Moment Polynomial Research Articles

An output feedback method to polynomial stabilization of hybrid pantograph stochastic systems with aperiodic sampling

This paper focuses on the polynomial stabilization problem of hybrid pantograph stochastic systems (HPSSs) with aperiodic sampling in the presence of randomly occurring transmission delay. Firstly, a sufficient condition is provided to ensure the existence, uniqueness and boundedness of solution to such systems. Next, taking advantage of the Lyapunov stability theory and the comparison method, the criteria of pth moment polynomial stability are established for the HPSSs under the feedback control and aperiodic intermittent control strategy, respectively. Finally, the proposed theoretical results of two control methods are simultaneously validated through a numerical example.

Almost sure polynomial stability and stabilization of stochastic differential systems with impulsive effects

In this paper, we first consider the almost surely and pth moment polynomial stability of stochastic differential equations with impulsive effects. Sufficient conditions are given to obtain the polynomial stability. Then the stabilization of stochastic differential systems is studied, which shows that impulse can make stochastic differential equations stable.

Asymptotic stability of neutral stochastic pantograph systems with Markovian switching

ABSTRACT This paper investigates the asymptotic stability of neutral stochastic pantograph systems with Markovian switching (NSPSMS). The stochastic LaSalle theorem has been set up to locate the attractive sets for NSPSMS without linear growth condition. Subsequently, combining the stochastic LaSalle theorem and uniformly continuous theory, almost surely asymptotic stability and pth moment asymptotic stability are analysed. Furthermore, we also present one new criterion on the pth moment polynomial stability and almost surely polynomial stability for NSPSMS, where the coefficients of the delay term are time invariable. Finally, three examples are provided to exhibit the efficiency of the proposed results.

Razumikhin Theorems on Polynomial Stability of Neutral Stochastic Pantograph Differential Equations with Markovian Switching

This paper investigates the polynomial stability of neutral stochastic pantograph differential equations with Markovian switching (NSPDEsMS). Firstly, under the local Lipschitz condition and a more general nonlinear growth condition, the existence and uniqueness of the global solution to the addressed NSPDEsMS is considered. Secondly, by adopting the Razumikhin approach, one new criterion on the qth moment polynomial stability of NSPDEsMS is established. Moreover, combining with the Chebyshev inequality and the Borel–Cantelli lemma, the almost sure polynomial stability of NSPDEsMS is examined. The results derived in this paper generalize the previous relevant ones. Finally, two examples are provided to illustrate the effectiveness of the theoretical work.

Pth moment polynomial input-to-state stability of switched neutral pantograph stochastic hybrid systems with Lévy noise

This article discusses pth () moment polynomial integral input-to-state stability (iISS) and -weighted integral input-to-state stability (-weighted iISS) of the switched neutral pantograph stochastic hybrid systems with Lévy noise (SNPSHSs-LN) provided that all dynamical subsystems are input-to-state stability (ISS). Under the linear growth hypothesis, the criteria of ISS-type characterisation are deduced by using random analysis techniques, Lyapunov stability theory and some useful inequality techniques on the basis of the dynamic conditions of the system itself, which is an improvement on existing results about neutral systems. Finally, the theoretical results are demonstrated with a numerical example.

Model tests on horizontal load-bearing and deformation resistance capacities of pile-bucket foundation in soft seabed soil

Large-diameter pile-bucket foundations are a new type of foundation used for offshore wind turbines. In this study, model tests were performed on pile-bucket foundations to investigate their load-bearing and deformation resistance capacities. The strains of the pile and bucket and the horizontal displacements at the loading point were measured using strain gauges and displacement transducers, respectively. The variations in moments of the pile and bucket with depth were calculated and fitted into polynomial functions of sixth to eighth orders. Based on the moment polynomials, the distributions of the horizontal displacements and soil reaction of the pile and bucket were determined according to the beam bending theory. Improvements in the load-bearing and deformation resistance capacities were quantitatively determined by comparing the test results for the pile-bucket foundation with those of the monopile foundation. The initial stiffness and ultimate horizontal bearing capacity of the pile-bucket foundation were 50% and 20%, respectively, higher than those of the monopile foundation. For sections within the bearing depth of the bucket, the ultimate soil reaction force of the pile-bucket foundation was 20 times that of the monopile foundation. However, a 10%–20% improvement in the ultimate soil reaction was observed for deeper sections.

Stability of numerical solutions for the stochastic pantograph differential equations with variable step size

In this paper, two different techniques are used to research the stability of numerical solutions with variable step size for stochastic pantograph differential equations. Based on the discrete semimartingale convergence theorem, the almost sure polynomial stability and the αth moment polynomial stability of the numerical solutions by the Euler–Maruyama (EM) and the backward Euler–Maruyama (BEM) methods with variable step size have been discussed. By using the discrete Razumikhin-type technique, the global αth moment asymptotically stability and the αth moment polynomial stability of the numerical solutions by the general numerical scheme with variable step size have been researched. Based on the results of the stability of the general numerical scheme, we discuss the stability of two special numerical solutions, namely the EM method and the BEM method. By comparing the two techniques (i.e. the discrete semimartingale convergence theorem and the discrete Razumikhin-type technique) on the research of the moment stability, we can easily get the latter is better. To illustrate the theoretical results, we give some examples to examine the almost sure polynomial stability and the αth moment polynomial stability of the numerical solutions by the EM and BEM schemes with variable step size.

