Eigenvalue Theory Research Articles

The Dynamical Behaviors of a Fractional-Order Malware Propagation Model in Information Networks

With the rapid development of communication and information technology, information networks have become increasingly common in our work and daily lives. The aim of this paper is to study the process of introducing a new fractional-order malware propagation model into information networks. Based on the matrix theory of eigenvalues, the conditions of local stability for the above model are discussed. In addition, taking time delay as the bifurcation parameter, Hopf bifurcation is considered. The correctness of our theoretical results is verified through numerical experiments.

A new algorithm for determining the minimum eigenvalue of irreducible Z-matrices

Abstract The theory of minimum eigenvalue of Z-matrices is applied to many fields, and there are many methods to estimate and calculate the minimum eigenvalue. In this paper, a new iterative algorithm is obtained by diagonal similar transformation by the relationship between Z-matrices and nonnegative matrices. The algorithm can obtain the minimum eigenvalue satisfying arbitrary accuracy requirements.

Global Stability of Phase-Change Neural Networks With Mixed Time Delays.

Phase-change memory (PCM) is a novel type of nonvolatile memory and is suitable for artificial neural synapses. This article investigates the Lagrange global exponential stability (LGES) of a class of PCNNs with mixed time delays. First, based on the conductivity characteristics of PCM, a piecewise equation is established to describe the electrical conductivity of PCM. By using the proposed piecewise equation to simulate the neural synapses, a novel PCNN with discrete and distributed time delays is proposed. Then, using comparative theory and fundamental inequalities, the LGES conditions based on the M -matrix are proposed in the sense of Filippov, and the exponential attractive set (EAS) is obtained based on M -matrix and external input. Moreover, the Lyapunov global exponential stability (GES) conditions of PCNNs without external input are obtained by using the inequality technique and eigenvalue theory, which is a form of M -matrix. Finally, two simulation examples are given to verify the validity of the obtained results.

Positive steady states in a two-species chemotaxis-competition system with signal-dependent diffusion and sensitivity

– In this paper, we consider the following stationary two-species chemotaxis-competition system with signal-dependent diffusion and sensitivity (0.1)−d1Δu=λu−u2−buv,x∈Ω,−div[d(w)∇v+χ(w)v∇w]=μv−v2−cuv,x∈Ω,−d3Δw=−κw+τu,x∈Ω,u=v=w=0,x∈∂Ωin a bounded smooth domain Ω⊂RN(N≥1), where λ,μ,b,c,d1,d3,κ,τ are positive constants, and (d(w),χ(w))∈[C1[0,∞)]2 with d(w),χ(w)>0 for all w≥0. Since there does not exist an immediate change variable that transforms (0.1) into a semilinear system when (0.1) is considered with d(w),α(w) being arbitrary functions in w, this makes the analysis of system (0.1) much more difficult. By the W2,p-estimate and the global bifurcation method, we obtain a bounded connected set of positive steady states joining the semitrivial solution of the form (u,0,w) with u,w>0 and the semitrivial solution of the form (0,v,0) with v>0, which provides the sufficient condition for the existence of positive steady states. Combining the eigenvalue theory, homogenization technique and various elliptic estimates, we also derive some sufficient conditions for the nonexistence of positive steady states.

Spatio-temporal adaptive algorithm using Hamming code based on 3D-MIMO wireless channel model

Space-time coding is an effective method for constructing communication systems while saving bandwidth, taking into account the shortage of frequency resources and power in fading channels, which is implemented based on the advantages of using adaptive antenna arrays. Currently, WiMAX radio access technology has become widespread; its use is justified in conditions of direct visibility between the base and mobile stations, but in a city with dense buildings and in the presence of organized interference, its use leads to an increase in the likelihood of bit errors. At the same time, the combination of OFDM (orthogonal frequency division multiplexing) and MIMO (spatial coding) technologies based on adaptive antenna arrays significantly reduces the probability of bit errors, and therefore increases the throughput of the communication system. The use of adaptive space-time devices at the outputs of antenna arrays makes it possible to select a signal path with maximum power and with a minimum error in receiving OFDM pilot subcarriers, even in the absence of direct visibility between the base and mobile stations and in conditions of active interference. In addition, the use of a 3D model of the communication channel will allow us to evaluate the effectiveness of the communication system model in real operating conditions. The goal of the work is to solve the problem of additionally increasing noise immunity in communication systems in conditions of numerous reflections caused by dense buildings in urban environments, using a combination of Hamming code, OFDM and an algorithm for adaptive spatio-temporal signal processing in receivers such as SISO (Singl Input – Singl Output), and MIMO (Multiply Input – Multiply Output), built on adaptive antenna arrays. The work calculated the weight coefficients of the adaptation algorithm using the theory of eigenvalues and eigenvectors of the spatial correlation matrix of signals at the outputs of the adaptive antenna arrays. Modeling of algorithms and a 3D model of the communication channel in Matlab using the Hamming code was carried out, as a result, an increase in the efficiency of error correction for the Hamming code was demonstrated, as well as a significant reduction in the probability of code errors and an increase in the performance of the communication system when using an adaptive algorithm. At the same time, the article uses the term “multipath” channel in contrast to the generally accepted, but incorrect “multibeam” channel, since multibeam can be a characteristic of an antenna system. The presented results confirm that when processing signals in a receiving system in a multipath channel with dense urban development, the use of adaptive spatiotemporal algorithms provides increased noise immunity and radio communication capacity, especially for image transmission when using unmanned aerial vehicles.

