Algebraic Criteria Research Articles

Multiple $\psi$ -Type Stability and Its Robustness for Recurrent Neural Networks With Time-Varying Delays

In this paper, the ψ -type stability and robustness of recurrent neural networks are investigated by using the differential inequality. By utilizing ψ -type functions combined with the inequality techniques, some sufficient conditions ensuring ψ -type stability and robustness are derived for linear neural networks with time-varying delays. Then, by choosing appropriate Lipschitz coefficient in subregion, some algebraic criteria of the multiple ψ -type stability and robust boundedness are established for the delayed neural networks with time-varying delays. For special cases, several criteria are also presented by selecting parameters with easy implementation. The derived results cover both ψ -type mono-stability and multiple ψ -type stability. In addition, these theoretical results contain exponential stability, polynomial stability, and μ -stability, and they also complement and extend some previous results. Finally, two numerical examples are provided to illustrate the effectiveness of the proposed criteria.

Nonchaotic Behavior in Quadratic Three-Dimensional Differential Systems with a Symmetric Jacobian Matrix

In this paper, we give an algebraic criterion to determine the nonchaotic behavior for polynomial differential systems defined in [Formula: see text] and, using this result, we give a partial positive answer for the conjecture about the nonchaotic dynamical behavior of quadratic three-dimensional differential systems having a symmetric Jacobian matrix. The algebraic criterion presented here is proved using some ideas from the Darboux theory of integrability, such as the existence of invariant algebraic surfaces and Darboux invariants, and is quite general, hence it can be used to study the nonchaotic behavior of other types of differential systems defined in [Formula: see text], including polynomial differential systems of any degree having (or not having) a symmetric Jacobian matrix.

Toward a contemporary quantitative model of punishment.

A direct-suppression, or subtractive, model of punishment has been supported as the qualitatively and quantitatively superior matching law-based punishment model (Critchfield, Paletz, MacAleese, & Newland, 2003; de Villiers, 1980; Farley, 1980). However, this conclusion was made without testing the model against its predecessors, including the original (Herrnstein, 1961) and generalized (Baum, 1974) matching laws, which have different numbers of parameters. To rectify this issue, we reanalyzed a set of data collected by Critchfield et al. (2003) using information theoretic model selection criteria. We found that the most advanced version of the direct-suppression model (Critchfield et al., 2003) does not convincingly outperform the generalized matching law, an account that does not include punishment rates in its prediction of behavior allocation. We hypothesize that this failure to outperform the generalized matching law is due to significant theoretical shortcomings in model development. To address these shortcomings, we present a list of requirements that all punishment models should satisfy. The requirements include formal statements of flexibility, efficiency, and adherence to theory. We compare all past punishment models to the items on this list through algebraic arguments and model selection criteria. None of the models presented in the literature thus far meets all of the requirements.

Discontinuity Propagation in Delay Differential-Algebraic Equations

The propagation of primary discontinuities in initial value problems for linear delay differential-algebraic equations (DDAEs) is discussed. Based on the (quasi-) Weierstra{\ss} form for regular matrix pencils, a complete characterization of the different propagation types is given and algebraic criteria in terms of the matrices are developed. The analysis, which is based on the method of steps, takes into account all possible inhomogeneities and history functions and thus serves as a worst-case scenario. Moreover, it reveals possible hidden delays in the DDAE and allows to study exponential stability of the DDAE based on the spectral abscissa. The new classification for DDAEs is compared to existing approaches in the literature and the impact of splicing conditions on the classification is studied.

New Algebraic Criteria for Global Exponential Periodicity and Stability of Memristive Neural Networks with Variable Delays

This paper concentrates on the problem of global exponential periodicity and stability of memristive neural networks with variable delays. By constructing the appropriate Lyapunov functionals and utilizing some inequality techniques, new algebraic criteria are proposed to guarantee the existence and global exponential stability of periodic solution of the considered system. In addition, the proposed theoretical results not only expand and complement the earlier publications, but also are easy to be checked with the parameters of system itself. A numerical example is given to demonstrate the effectiveness of our results.

