Class Of Second Order Research Articles

Algebraic Conditions for Conformal Superintegrability in Arbitrary Dimension

We consider second order (maximally) conformally superintegrable systems and explain how the definition of such a system on a (pseudo-)Riemannian manifold gives rise to a conformally invariant interpretation of superintegrability. Conformal equivalence in this context is a natural extension of the classical (linear) Stäckel transform, originating from the Maupertuis-Jacobi principle. We extend our recently developed algebraic geometric approach for the classification of second order superintegrable systems in arbitrarily high dimension to conformally superintegrable systems, which are presented via conformal scale choices of second order superintegrable systems defined within a conformal geometry. For superintegrable systems on constant curvature spaces, we find that the conformal scales of Stäckel equivalent systems arise from eigenfunctions of the Laplacian and that their equivalence is characterised by a conformal density of weight two. Our approach yields an algebraic equation that governs the classification under conformal equivalence for a prolific class of second order conformally superintegrable systems. This class contains all non-degenerate examples known to date, and is given by a simple algebraic constraint of degree two on a general harmonic cubic form. In this way the yet unsolved classification problem is put into the reach of algebraic geometry and geometric invariant theory. In particular, no obstruction exists in dimension three, and thus the known classification of conformally superintegrable systems is reobtained in the guise of an unrestricted univariate sextic. In higher dimensions, the obstruction is new and has never been revealed by traditional approaches.

EXISTENCE OF PERIODIC SOLUTIONS FOR TWO CLASSES OF SECOND ORDER <inline-formula><tex-math id="M1">$ P $</tex-math></inline-formula>-LAPLACIAN DIFFERENTIAL EQUATIONS

In this paper, by using the Man$ \acute{a} $sevich-Mawhin theorem on continuity of the topological degree, we prove the existence of periodic solutions for two classes of second order $ p $-Laplacian polynomial differential equations. Finally, some examples are given to show applications of the conclusions.

On the oscillation and non-oscillation of solutions of forced second order differential equations

The oscillatory behavior of solutions of a class of second order forced non-linear differential equations is discussed. Several oscillation and non-oscillation criteria are established using Riccati transformations technique. Four examples are given to illustrate our results.

A regularity result for a class of non-uniformly elliptic operators

We obtain an explicit Hölder regularity result for viscosity solutions of a class of second order fully nonlinear equations led by operators that are neither convex/concave nor uniformly elliptic.

Stability of a Certain Class of Second Order Switched Systems

Abstract This paper present study of stability of a special two-dimensional system with switching. Its subsystems have second order and are Hurwitz. The first subsystem is linear system presented in Frobenius form and the second one has the same eigenvalues as first one but has swapped states and corresponding realization. The considered problem is to find all permissible eigenvalues that provide asymptotic stability to the switched system under any switching signal and non-zero initial condition. The study is based on common Lyapunov functions which existence is checked using known approach that consists in studying the stability of a convex combination of the subsystems matrices.

EXISTENCE AND MULTIPLICITY OF POSITIVE PERIODIC SOLUTIONS FOR A CLASS OF SECOND ORDER DAMPED FUNCTIONAL DIFFERENTIAL EQUATIONS WITH MULTIPLE DELAYS

<abstract> By using the Krasnoselskii fixed point theorem, sufficient conditions are obtained for the existence and multiplicity of positive periodic solutions for a class of second order damped functional differential equations with multiple delays. Our results are a further expansion of the previous research results. </abstract>

Second order estimates for fully nonlinear elliptic equations with gradient terms on closed Riemannian manifolds

Establishing a priori estimates for second derivatives is a key ingredient in the study of fully nonlinear elliptic equations,which is the focus of this paper. We consider a class of second order fully nonlinear equations on closed Riemannian manifolds. In terms of the tangent cone at infinity to the level sets of an associated function which may depend on the solution and its gradient, we introduce a condition to derive second order estimates for solutions of the equation.

Edge photocurrent driven by terahertz electric field in bilayer graphene

We report on the observation of edge electric currents excited in bi-layer graphene by terahertz laser radiation. We show that the current generation belongs to the class of second order in electric field phenomena and is controlled by the orientation of the THz electric field polarization plane. Additionally, applying a small magnetic field normal to the graphene plane leads to a phase shift in the polarization dependence. Increasing the magnetic field strength, the current starts to exhibit 1/B-magnetooscillations with a period consistent with that of the Shubnikov-de-Haas effect and amplitude by an order of magnitude larger as compared to the current at zero magnetic field measured under the same conditions. The microscopic theory developed shows that the current is formed in the edges vicinity limited by the mean-free path of carriers and the screening length of the high-frequency electric field. The current originates from the alignment of the free carrier momenta and dynamic accumulation of charge at the edges, where the P-symmetry is naturally broken. The observed magnetooscillations of the photocurrent are attributed to the formation of Landau levels.

Analysis About a Class of Second Order Damping Dynamic Model Based on The Runge-Kutta Algorithm

Analysis About a Class of Second Order Damping Dynamic Model Based on The Runge-Kutta Algorithm

ANALYSIS OF STABILITY OF ONE CLASS OF NUMERICAL METHODS OF SECOND ORDER

The paper deals with the analysis of the stability of a class of second order combined numerical methods based on the trapezoid method and the difference formula, what obtained by the author. The stability conditions for this class of methods are obtained by the example of conservative systems.