Optimization of some types of Rényi divergences between unitary orbits

In this article, we investigate explicitly the minimal and maximal values of the α-z-Rényi divergences and some other types of Rényi divergences between unitary orbits. Our main tools are the majorization theory of eigenvalues of matrices, and some matrix inequalities such as Golden–Thompson's and Araki–Lieb–Thirring's inequalities.

Negative variance components and intercept-slope correlations greater than one in magnitude: How do such "non-regular" random intercept and slope models arise, and what should be done when they do?

Statistical models with random intercepts and slopes (RIAS models) are commonly used to analyze longitudinal data. Fitting such models sometimes results in negative estimates of variance components or estimates on parameter space boundaries. This can be an unlucky chance occurrence, but can also occur because certain marginal distributions are mathematically identical to those from RIAS models with negative intercept and/or slope variance components and/or intercept-slope correlations greater than one in magnitude. We term such parameters "pseudo-variances" and "pseudo-correlations," and the models "non-regular." We use eigenvalue theory to explore how and when such non-regular RIAS models arise, showing: (i) A small number of measurements, short follow-up, and large residual variance increase the parameter space for which data (with a positive semidefinite marginal variance-covariance matrix) are compatible with non-regular RIAS models. (ii) Non-regular RIAS models can arise from model misspecification, when non-linearity in fixed effects is ignored or when random effects are omitted. (iii) A non-regular RIAS model can sometimes be interpreted as a regular linear mixed model with one or more additional random effects, which may not be identifiable from the data. (iv) Particular parameterizations of non-regular RIAS models have no generality for all possible numbers of measurements over time. Because of this lack of generality, we conclude that non-regular RIAS models can only be regarded as plausible data-generating mechanisms in some situations. Nevertheless, fitting a non-regular RIAS model can be acceptable, allowing unbiased inference on fixed effects where commonly recommended alternatives such as dropping the random slope result in bias.

Dynamics of a Fokker–Planck type diffusion epidemic model with general incidence and relapse

In this paper, a spatially heterogeneous SIRI epidemic model with general morbidity is studied, which conforms to the Fokker–Planck type diffusion law. The basic reproduction number R0 of the model is introduced, and the threshold dynamics based on R0 is discussed. In particular, with the help of elliptic eigenvalue theory, we determine the asymptotic profiles of the endemic equilibrium when the diffusion rate of susceptible individuals approaches zero or infinity.

On the role of advection in a spatial epidemic model with general boundary conditions

This paper is concerned with a reaction-diffusion-advection model for vector-borne disease with general boundary conditions and general incidences. Due to the boundary conditions, we first apply the eigenvalue theory of elliptic system to prove the existence and uniqueness of steady state for the model. The well-posedness of the model is established using an induction argument. By overcoming the difficulty of the associated elliptic eigenvalue problem, we originally derive the variational expression of the basic reproduction ratio R0. The asymptotic profiles and monotonicity of R0 with respect to the mobility and advection rates are investigated following the variational characterization of R0. Furthermore, the spatial dynamics of the model with Robin type boundary condition are categorized via classifying the level set of R0. This work provides new clues for further research on the spread of epidemics in open advective environments.