Free and Properly Discontinuous Actions of Groups on Homotopy 2n-spheres

Let G be a group acting freely, properly discontinuously and cellularly on some finite dimensional CW-complex Σ(2n) which has the homotopy type of the 2n-sphere 𝕊2n. Then, that action induces a homomorphism G → Aut(H2n(Σ(2n))). We classify all pairs (G, φ), where G is a virtually cyclic group and φ: G → Aut(ℤ) is a homomorphism, which are realizable in the way above and the homotopy types of all possible orbit spaces as well. Next, we consider the family of all groups which have virtual cohomological dimension one and which act on some Σ(2n). Those groups consist of free groups and semi-direct products F ⋊ ℤ2 with F a free group. For a group G from the family above and a homomorphism φ: G → Aut(ℤ), we present an algebraic criterion equivalent to the realizability of the pair (G, φ). It turns out that any realizable pair can be realized on some Σ(2n) with dim Σ(2n) ≤ 2n + 1.

三维薄结构热传导问题分层二次元方程的多水平方法

When the finite element method is applied to analyze the 3D thin-walled structures, some thin hexahedral elements are usually used in order to reduce the number of elements, and the corresponding higher-order elements are preferred since they have some obvious advantages in the calculation accuracy, the anti-distortion degree and so on. However, they have much higher computational complexity than the lower-order (e.g., linear) elements and the coefficient matrix of the linear algebraic equation system is severely ill-conditioned. The convergence of the commonly used solvers will deteriorate with the increasing size of the problem. An efficient and robust multi-level method was presented for the hierarchical quadratic discretizations of 3D thin-walled structures through combination of two special local block Gauss-Seidel smoothers and the DAMG algorithm based on the distance matrix. Since a hierarchical basis is used, those algebraic criteria are not needed to judge the relationships between the unknown variables and the geometric node types, and the grid transfer operators are also trivial. This makes it easy to find the coarse level (linear element) matrix derived directly from the fine level matrix, and thus the overall efficiency is greatly improved. The numerical results verify the efficiency and robustness of the proposed method.

Multiple μ-stability analysis for memristor-based complex-valued neural networks with nonmonotonic piecewise nonlinear activation functions and unbounded time-varying delays

Abstract This paper concentrates on the coexistence and dynamical behaviors of multiple equilibrium points for a class of memristor-based complex-valued neural networks (MCVNNs) with non-monotonic piecewise nonlinear activation functions and unbounded time-varying delays. Based on the geometrical configuration of activation functions, by utilizing the Intermediate Value Theorem and other analytical tools, some novel algebraic criteria are proposed to guarantee the coexistence of 25n equilibrium points in which 9n equilibrium points are locally μ-stable for such MCVNNs. As a direct application of these results, some criteria that assure the multiple exponential stability, multiple power stability, multiple log-stability and multiple log–log-stability are established. The proposed results show that the complex-valued neural networks introduced in this paper can have greater storage capacity than the real-valued ones. Finally, numerical example is presented to demonstrate the validity and feasibility of the obtained theoretical findings.

Finite-Time Stabilization and Adaptive Control of Memristor-Based Delayed Neural Networks.

Finite-time stability problem has been a hot topic in control and system engineering. This paper deals with the finite-time stabilization issue of memristor-based delayed neural networks (MDNNs) via two control approaches. First, in order to realize the stabilization of MDNNs in finite time, a delayed state feedback controller is proposed. Then, a novel adaptive strategy is applied to the delayed controller, and finite-time stabilization of MDNNs can also be achieved by using the adaptive control law. Some easily verified algebraic criteria are derived to ensure the stabilization of MDNNs in finite time, and the estimation of the settling time functional is given. Moreover, several finite-time stability results as our special cases for both memristor-based neural networks (MNNs) without delays and neural networks are given. Finally, three examples are provided for the illustration of the theoretical results.Finite-time stability problem has been a hot topic in control and system engineering. This paper deals with the finite-time stabilization issue of memristor-based delayed neural networks (MDNNs) via two control approaches. First, in order to realize the stabilization of MDNNs in finite time, a delayed state feedback controller is proposed. Then, a novel adaptive strategy is applied to the delayed controller, and finite-time stabilization of MDNNs can also be achieved by using the adaptive control law. Some easily verified algebraic criteria are derived to ensure the stabilization of MDNNs in finite time, and the estimation of the settling time functional is given. Moreover, several finite-time stability results as our special cases for both memristor-based neural networks (MNNs) without delays and neural networks are given. Finally, three examples are provided for the illustration of the theoretical results.