Relationship Between Adherence to Anti-Diabetic, Anti-Hypertensive, and Lipid Lowering Medications and Healthcare Utilization and Expenditure in a Medicare Supplemental Sample

A new discrete spline method is developed to solve the two point fractional Bagley–Torvik equation.The method is based on discrete spline function and a nonstandard Grünwald–Letnikov difference (NSGD) and the weighted and shifted Grünwald–Letnikov difference (WSGD) operators to approximate the fractional derivative. Bounds for Grünwald–Letnikov weights are considered. Convergence analysis is discussed and a class of second order and third order methods are obtained. Illustrative examples are presented to validate the practical usefulness of the methods.

一类二阶非线性微分方程解的零点存在性

一类二阶非线性微分方程解的零点存在性

Periodic solutions of discontinuous damped Duffing equations

In this paper, we are concerned with the problem of periodic solutions for a class of second order periodically forced nonlinear damped Duffing equations with a discontinuity line x′′+cx′+g(x)=e(t), where g(x) is discontinuous and e∈C0(R,R) is T-periodic. We first show that the periodic solutions can only be harmonic and subharmonic solutions. Secondly, note that g(x) is an abstract function, so the explicit solutions cannot be computed and then known theory cannot be applied directly. Therefore, we give geometric properties of solutions for the discontinuous forced damped Duffing equation. Then by the Poincaré–Bohl theorem we investigate the existence and uniqueness of crossing harmonic and subharmonic solutions of the equation.

Fractional Order Sliding Mode Control of a Class of Second Order Perturbed Nonlinear Systems: Application to the Trajectory Tracking of a Quadrotor

A Fractional Order Sliding Mode Control (FOSMC) is proposed in this paper for an integer second order nonlinear system with an unknown additive perturbation term. A sufficient condition is given to assure the attractiveness to a given sliding surface where trajectory tracking is assured, despite the presence of the perturbation term. The control scheme is applied to the model of a quadrotor vehicle in order to have trajectory tracking in the space. Simulation results are presented to evaluate the performance of the control scheme.

On a Class of Fully Nonlinear Elliptic Equations Containing Gradient Terms on Compact Hermitian Manifolds

AbstractIn this paper we study a class of second order fully nonlinear elliptic equations containing gradient terms on compact Hermitian manifolds and obtain a priori estimates under proper assumptions close to optimal. The analysis developed here should be useful to deal with other Hessian equations containing gradient terms in other contexts.

Strict monotonicity of principal eigenvalues of elliptic operators in [formula omitted] and risk-sensitive control

This paper studies the eigenvalue problem on Rd for a class of second order, elliptic operators of the form Lf=aij∂xi∂xj+bi∂xi+f, associated with non-degenerate diffusions. We show that strict monotonicity of the principal eigenvalue of the operator with respect to the potential function f fully characterizes the ergodic properties of the associated ground state diffusion, and the unicity of the ground state, and we present a comprehensive study of the eigenvalue problem from this point of view. This allows us to extend or strengthen various results in the literature for a class of viscous Hamilton–Jacobi equations of ergodic type with smooth coefficients to equations with measurable drift and potential. In addition, we establish the strong duality for the equivalent infinite dimensional linear programming formulation of these ergodic control problems. We also apply these results to the study of the infinite horizon risk-sensitive control problem for diffusions, and establish existence of optimal Markov controls, verification of optimality results, and the continuity of the controlled principal eigenvalue with respect to stationary Markov controls.

A∞ implies NTA for a class of variable coefficient elliptic operators

We consider a certain class of second order, variable coefficient divergence form elliptic operators, in a uniform domain Ω with Ahlfors regular boundary, and we show that the A∞ property of the elliptic measure associated to any such operator and its transpose imply that the domain is in fact NTA (and hence chord-arc). The converse was already known, and follows from work of Kenig and Pipher [30].

A novel integrated approach for path following and directional stability control of road vehicles after a tire blow-out

The path following and directional stability are two crucial problems when a road vehicle experiences a tire blow-out or sudden tire failure. Considering the requirement of rapid road vehicle motion control during a tire blow-out, this article proposes a novel linearized decoupling control procedure with three design steps for a class of second order multi-input-multi-output non-affine system. The evaluating indicators for controller performance are presented and a performance related control parameter distribution map is obtained based on the stochastic algorithm which is an innovation for non-blind parameter adjustment in engineering implementation. The analysis on the robustness of the proposed integrated controller is also performed. The simulation studies for a range of driving conditions are conducted, to demonstrate the effectiveness of the proposed controller.

Discrete spline methods for solving two point fractional Bagley–Torvik equation

A new discrete spline method is developed to solve the two point fractional Bagley–Torvik equation.The method is based on discrete spline function and a nonstandard Grünwald–Letnikov difference (NSGD) and the weighted and shifted Grünwald–Letnikov difference (WSGD) operators to approximate the fractional derivative. Bounds for Grünwald–Letnikov weights are considered. Convergence analysis is discussed and a class of second order and third order methods are obtained. Illustrative examples are presented to validate the practical usefulness of the methods.

Dynamics of second order in time evolution equations with state-dependent delay

We deal with a class of second order in time nonlinear evolution equations with state-dependent delay. This class covers several important PDE models arising in the theory of nonlinear plates. We first prove well-posedness in a certain space of functions which are C1 in time. In contrast with the first order models with discrete state-dependent delay this result does not require any compatibility conditions. The solutions constructed generate a dynamical system in a C1-type space over delay time interval. Our main result shows that this dynamical system possesses compact global and exponential attractors of finite fractal dimension. To obtain this result we adapt the recently developed method of quasi-stability estimates.