, it is shown that a hybrid algorithm can either verify or refine the optimal value obtained by a genetic algorithm.