Quadratic Integrals Research Articles

Modeling Methodology Suitable for Converters Integrating Renewables: The QMQI Method

This paper presents a new approach for time-domain analyses using model quadratization quadratized model and subsequent quadratic integration of the quadratized dynamic-models quadratized model-quadratic integration (QMQI) method. The modeling methodology is suitable for analysis of power systems with nonlinear components and switching subsystems. The quadratic integration (QI) method has been demonstrated to be more numerically stable, robust, and accurate than trapezoidal integration (TI), one of the most popularly used methods for power transient analyses. The model quadratization enables nonlinearities to be modeled with equations of nonlinearities no higher than second order without approximations or simplifications. Upon QI, the resulting companion form is a model with nonlinearities no higher than second order (quadratic companion form). Newton's method is employed for the solution of the quadratic companion form equations. The numerical convergence of the overall method is robust and fast. The performance of the QMQI method is quantified with a couple of systems of nonlinear components and power electronics. The method can be expanded to cubic integration (CI) as well as higher-order integration methods. This paper compares the QMQI method with the TI and the CI, for the purpose of determining the pros and cons as higher (or lower)-order integration methods are used.

The On-Line Monitoring Method for Drainage Wire Galloping in High Voltage Overhead Power Transmission Line

In order to analysis wave characteristic of drainage wire, considering the shortcoming of existed measuring systems, including lack of accuracy and heavy load of data transmission, and combining with the PDR system which has been widely used in absolute position of a pedestrian, the on-line monitoring method for drainage wire galloping is proposed, which is based on accelerometer and gyroscope. Min-vector of angular velocity is applied to capture the onset/end of each gallop. ZUPT algorithm is implemented to correct the galloping velocity at onset when errors existed. Cluster sampling and quadratic integral are used to measure Galloping parameters accurately. The galloping orbit is also displayed dynamically. Finally, practical experiments are carried out. The feasibility and availability of the on-line monitoring method based on PDR system are demonstrated. The method can measure wave amplitude, wave speed and wave trajectory precisely, and offer technical support for in-depth research of drainage wire galloping.

Existence results for nonlinear quadratic integral equations of fractional order in Banach algebra

Abstract We present an existence theorem for at least one continuous solution for a nonlinear quadratic functional integral equation of fractional order. Also, a general quadratic integral of fractional order will be considered.

Quadratic first integrals of kinetic differential equations

Classes of kinetic differential equations are delineated which do have a quadratic first integral, and classes which can not have one. Example reactions corresponding to the obtained kinetic differential equations are shown, and a few figures showing the trajectories of the corresponding systems are also included. Connections to other areas are mentioned and unsolved problems collected. The new results are theoretical, although computational tools are heavily used. Applications from biology and combustion theory will come later.

Canonical quantization of static spherically symmetric geometries

The conditional symmetries of the reduced Einstein–Hilbert action emerging from a static, spherically symmetric geometry are used as supplementary conditions on the wave function. Based on their integrability conditions, only one of the three existing symmetries can be consistently imposed, while the unique Casimir invariant, being the product of the remaining two symmetries, is calculated as the only possible second condition on the wave function. This quadratic integral of motion is identified with the reparametrization generator, as an implication of the uniqueness of the dynamical evolution, by fixing a suitable parametrization of the r-lapse function. In this parametrization, the determinant of the supermetric plays the role of the mesure. The combined Wheeler – DeWitt and linear conditional symmetry equations are analytically solved. The solutions obtained depend on the product of the two "scale factors".

Quantization of the Schwarzschild geometry

The conditional symmetries of the reduced Einstein-Hilbert action emerging from a static, spherically symmetric geometry are used as supplementary conditions on the wave function. Based on their integrability conditions, only one of the three existing symmetries can be consistently imposed, while the unique Casimir invariant, being the product of the remaining two symmetries, is calculated as the only possible second condition on the wave function. This quadratic integral of motion is identified with the reparametrization generator, as an implication of the uniqueness of the dynamical evolution, by fixing a suitable parametrization of the r-lapse function. In this parametrization, the determinant of the supermetric plays the role of the mesure. The combined Wheeler – DeWitt and linear conditional symmetry equations are analytically solved. The solutions obtained depend on the product of the two "scale factors".