Geometric topology of CR-warped products in six-dimensional sphere

<p>In this paper, we prove that a three-dimensional CR-warped product submanifold of a nearly Kaehler six-dimensional sphere, under some restrictions, contains finite fundamental groups. Using another approach from the theory of eigenvalues and also the gradient of Ricci curvature, we obtain similar results.</p>

On adjacency operators of locally finite graphs

A graph $\Gamma$ is called locally finite if, for each vertex $v\in \Gamma$, the set $\Gamma(v)$ of its adjacent vertices is finite. For an arbitrary locally finite graph $\Gamma$ with vertex set $V(\Gamma)$ and an arbitrary field $F$, let $F^{V(\Gamma)}$ be the vector space over $F$ of all functions $V(\Gamma) \to F$ (with natural componentwise operations) and let $A^{(\mathrm{alg})}_{\Gamma,F}$ be the linear operator $F^{V(\Gamma)} \to F^{V(\Gamma)}$ defined by $(A^{(\mathrm{alg})}_{\Gamma,F}(f))(v) = \sum_{u \in \Gamma(v)}f(u)$ for all $f \in F^{V(\Gamma)}$, $v \in V(\Gamma)$. In the case of a finite graph $\Gamma$, the mapping $A^{({\mathrm{alg}})}_{\Gamma,F}$ is the well-known operator defined by the adjacency matrix of the graph $\Gamma$ (over $F$), and the theory of eigenvalues and eigenfunctions of such operators is a well developed part of the theory of finite graphs (at least, in the case $F = \mathbb{C}$). In the present paper, we develop the theory of eigenvalues and eigenfunctions of the operators $A^{({\mathrm{alg}})}_{\Gamma,F}$ for infinite locally finite graphs $\Gamma$ (however, some results that follow may present certain interest for the theory of finite graphs) and arbitrary fields $F$, even though in the present paper special emphasis is placed on the case of a connected graph $\Gamma$ with uniformly bounded degrees of vertices and $F = \mathbb{C}$. The previous attempts in this direction were not, in the author's opinion, quite satisfactory in the sense that they have been concerned only with eigenfunctions (and corresponding eigenvalues) of rather special type.

Cooperative Spectrum Sensing Algorithm Based on Eigenvalue Fusion

A novel algorithm is introduced to improve collaborative spectrum sensing under low cognitive capabilities and insufficient signal-to-noise ratio. The algorithm is based on the difference of random matrix eigenvalues and uses the theory of random eigenvalues and the extreme distribution of the minimum eigenvalue. It makes use of the average, both arithmetic and geometric, as well as the minimum and maximum values of eigenvalues as the detection metric. It calculates the fusion power parameter through local energy spectrum sensing. Simulation results demonstrate that the algorithm outperforms the DMM algorithm and the NMME algorithm under users with low cognitive capabilities and Insufficient signal-to-noise ratio, making it more suitable for low signal-to-noise ratio environments.

Topology uniformity pinning control for multi-agent flocking

The optimal selection of pinning nodes for multi-agent flocking is a challenging NP-hard problem. Current pinning node selection strategies mainly rely on centrality measures of complex networks, which lack rigorous mathematical proof for effective flocking control. This paper proposes a pinning node selection strategy based on matrix eigenvalue theory. First, the effect of the pinning node on the eigenvalue of the Laplacian matrix is analyzed. Then, a synchronization index representing the topology uniformity of the multi-agent system is proposed to exert maximum influence on the system synchronizability. A practical optimal pinning node selection method based on the synchronization index is proposed and analyzed using the eigenvalue perturbation method. Finally, simulations demonstrate that the convergence rate of the system obtained using the optimal synchronizability pinning node selection method is better than that achieved with the maximum degree centrality node selection strategy.

Theoretical and experimental investigation on the stall inception of low-speed axial compressors with swept rotor blades

The effect of rotor blade sweep on the flow stability of low-speed axial compressors is investigated by means of theoretical model, numerical simulations and experimental measurements. Four compressors with forward or backward swept rotors are carefully designed by modifying the tip element location of the baseline rotor. Steady simulation results show that the designed aerodynamic sweep in this work hardly changes the pressure rise and efficiency characteristics of the compressor, which ensures the comparability between these compressors regarding flow stability. The developed stall inception prediction model based on global linear stability analysis and eigenvalue theory is employed to evaluate the stall boundary of compressors after blade sweep designs, and results indicate that the forward sweep slightly improve the flow stability, while the backward sweep has significant deterioration effect. What's more, the circumferential propagation frequency of the perturbation wave at stall onset changes little after the sweep design. The steady and dynamic measurements in experiments for different rotor blades verifies the theoretical conclusions and the prediction accuracy of the theoretical model. Through steady and unsteady flow field analysis, it is found that the forward sweep reduces the diffusion factor, elementary work and blade loading at the rotor tip region, while the backward sweep has the opposite effect and enhances the interaction between the tip leakage flow and the primary flow, which is deemed the contributors to their effect on flow stability.