Razumikhin-type theorems on polynomial stability of hybrid stochastic systems with pantograph delay

The main aim of this paper is to investigate the polynomial stability of hybrid stochastic systems with pantograph delay (HSSwPD). By using the Razumikhin technique and Lyapunov functions, we establish several Razumikhin-type theorems on the \begin{document}$ p $\end{document} th moment polynomial stability and almost sure polynomial stability for HSSwPD. For linear HSSwPD, sufficient conditions for polynomial stability are presented.

Razumikhin-type technique on stability of exact and numerical solutions for the nonlinear stochastic pantograph differential equations

In this paper, we establish Razumikhin-type theorems on $$\alpha $$ th moment polynomial stability of exact solution for the stochastic pantograph differential equations, which improves the existing stochastic Razumikhin-type theorems. By using discrete Razumikhin-type technique, we construct conditions for the stability of general numerical scheme of the stochastic pantograph differential equations (SPDEs). The stabilities mainly conclude the global $$\alpha $$ th moment asymptotically stability and $$\alpha $$ th moment polynomial stability. Using the conditions constructed for the stability of the numerical solutions, we discuss the stability of two special numerical methods, namely the Euler–Maruyama method and the backward Euler–Maruyama method. Finally, an example is given to illustrate the consistence with the theoretical results on $$\alpha $$ th moment polynomial stability.

Computing the moment polynomials of the zeta function

We describe a method to accelerate the numerical computation of the coefficients of the polynomials P k ( x ) P_k(x) that appear in the conjectured asymptotics of the 2 k 2k -th moment of the Riemann zeta function. We carried out our method to compute the moment polynomials for k ≤ 13 k \leq 13 , and used these to experimentally test conjectures for the moments up to height 10 8 10^8 .

Uniform asymptotics for the full moment conjecture of the Riemann zeta function

Conrey, Farmer, Keating, Rubinstein, and Snaith, recently conjectured formulas for the full asymptotics of the moments of L -functions. In the case of the Riemann zeta function, their conjecture states that the 2 k -th absolute moment of zeta on the critical line is asymptotically given by a certain 2 k -fold residue integral. This residue integral can be expressed as a polynomial of degree k 2 , whose coefficients are given in exact form by elaborate and complicated formulas. In this article, uniform asymptotics for roughly the first k coefficients of the moment polynomial are derived. Numerical data to support our asymptotic formula are presented. An application to bounding the maximal size of the zeta function is considered.

Local and global features based image retrieval system using orthogonal radial moments

Orthogonal radial moments such as Zernike moments, pseudo Zernike moments, and orthogonal Fourier Mellin moments have been studied extensively in the literature. In conventional methods of moment computation, the entire image intensity function is projected upon orthogonal moment polynomials to compute the global features. In this paper, we provide a novel methodology for the computation of these moments, which formulate them as effective local descriptors. The moment functions of edge images are computed to determine the local change in images, which provide the local aspects of the image. Thus, the obtained local and global moment features are combined to evaluate their performance on the image retrieval system. The moments ZMs, PZMs, and OFMMs, are compared in terms of their image retrieval effectiveness. The experiments are performed on various databases to examine the performance of the system in diverse circumstances such as images affected with noise, partial occlusion, distortion, complex structure, etc. The experiments results reveal that the proposed system outperforms the existing recent approaches to moments computation for image retrieval.

Zeons, Permanents, the Johnson Scheme, and Generalized Derangements

Starting with the zero-square “zeon algebra,” the connection with permanents is shown. Permanents of submatrices of a linear combination of the identity matrix and all-ones matrix lead to moment polynomials with respect to the exponential distribution. A permanent trace formula analogous to MacMahon's master theorem is presented and applied. Connections with permutation groups acting on sets and the Johnson association scheme arise. The families of numbers appearing as matrix entries turn out to be related to interesting variations on derangements. These generalized derangements are considered in detail as an illustration of the theory.

Uniform asymptotics of the coefficients of unitary moment polynomials

Keating and Snaith showed that the 2 k th absolute moment of the characteristic polynomial of a random unitary matrix evaluated on the unit circle is given by a polynomial of degree k 2 . In this article, uniform asymptotics for the coefficients of that polynomial are derived, and a maximal coefficient is located. Some of the asymptotics are given in an explicit form. Numerical data to support these calculations are presented. Some apparent connections between the random matrix theory and the Riemann zeta function are discussed.

Influence of foundation on pillar grounding system's characteristics

PurposeThe purpose of this paper is to estimate influence of the pillar concrete foundation approximated by a semiconducting semi‐sphere on the pillar grounding system.Design/methodology/approachUnknown current distributions and impedance of pillar grounding system, which consists of a ferro armature modelled as a single electrode inside concrete foundation, ring electrode and two earthing conductors, are determined in this paper. Concrete foundation is approximated by a semi‐spherical semiconducting inhomogeneity. The expressions for electric scalar potential are formed using the quasi‐stationary image theory, including a recently proposed Green's function for the point source in the presence of a semiconducting sphere. Based on the quasi‐stationary antenna model, unknown currents are determined solving the system of integral equations using the moment method and polynomial approximation for current distributions.FindingsThe influence of inhomogeneity is not negligible for real values of specific conductivity of the inhomogeneity domain. The same goes for earthing conductor influence, especially the one on electric scalar potential distribution on the ground surface.Practical implicationsBesides the analyzed system, the model is applicable for solving for example a grounding system in the vicinity of large holes in the ground (pond and small lake) filled with water, treated as semi‐sphere or near vertical container (silage and reservoir) having semi‐spherical basis with a lower one buried in the ground.Originality/valueObtained results give possibility to analyze influence of semiconducting ground inhomogenities, which can be treated as semi‐spherically shaped ones, on grounding system. It results in better accordance of projected and real grounding system's characteristics.