Characteristic Matrix Equation Research Articles

A discussion of stability analysis for systems of differential equations with multiple and distributed delays

In this article, we consider a class of systems of multiple delay differential equations (MDDEs). We first define a characteristic matrix equation that can be used to analyze the stability of the equilibrium of a system of MDDEs. Then we construct a matrix based on the coefficients of the characteristic matrix equation and use the spectrum of this matrix to derive necessary and sufficient conditions for the system to be stable. Next we discuss a comparison of the stability equivalency between a system of delay differential equations (DDEs) to the system of MDDEs and relate our results to distributed delay systems (DDSs). Numerical examples are given to justify our theory.

Influence of ion structure and solvent electric dipole on ultrananoporous supercapacitor: a lattice model study

A five-state modified BEG model is proposed for describing electrolyte in polar solvent in cylindrical electrode pore in which only one row of particles can be stationed. Two properties the closest to the actual situation of the aqueous electrolyte are integrated into the model: cation is dimer comprised of two tangentially tethered hard spheres of different diameters: one is charged, while the other is neutral, and polar solvent is modeled as neutral hard sphere with an electric dipole. The system partition function is obtained by numerically solving maximum real root of characteristic equation of Kramers-Wannier transfer matrix. The model is used to investigate effects of ion structure and electric valence, solvent electric dipole, and cylindrical pore radius on specific differential capacitance and specific energy storage of the resulting supercapacitor. Main findings are briefly described below. (i) The dimer counter-ion electric valence does not exert obvious influence on the saturation value of the but reduces the threshold electrode potential strength beyond which the value is just reached. (ii) Whether the dimer counter-ion is monovalent or bivalent, the peak height and the value tend to rise with the dimer neutral site size decreasing, but influence of the value on the value is very small. Only when the value is small enough, the value rises a little with decrease of the value. (iii) Increase of solvent electric dipole moment (whether by increasing the dipole charge or dipole length) always raises the value, and enlarges width of the pure solvent domain or zero and zero domain around the zero electrode potential, but does not change the value obviously. (iv) With the pore size increasing, the value reduces and the value increases. So, using small size pore is favorable to improve the supercapacitor performance from the perspective of energy storage and electrode potential applied. (v) The three parameters: dipole charge, and dipole length, are cooperative in influencing the value.

EIGENTIME IDENTITY OF THE WEIGHTED KOCH NETWORKS

The eigenvalues of the transition matrix of a weighted network provide information on its structural properties and also on some relevant dynamical aspects, in particular those related to biased walks. Although various dynamical processes have been investigated in weighted networks, analytical research about eigentime identity on such networks is much less. In this paper, we study analytically the scaling of eigentime identity for weight-dependent walk on small-world networks. Firstly, we map the classical Koch fractal to a network, called Koch network. According to the proposed mapping, we present an iterative algorithm for generating the weighted Koch network. Then, we study the eigenvalues for the transition matrix of the weighted Koch networks for weight-dependent walk. We derive explicit expressions for all eigenvalues and their multiplicities. Afterwards, we apply the obtained eigenvalues to determine the eigentime identity, i.e. the sum of reciprocals of each nonzero eigenvalues of normalized Laplacian matrix for the weighted Koch networks. The highlights of this paper are computational methods as follows. Firstly, we obtain two factors from factorization of the characteristic equation of symmetric transition matrix by means of the operation of the block matrix. From the first factor, we can see that the symmetric transition matrix has at least [Formula: see text] eigenvalues of [Formula: see text]. Then we use the definition of eigenvalues and eigenvectors to calculate the other eigenvalues.