On the two-dimensional classical motion of a charged particle in an electromagnetic field with an additional quadratic integral of motion

The problem of quadratic Hamiltonians with an electromagnetic field commuting in the sense of the standard Poisson brackets has been considered. It has been shown that, as in the quantum case, any such pair can be reduced to the canonical form, which makes it possible to construct the complete classification of the solutions in the class of meromorphic solutions for the main function of one variable. The transformation to the canonical form is performed through the change of variables to the Kovalevskaya-type variables, which is similar to that in the theory of integrable tops. This transformation has been considered for the two-dimensional Hamiltonian of a charged particle with an additional quadratic integral of motion.

Conditional symmetries and the canonical quantization of constrained minisuperspace actions: The Schwarzschild case

A conditional symmetry is defined, in the phase space of a quadratic in velocities constrained action, as a simultaneous conformal symmetry of the supermetric and the superpotential. It is proven that such a symmetry corresponds to a variational (Noether) symmetry. The use of these symmetries as quantum conditions on the wave function entails a kind of selection rule. As an example, the minisuperspace model ensuing from a reduction of the Einstein–Hilbert action by considering static, spherically symmetric configurations and r as the independent dynamical variable is canonically quantized. The conditional symmetries of this reduced action are used as supplementary conditions on the wave function. Their integrability conditions dictate, at the first stage, that only one of the three existing symmetries can be consistently imposed. At a second stage one is led to the unique Casimir invariant, which is the product of the remaining two, as the only possible second condition on Ψ. The uniqueness of the dynamical evolution implies the need to identify this quadratic integral of motion to the reparametrization generator. This can be achieved by fixing a suitable parametrization of the r-lapse function, exploiting the freedom to arbitrarily rescale it. In this particular parametrization the measure is chosen to be the determinant of the supermetric. The solutions to the combined Wheeler–DeWitt and linear conditional symmetry equations are found and seen to depend on the product of the two “scale factors”.

Asymptotic Stability of Solutions to a Nonlinear Urysohn Quadratic Integral Equation

Here, we prove the existence of -nondecreasing solution to a nonlinear quadratic integral equation of Urysohn type by applying the technique of weak noncompactness. Also, the asymptotic stability of solutions for that quadratic integral equation is studied.

Higher order superintegrability of separable potentials with a new approach to the Post–Winternitz system

The higher order superintegrability of separable potentials is studied. It is proved that these potentials possess (in addition to the two quadratic integrals) a third integral of higher order in the momenta that can be obtained as the product of powers of two particular rather simple complex functions. Some systems related to the harmonic oscillator, such as the generalized SW system and the TTW system, were studied in previous papers; now a similar analysis is presented for superintegrable systems related to the Kepler problem. In this way, a new proof of the superintegrability of the Post–Winternitz system is presented and the explicit expression for the integral is obtained. Finally, the relations between the superintegrable systems with quadratic constants of motion (separable in several different coordinate systems) and the superintegrable systems with higher order constants of motion are analyzed.

Green function diagonal for a class of heat equations

A construction of the heat kernel diagonal is considered as element of generalized zeta function theory, which gradient at the origin defines determinant of a differential operator in a technique for regularizing quadratic path integral. Some classes of explicit expressions of the Green function in the case of finite-gap potential coefficient of the heat equation are constructed. An algorithm and program for Mathematica are presented for a subclass directly linked to hyperelliptic functions.

Positive continuous solution of a quadratic integral equation of fractional orders

We are concerned here with the existence of a unique positive continuous solution for the quadratic integral of fractional orders 1 1 1 2 0 0 ( ) ( ) ( ) ( ) ( , ( )) . ( , ( )) , ( ) ( ) t t t s t s x t a t f s x s ds f s x s ds t I                 where f1 and f2 are Caratheodory functions. As an application the Cauchy problems of fractional order differential equation * ( ) ( , ( )), 0 D x t f t x t t    with one of the two initial values (0) 0 x  or 1 ( ) 0 I x t    will be studied. Some examples are considered as applications of our results.

О некоторых специальных решениях уравнения Эйзенхарта

In this note we study a 6-dimensional pseudo-Riemannian space V (gij) with the signature [+ + − − −−], which admits projective motions, i.e., continuous transformation groups preserving geodesics. A general method of determining pseudo-Riemannian spaces admitting some nonhomothetic projective groupGr was developed by A.V. Aminova. A.V. Aminova classified all Lorentzian manifolds of dimension greater than three admitting nonhomothetic projective or affine infinitesimal transformations. The problem of classification is not solved for pseudo-Riemannian spaces with arbitrary signature. In order to find a pseudo-Riemannian space admitting a nonhomothetic infinitesimal projective transformation, one has to integrate Eisenhart equation hij,k = 2gijφ,k + gikφ,j + gjkφ,i. Pseudo-Riemannian manifolds for which there exist nontrivial solutions hij = cgij to the Eisenhart equation are called h-spaces. It is known that the problem of describing such spaces depends on the type of the h-space, i.e., on the type of the bilinear form LXgij determined by the characteristic of the λ-matrix (hij−λgij). The number of possible types depends on the dimension and the signature of an h-space In this work we find the metric and determine quadratic first integrals of the corresponding geodesic lines equations for 6-dimensional h-spaces of the type [(21 . . . 1)(21 . . . 1) . . . (1 . . . 1)].