Fixed points of [formula omitted]-valued maps, the fixed point property and the case of surfaces—A braid approach

We study the fixed point theory of n-valued maps of a space X using the fixed point theory of maps between X and its configuration spaces. We give some general results to decide whether an n-valued map can be deformed to a fixed point free n-valued map. In the case of surfaces, we provide an algebraic criterion in terms of the braid groups of X to study this problem. If X is either the k-dimensional ball or an even-dimensional real or complex projective space, we show that the fixed point property holds for n-valued maps for all n≥1, and we prove the same result for even-dimensional spheres for all n≥2. If X is the 2-torus, we classify the homotopy classes of 2-valued maps in terms of the braid groups of X. We do not currently have a complete characterisation of the homotopy classes of split 2-valued maps of the 2-torus that contain a fixed point free representative, but we give an infinite family of such homotopy classes.

Adaptive formation control of networked Lagrangian systems with a moving leader

This paper investigates the formation control problem of networked Lagrangian systems with a moving leader under the directed network topology. A special form of geometric pattern is introduced to design the desired formation for such systems. Three adaptive control strategies are proposed for the networked Lagrangian systems to achieve the formation for the cases of the absence and presence of time delays. Some simple yet general algebraic criteria are developed to ensure that the networked Lagrangian systems can always achieve desired geometric formation. Furthermore, the effect of communication time delays on the performance of formation control is numerically investigated. Finally, numerical examples are given to illustrate and visualize the effectiveness of the theoretical results.

Moment map flows and the Hecke correspondence for quivers

In this paper we investigate the convergence properties of the upwards gradient flow of the norm-square of a moment map on the space of representations of a quiver. The first main result gives a necessary and sufficient algebraic criterion for a complex group orbit to intersect the unstable set of a given critical point. Therefore we can classify all of the isomorphism classes which contain an initial condition that flows up to a given critical point. As an application, we then show that Nakajima's Hecke correspondence for quivers has a Morse-theoretic interpretation as pairs of critical points connected by flow lines for the norm-square of a moment map. The results are valid in the general setting of finite quivers with relations.

Classification of a Subclass of Two-Dimensional Lattices via Characteristic Lie Rings

The main goal of the article is testing a new classification algorithm. To this end we apply it to a relevant problem of describing the integrable cases of a subclass of two-dimensional lattices. By imposing the cut-off conditions $u_{-1}=c_0$ and $u_{N+1}=c_1$ we reduce the lattice $u_{n,xy}=\alpha(u_{n+1},u_n,u_{n-1})u_{n,x}u_{n,y}$ to a finite system of hyperbolic type PDE. Assuming that for each natural $N$ the obtained system is integrable in the sense of Darboux we look for $\alpha$. To detect the Darboux integrability of the hyperbolic type system we use an algebraic criterion of Darboux integrability which claims that the characteristic Lie rings of such a system must be of finite dimension. We prove that up to the point transformations only one lattice in the studied class passes the test. The lattice coincides with the earlier found Ferapontov-Shabat-Yamilov equation. The one-dimensional reduction $x=y$ of this lattice passes also the symmetry integrability test.

Matrix measure based exponential stabilization for complex-valued inertial neural networks with time-varying delays using impulsive control

In this paper, the problem on the exponential stabilization of complex-valued inertial neural networks with time-varying delays via impulsive control is studied. By virtue of an appropriate variable transformation, the original inertial neural network is transformed into the first order complex-valued differential system. Based on matrix measure and applying impulsive differential inequality, some easily verifiable algebraic criteria on delay-dependent conditions are derived to ensure the global exponential stabilization for the addressed neural networks using impulsive control. Moreover, the different unstable equilibrium point can also be exponentially stabilized by using the different impulsive controllers and the exponential convergence rate index is also estimated. Finally, two numerical examples with simulations are presented to show the effectiveness of the obtained results.

Hopf Bifurcation Control for Rolling Mill Multiple-Mode-Coupling Vibration Under Nonlinear Friction

Rolling mill system may lose its stability due to the change of lubrication conditions. Based on the rolling mill vertical–torsional–horizontal coupled dynamic model with nonlinear friction considered, the system stability domain is analyzed by Hopf bifurcation algebraic criterion. Subsequently, the Hopf bifurcation types at different bifurcation points are judged. In order to restrain the instability oscillation induced by the system Hopf bifurcation, a linear and nonlinear feedback controller is constructed, in which the uncoiling speed of the uncoiler is selected as the control variable, and variations of tensions at entry and exit as well as system vibration responses are chosen as feedback variables. On this basis, the linear control of the controller is studied using the Hopf bifurcation algebraic criterion. And the nonlinear control of the controller is studied according to the center manifold theorem and the normal form theory. The results show that the system stability domain can be expanded by reducing the linear gain coefficient. Through choosing an appropriate nonlinear gain coefficient, the occurring of the system subcritical bifurcation can be suppressed. And system vibration amplitudes reduce as the increase of the nonlinear gain coefficient. Therefore, introducing the linear and nonlinear feedback controller into the system can improve system dynamic characteristics significantly. The production efficiency and the product quality can be guaranteed as well.