Tunable bandgaps in a deployable metamaterial

In this manuscript, we investigate deployable structures (such as solar arrays) and origami-inspired foldable structures as metamaterials capable of tunable wave manipulation. Specifically, we present a metamaterial whose bandgaps can be modulated by changing the fold angle of adjacent panels. The repeating unit cell of the structure consists of a beam (representing a panel) and a torsional spring (representing the folding mechanism). Two important cases are considered. Firstly, the fold angle (angle between adjacent beams), Ψ, is zero and only flexural waves propagate. In the second case, the fold angle is greater than zero (Ψ>0). This causes longitudinal and transverse vibration to be coupled. FEM models are used to validate both these analyses.Increasing the fold angle was found to inflict notable changes to the wave transmission characteristics of the structure. In general, increasing the fold angles caused the bandwidth of bandgaps to increase. For the lowest four bandgaps we found bandwidth increases of 252%, 177%, 230% and 163% respectively at Ψ=90deg (relative to the bandwidths at Ψ=0). In addition, non-trivial increases in bandwidth of the odd-numbered bandgaps occurs even at small fold angles-the bandwidth for the first and third bandgaps effectively double in size (increase by 100%) at Ψ=20deg relative to those at Ψ=0. This could have ramifications in the context of tunable wave manipulation and adaptive filtering.In addition, by expanding out the characteristic equation of transfer matrix for the straight structure, we prove that the upper band edge of the nth bandgap will always equal the nth simply supported natural frequency of the constituent beam. Further, we found that the ratio (EIkt) is a pertinent parameter affecting the bandwidth of bandgaps. For low values of the ratio, effectively, no bandgap exists. For higher values of the ratio (EIkt), we obtain a relatively large bandgap over which no waves propagate. This can have ramifications for the design of foldable structures. As an alternative to impedance-based structural health monitoring, these insights can aid in health monitoring of deployable structures by tracking the bandwidth of bandgaps which can provide clues about the mechanical parameters of the structure.

On five-dimensional massive (bi-)gravity

Main results of our recent investigations on five-dimensional scenarios of massive (bi-)gravity will be summarized in this article. In particular, we will show how to construct higher dimensional massive graviton terms from the characteristic equation of square matrix, which is a consequence of the Cayley-Hamilton theorem. Then, we will show whether massive graviton terms of five-dimensional massive (bi-)gravity behave as effective cosmological constants for a number of physical metrics compatible with fiducial ones such as the Friedmann-Lemaitre-Robertson-Walker, Bianchi type I, and Schwarzschild-Tangherlini metrics. Finally, we will show the corresponding cosmological solutions for the five-dimensional massive (bi-)gravity.

ДИНАМИЧЕСКИЙ РАСЧЁТ ФЕРМЫ ПОКРЫТИЯ ПРИ СВОБОДНЫХ КОНСТРУКТИВНО НЕЛИНЕЙНЫХ КОЛЕБАНИЯХ

A mathematical model of construction of the total mechanical energy of a discrete statically loaded dissipative system in the process of free oscillations in the form the time analysis method, which is based on the study of the characteristic quadratic matrix equation, was given. The features of the behavior of the parameters of the dynamic response when the elastic structurally nonlinear system to a new state was shown. It is also given the expressions to determine the kinematic (displacements, velocities, accelerations) and force (restoring, dissipative and inertial forces) parameters of the dynamic response. The criteria for assessing the load bearing capacity of the structure at the non-linear operation was suggested. Using a mathematical model is illustrated by the example of the calculation of flat steel statically indeterminate roof truss during the process of free oscillations, which was caused by a sudden failure of one of the bearing elements. Turning off the rod caused by the loss of strength as a result of chemical cracking due to corrosion effect.

The QR Method for Determining All Eigenvalues of Real Square Matrices

Eigenvalues are special sets of scalars associated with a given matrix. In other words for a given matrix A, if there exist a non-zero vector V such that, AV= λV for some scalar λ, then λ is called the eigenvalue of matrix A with corresponding eigenvector V. The set of all nxm matrices over a field F is denoted by Mnm (F). If m = n, then the matrices are square, and which is denoted by Mn (F). We omit the field F = C and in this case we simply write Mnm or Mn as appropriate. Each square matrix AϵMnm has a value in R associated with it and it is called its determinant which is use full for solving a system of linear equation and it is denoted by det (A). Consider a square matrix AϵMn with eigenvalues λ, and then by definition the eigenvectors of A satisfy the equation, AV = λV, where v={v1, v2, v3…………vn}. That is, AV=λV is equivalent to the homogeneous system of linear equation (A-λI) v=0. This homogeneous system can be written compactly as (A-λI) V = 0 and from Cramer’s rule, we know that a linear system of equation has a non-trivial solution if and only if its determinant is zero, so the solution λ is given by det (A-λI) =0. This is called the characteristic equation of matrix A and the left hand side of the characteristic equation is the characteristic polynomial whose roots are equals to λ.

Critical Exponents in a Doubly Degenerate Nonlinear Parabolic System with Inner Absorptions

This paper deals with critical exponents for a doubly degenerate nonlinear parabolic system coupled via local sources and with inner absorptions under null Dirichlet boundary conditions in a smooth bounded domain. The author first establishes the comparison principle and local existence theorem for the above problem. Then under appropriate hypotheses, the author proves that the solution either exists globally or blows up in finite time depends on the initial data and the relations of the parameters in the system. The critical exponent of the system is simply described via a characteristic matrix equation introduced.