Stability and HOPF Bifurcation Analysis of Periodic Solutions of a Duffing Equation

In this study, the stability and Hopf bifurcation analysis of periodic solutions of Duffing equations were considered. Also other types of bifurcation like the Saddle-node, Transcritical and Pitchkfork were also studied. The eigen value, Jacobian and Floquet theory were used to analyse both the stability and Hopf bifurcations of the periodic solutions of the equilibrium points. The results showed that equilibrium points have at most three -periodic solutions under a strong damped conditions due to the cubic nonlinearities. The bifurcation points showed; one critical, another subcritical and the third point showed that the homoclinic was present when the damping coefficient is zero. Furthermore, the presence of strange attractors varied with the driving force and damping. The MATCAD software was used to illustrate the numerical behaviour of the solution which extend some results in the literature.

Full characterization of vector eigenstates in symmetrically confined systems with photonic spin-orbit coupling.

The photonic spin-orbit (SO) coupling is a widely investigated effect in optical microcavities leading to various interesting physical phenomena and potential applications. We report the full sets of eigenenergies and eigenstates in a symmetrically confined potential under the effect of SO coupling induced by the transverse-electric transverse-magnetic (TE-TM) splitting, which are derived analytically via the degenerate perturbation theory. We obtained the eigenenergies and the eigenstates from the 1st to the 6th orders of excited manifold, and demonstrate unambiguously that universal rules governing the mode formation exist in such complicated photonic systems, making the modes exhibiting the features of solid and hollow skyrmions as well as spin vortices. We show that these eigenstates can be described by the SO coupled hyperspheres that can be decomposed into a series of higher-order Poincare spheres. Our results significantly extend the area of microcavity spin-optronics to the general theory of eigenvalues in confined systems, and provide an efficient theoretical frame for the information processing using microcavity-based high-dimensional vector states.

Dynamics of an unstirred chemostat model with Beddington–DeAngelis functional response

This paper deals with an unstirred competitive chemostat model with the Beddington–DeAngelis functional response. With the help of the linear eigenvalue theory and the monotone dynamical system theory, we establish a relatively clear dynamic classification of this system in terms of the growth rates of two species. The results indicate that there exist several critical curves, which may classify the dynamics of this system into three scenarios: 1) extinction; 2) competitive exclusion; and 3) coexistence. Comparing with the classical chemostat model [26], our theoretical results reveal that under the weak–strong competition cases, the role of intraspecific competition can lead to species coexistence. Moreover, the simulations suggest that under different competitive cases, coexistence can occur for suitably small diffusion rates and some intermediate diffusion rates. These new phenomena indicate that the intraspecific competition and diffusion have a great influence on the dynamics of the unstirred chemostat model of two species competing with the Beddington–DeAngelis functional response.

The principal eigenvalue for partially degenerate and periodic reaction-diffusion systems with time delay

In this paper, we first establish the theory of principal eigenvalues for a large class of partially degenerate, linear and periodic parabolic cooperative systems with time delay. Then we apply these theoretical results to study the global dynamics of a blacklegged tick Ixodes scapularis population model. To present a threshold-type result in terms of basic reproduction ratio R0 for such a model, we also extend the earlier theory of R0 to abstract functional differential equations with time-delayed internal transition.

Bifurcation and stability of a diffusive predator-prey model with the fear effect and time delay.

The predator-prey system can induce wealth properties with fear effects. In this paper, we propose a diffusive predator-prey model where the influence of fear effects and time delay is considered, under the Dirichlet boundary condition. It follows from the Lyapunov-Schmidt reduction method that there exists a non-homogeneous steady-state solution of the system and the specific expressions are also given. By the aid of bifurcation theory and eigenvalue theory, we also investigate the existence/non-existence and the stability of Hopf bifurcation under three different conditions of bifurcation parameters. Furthermore, the effects of the fear on population density, stability, and Hopf bifurcation are also considered and the results show that the increase of fear effects will reduce the population density, and Hopf bifurcation is more likely difficult to undergo as k increases under some conditions.

Eigenvalue problems for singular p-Monge-Ampère equations

Using the eigenvalue theory of completely continuous operators, we are concerned with the determining values of μ, in which there are nontrivial radial solutions to the singular p-Monge-Ampère problem{det(D(|Du|p−2Du))=μh(|x|)f(−u)inΩ,u=0on∂Ω, where μ>0 is a parameter, Ω is the open unit ball in Rn. In addition, as an application, we derive some criteria for the existence of nontrivial radial solutions to the power-type problem of p-Monge-Ampère equations. The dependence of nontrivial radial solution uμ on the parameter μ is also considered.