Stability and synchronization of fractional-order memristive neural networks with multiple delays

The paper presents theoretical results on the global asymptotic stability and synchronization of a class of fractional-order memristor-based neural networks (FMNN) with multiple delays. First, the asymptotic stability of fractional-order (FO) linear systems with single or multiple delays is discussed. Delay-independent stability criteria for the two types of systems are established by using the maximum modulus principle and the spectral radii of matrices. Second, new testable algebraic criteria for ensuring the existence and global asymptotic stability of the system equilibrium point are obtained by employing the Kakutani’s fixed point theorem of set-valued maps, the comparison theorem, and the stability criterion for FO linear systems with multiple delays. Third, the synchronization criterion for FMNN is presented based on the linear error feedback control. Finally, numerical examples are given demonstrating the effectiveness of the proposed results.

Clustering resonance effects in the electronic energy spectrum of tridiagonal Fibonacci quasicrystals

In this work, we show that the fundamental structure of the electronic energy spectrum of binary Fibonacci quasicrystals can be decomposed in terms of two main contributions, stemming from two related characteristic symmetries. The algebraic approach, we introduce allows us for a unified and systematic description of the energy spectrum finer structure details in terms of block matrices commutators properties, within the framework of a renormalization approach based on transfer matrices. Close analytical expressions can be explicitly obtained by exploiting some algebraic properties of these blocks commutators. In particular, the overall main features of the electronic energy spectrum structure are related to the roots of a series of polynomials of the form _[PF_(j)](E), where the subscript denotes the degree of the polynomial, which is given by a Fibonacci number F_j. These polynomials, in turn, are derived in a systematic way from commutators involving progressively longer palindromic fundamental blocks containing two types of basic renormalized matrices. In this way, we can classify the resonance energies defining the different fragmentation patterns of the energy spectrum on the basis of purely algebraic criteria. The transmission coefficient of these resonant states is always bounded below, although their related Landauer conductance values may range from highly conductive to highly resistive ones, depending on the system length. The obtained results significantly contribute to gain a better understanding of the symmetry principles governing the fragmentation process leading to the characteristic nested structure of the energy spectrum.

A note on: Quasi-linear quotients and shellability of pure simplicial complexes

ABSTRACTThe purpose of this note is to point out a careless error in the algebraic criterion of shellability of a pure simplicial complex Δ given in [1].

Deterministic submanifolds and analytic solution of the quantum stochastic differential master equation describing a monitored qubit

This paper studies the stochastic differential equation (SDE) associated with a two-level quantum system (qubit) subject to Hamiltonian evolution as well as unmonitored and monitored decoherence channels. The latter imply a stochastic evolution of the quantum state (density operator), whose associated probability distribution we characterize. We first show that for two sets of typical experimental settings, corresponding either to weak quantum non-demolition measurements or to weak fluorescence measurements, the three Bloch coordinates of the qubit remain confined to a deterministically evolving surface or curve inside the Bloch sphere. We explicitly solve the deterministic evolution, and we provide a closed-form expression for the probability distribution on this surface or curve. Then we relate the existence in general of such deterministically evolving submanifolds to an accessibility question of control theory, which can be answered with an explicit algebraic criterion on the SDE. This allows us to show that, for a qubit, the above two sets of weak measurements are essentially the only ones featuring deterministic surfaces or curves.

Finite-time synchronization of inertial memristive neural networks with time-varying delays via sampled-date control

In this paper, the finite-time synchronization issues for inertial memristive neural networks with time-varying delay using hybrid feedback controller with sampled-date term are investigated. Firstly, by constructing a proper variable substitution,the original system is transformed into the first order differential system. Next, constructing a useful Lyapunov function, a suitable controller and using inequality techniques, some new testable algebraic criterion are obtained for ensuring the finite-time synchronization goal. Moreover, we discussed deeply on the relation between the parameter ξi and estimated value of settling time in the different cases, and we can obtain the minimum estimated value of settling time. Finally, the effectiveness of the theoretical results have been illustrated via two numerical examples.