Stability and bifurcation of epidemic spreading on adaptive network

Adaptive network is characterized by feedback loop between states of nodes and topology of the network. In this paper, for adaptive epidemic spreading model, epidemic spreading dynamics is studied by using a nonlinear differential dynamic system. The local stability and bifurcation behavior of the equilibrium in this network model are investigated and all kinds of bifurcation point formula are obtained by analyzing its corresponding characteristic equation of Jacobian matrix of the nonlinear system. It is shown that, when the epidemic threshold is less than epidemic persistence threshold R00c, the disease always dies out and the disease-free equilibrium is asymptotically locally stable. If R0c01, a backward bifurcation leading to bistability possibly occurs, and there are possibly three equilibria: a stable disease-free equilibrium, a larger stable endemic equilibrium, and a smaller unstable endemic equilibrium. If R01, the disease is uniformly persistent and only one endemic equilibrium is asymptotically locally stable. It is also found that the system has saddle-node bifurcation, transcritical bifurcation, and Hopf bifurcation. Numerical simulations are given to verify the results of theoretical analysis.

Eigenpairs of a coupled rectangular cavity and its fundamental properties

This paper deals with the eigenvalue problem of a coupled rectangular cavity comprising five rigid walls and one flexible panel frequently employed in much literature. It is the purpose of this paper to derive explicitly the eigenpairs of the coupled cavity, which are yet to be found. First, the coupling orthogonality conditions the eigenpairs need to satisfy are derived, thereby enabling the verification of the eigenpairs newly sought or already existent. Using the coupling orthogonality conditions, the modal equation of the coupled cavity system is then obtained, permitting one to deal with a forced response of the coupled cavity. It is shown that the eigenfunctions governing the dynamics of the sound field are expressed as the infinite sum of degenerate eigenfunctions. The characteristic matrix equation is then derived, specifying the eigenpairs of the coupled cavity. In order to investigate the fundamental properties of the eigenpairs derived, a numerical analysis is conducted, revealing the presence of evanescent modes in addition to the conventional standing wave modes. Finally, an experiment is carried out, verifying the validity of the eigenpairs derived in the article.

Hyperspectral Imaging Target Detection Based on Improved Kernel Principal Component Analysis

Abstract The kernel principal component analysis (KPCA) algorithm has been extensively used in target detection and classification for hyperspectral imaging. Kernel parameters are a key component of KPCA, but no optimal method for selecting appropriate parameters has been proposed. We study the largest eigenvalue and sum of the eigenvalues of characteristic equation of kernel matrix, and then put forward a proposal for parameter selection. Experiments show that the proposal enables best parameter selection.

KINEMATIC ANALYSIS AND PROTOTYPE OF A METAMORPHIC ANTHROPOMORPHIC HAND WITH A RECONFIGURABLE PALM

A novel metamorphic anthropomorphic hand is for the first time introduced in this paper. This robotic hand has a reconfigurable palm that generates changeable topology and augments dexterity and versatility of the hand. Structure design of the robotic hand is presented and based on mechanism decomposition kinematics of the metamorphic anthropomorphic hand is characterized with closed-form solutions leading to the workspace investigation of the robotic hand. With characteristic matrix equation, twisting motion of the metamorphic robotic hand is investigated to reveal both dexterity and manipulability of the metamorphic hand. Through a prototype, grasping and prehension of the robotic hand are tested to illustrate characteristics of the new metamorphic anthropomorphic hand.

任意多包层光纤模场的数值解法

The non-linear differential method was applied to the first-order and second-order differential appoximation.After non-linear grid dividing of radius and refractive index distribution of any multi-layer fiber,the numeralization of Helmholtz equation was obtained.By solving the characteristic matrix equation using Matlab,accurate mode field distributions of single mode fiber and multimode fiber were obtained,and the normalized mode field of five cladding dispersion flattened fiber was calculated.The results show that the non-linear differential method ensures the accuracy,and also improves the computational speed greatly,which lays a foundation for further study of other characteristics of optical fiber.