1A2-Q08 A Study of Force Control by Using Dual Arm Robot(Robots for Works (1))

A seven -link dual-manipulator robot with three-fingered hands is proposed with application of vision and impedance controls to produce human-like skills capability for the manipulation of automatic works. First, a stereo camera fixed on top of the robot provides an overview of position of objects placed on a worktable to move the arm closer to the objects. Two case studies have been assigned; in the first case study, the control system of robot hand based on LQI (Linear quadratic integral) control was designed to operate the hand for autonomous work, while the second case, a task of opening a beer bottle was designed to apply this method in the position based impedance control.

The gap phenomenon in the dimension study of finite type systems

Abstract Several examples of gaps (lacunes) between dimensions of maximal and submaximal symmetric models are considered, which include investigation of number of independent linear and quadratic integrals of metrics and counting the symmetries of geometric structures and differential equations. A general result clarifying this effect in the case when the structure is associated to a vector distribution, is proposed.

Elastoplastic boundary element method for shear deformable shells

The boundary element formulation for elastoplastic analysis of shear deformable shells is presented. The numerical implementation is achieved using quadratic elements and domain integrals due to the shell curvature are transformed to the boundary. The integrals involving plastic region are computed using internal cells.

Driven cofactor systems and Hamilton–Jacobi separability

This is a continuation of the work initiated in Sarlet and Vanbiervliet (2008 J. Phys. A: Math. Theor. 41 042001) on the so-called driven cofactor systems, which are partially decoupling second-order differential equations of a special kind. The main purpose in that work was to obtain an intrinsic, geometrical characterization of such systems, and to explain the basic underlying concepts in a brief note. In this paper, we address the more intricate part of the theory. It involves in the first place understanding all details of an algorithmic construction of quadratic integrals and their involutivity. It secondly requires explaining the subtle way in which suitably constructed canonical transformations reduce the Hamilton–Jacobi problem of the (a priori time-dependent) driven part of the system into that of an equivalent autonomous system of Stäckel type.

High Weak Order Methods for Stochastic Differential Equations Based on Modified Equations

Inspired by recent advances in the theory of modified differential equations, we propose a new methodology for constructing numerical integrators with high weak order for the time integration of stochastic differential equations. This approach is illustrated with the constructions of new methods of weak order two, in particular, semi-implicit integrators well suited for stiff (mean-square stable) stochastic problems, and implicit integrators that exactly conserve all quadratic first integrals of a stochastic dynamical system. Numerical examples confirm the theoretical results and show the versatility of our methodology.

Closed-Loop Measurements of Iso-Response Stimuli Reveal Dynamic Nonlinear Stimulus Integration in the Retina

Neurons often integrate information from multiple parallel signaling streams. How a neuron combines these inputs largely determines its computational role in signal processing. Experimental assessment of neuronal signal integration, however, is often confounded by cell-intrinsic nonlinear processes that arise after signal integration has taken place. To overcome this problem and determine how ganglion cells in the salamander retina integrate visual contrast over space, we used automated online analysis of recorded spike trains and closed-loop control of the visual stimuli to identify different stimulus patterns that give the same neuronal response. These iso-response stimuli revealed a threshold-quadratic transformation as a fundamental nonlinearity within the receptive field center. Moreover, for a subset of ganglion cells, the method revealed an additional dynamic nonlinearity that renders these cells particularly sensitive to spatially homogeneous stimuli. This function is shown to arise from a local inhibition-mediated dynamic gain control mechanism.

Phase portraits and invariant straight lines of cubic polynomial vector fields having a quadratic rational first integral

In this paper we classify all cubic polynomial dierential systems having a first integral of degree two. In other words we characterize all the global phase portraits of the cubic polynomial dierential systems having all their orbits contained in conics. We also determine their configurations of invariant straight lines. We show that there are exactly 36 topologically dierent phase