Estimates of accuracy of approximate solutions of some combined modular elastic continuum systems with discrete interaction

At present, it happens increasingly often that one needs to study complicated modular systems with discrete interaction between the units. If a system contains a distributed unit, then it is said to be combined. One method for studying such systems is to analyze their frequency models; in this case, it is sometimes required to calculate the Green functions of the distributed unit and study the block structures generated by a model of a system with superelements. A technique for studying such systems was developed in [1] and is called the factorized perturbation method; in foreign literature, this technique is called the Green function method (e.g., see [2]). The main idea of such methods is first to construct an equivalent system in the characteristic space that arises in the discrete macrostructure of the original system as a result of the discrete interaction between the units of the latter and then to reduce the characteristic solution to the original solution by using a simple transfer relation. A typical characteristic of structural methods is their algorithmic universality, which is independent of the special form of the discrete microstructure and requires solving a characteristic matrix equation. To obtain approximate solutions by these methods, one should approximately calculate the Green functions of the distributed units and explicitly indicate their operating frequency ranges ( see [3, 4]). The concept of structural methods is to obtain a system solution by using a priori studied elements. There are a variety of methods for studying combined modular systems. In view of this, the accuracy of the approximate solutions, which are also obtained by numerous methods, should be estimated from the algorithmic standpoint. At present, this is done at the level of practical-empirical considerations like doubling the number of grid points for the analysis of the distributed unit, taking into account additional vibration modes in the Green functions, etc. In the present paper, we describe a structural method for solving elastic one-dimensional distributed systems with discrete interaction and present an efficient scalar a priori estimate and a universal a posteriori estimate for the accuracy of an approximate solution, which can be used in many approximate methods, e.g., FEM [5].

A nonlinear diffusion system with convection

This paper deals with a coupled nonlinear diffusion system with convection. A characteristic matrix equation covering all the six nonlinear exponents of the system is introduced to get a simple description for the blow-up criteria of the system. In addition, the influence of the four coefficients of the diffusion and the convection as well as the geometry of Ω on global boundedness or not of the solutions is shown clearly, especially for the situation with the critical sign of the characteristic parameters.

Quasi-spherical coordinate systems for three-dimensional stress singularity eigenproblems

To avoid the inherent singularity of conventional spherical coordinates at their poles, quasi-spherical coordinate systems are developed. Using these systems, a finite element procedure is developed to determine the displacement eigensolutions at three-dimensional vertices in which the displacement and stress are proportional to the ( λ + 1 ) th and λ th powers of the distance from the vertices, respectively. The resulting global equation is a second-order characteristic matrix equation. Several demonstrating problems are investigated. Satisfactory results are obtained. Unlike the previous attempts by singular transformation technique, the present predictions are insensitive to the numerical integration order.

Critical exponents in a parabolic system with inner absorption and coupled nonlinear boundary flux

This paper deals with critical exponents for a parabolic system with inner absorption and coupled nonlinear boundary flux. A detail analysis is given to the interactions among the multi-nonlinearities in the system. The critical exponent of the system is simply described via a characteristic matrix equation introduced.

Enhancing the natural frequency doublet splitting in almost periodic azimuthally corrugated cavities

A symmetry of a corrugated periodic structure can be broken due to loading or/and fabrication defects so that frequency doublets may appear, whose frequencies are just slightly splitted. In this letter, we study the potentials to enhance this splitting by enhancing the asymmetry of the cavity. This is provided by a single (loaded) resonator which differs in its geometry from other resonators. The initial causes of the asymmetry are not taken into account. The used characteristic matrix equation is obtained by a mode-matching technique. It is shown that the doublet splitting can reach several percent due to a convenient choice of the geometry of the loaded resonator.

A finite element approach for computing edge singularities in piezoelectric materials

By using the eigenfunction expansion technique and the weak form of the governing equations for prismatic sectorial domains composed of piezoelectrics and air, an one-dimensional finite element procedure is formulated for computing the eigensolutions of the electromechanical field problem. Generalized displacement and electric potential are taken to be the nodal variables. The resulting global equation is a second order characteristic matrix equation. Validity of the formulation is verified by comparing the computed results with the existing solutions for impermeable cracks and interfacial cracks. Configurations which are of practical interest including conducting cracks, permeable and impermeable notches are studied.

A simple finite element formulation for computing stress singularities at bimaterial interfaces

By using the weak form of the governing equations for sectorial bimaterial domains and assuming that the displacement field is proportional to the ( λ+1)th power of the distance from the singular stress point, a second-order characteristic matrix equation on λ is derived by a one-dimensional finite element formulation that only discretizes the domain circumferentially. Numerical examples covering a variety of interfacial singularities are presented to demonstrate the efficacy of the formulation. Accurate solutions are yielded by very few elements whereas convergence can be attained by either h- or p-refinement. The related procedures are programmed in a short MAPLE worksheet given in the appendix.