Higher-order Derivative Terms Research Articles

Classical string field mechanics with non-standard Lagrangians

We reformulate the string classical mechanics by substituting the standard Lagrangian by a non-standard exponential Lagrangian where higher-order derivative terms occur naturally in the equations of motion. Our motivation is based on the accumulating evidence that higher-order derivatives play a leading role in string field theories. Since non-standard Lagrangians generate higher-order derivatives in a usual way, it will be of interest to explore their roles in classical string field mechanics. It was observed that replacing standard by non-standard Lagrangians gives another possibility to obtain new aspects which may have interesting physical effects.

Higher Order Derivative Characterization for Fock-Type Spaces

In this paper, we give a characterization for the Fock-type space \({\mathcal{F}_{\alpha}^{\infty}(\mathbb{C}^N)}\) in terms of higher order derivatives of f and behaviors of local integral means of those derivatives. The space \({\mathcal{F}_{\alpha}^{\infty}(\mathbb{C}^N)}\) has the closed subspace \({\mathcal{F}_{\alpha, 0}^{\infty}(\mathbb{C}^N)}\). We also characterize this subspace via higher order derivatives. As an application we study the boundedness and compactness of the extended Cesaro operator Tg on \({\mathcal{F}_{\alpha}^{\infty}(\mathbb{C}^N)}\) and \({\mathcal{F}_{\alpha, 0}^{\infty}(\mathbb{C}^N)}\).

Steady-state bifurcation analysis of a strong nonlinear atmospheric vorticity equation

The atmospheric vorticity equation studied in the present paper is a simplified form of the atmospheric circulation model introduced by Charney and DeVore (1979) [4] on the existence of multiple steady states to the understanding of the persistence of atmospheric blocking. The fluid motion defined by the equation is driven by a zonal thermal forcing and an Ekman friction forcing measured by κ. It is proved that the steady-state solution is globally unique for large κ values while multiple steady-state solutions branch off the basic steady-state solution for κ<κcrit where the critical value κcrit is less than one. Without involvement of viscosity, the equation has strong non-linear property as its non-linear part contains the highest order derivative term. Steady-state bifurcation analysis is essentially based on the compactness, which can be simply obtained for semilinear equations such as the Navier–Stokes equations but is not available for the strong nonlinear vorticity equation in the Euler formulation. Therefore the Lagrangian formulation of the equation is employed to gain the required compactness.

Black holes in higher derivative gravity.

Extensions of Einstein gravity with higher-order derivative terms arise in string theory and other effective theories, as well as being of interest in their own right. In this Letter we study static black-hole solutions in the example of Einstein gravity with additional quadratic curvature terms. A Lichnerowicz-type theorem simplifies the analysis by establishing that they must have vanishing Ricci scalar curvature. By numerical methods we then demonstrate the existence of further black-hole solutions over and above the Schwarzschild solution. We discuss some of their thermodynamic properties, and show that they obey the first law of thermodynamics.

Massive spin-2 particles via embedment of the Fierz–Pauli equations of motion

Here we obtain alternative descriptions of massive spin-2 particles by an embedding procedure of the Fierz–Pauli equations of motion. All models are free of ghosts at the quadratic level, although most of them are of a higher order in derivatives. The models that we obtain can be nonlinearly completed in terms of a dynamic and a fixed metric. They include some f(R) massive gravities recently considered in the literature. In some cases there is an infrared (no derivative) modification of the Fierz–Pauli mass term altogether, with higher order terms in derivatives. The analytic structure of the propagator of the corresponding free theories is not affected by the extra terms in the action, as compared to the usual second order Fierz–Pauli theory.

Neurobiological degeneracy: Supporting stability, flexibility and pluripotentiality in complex motor skill

This paper investigated neurobiological degeneracy of the motor system that emerged as a function of levels of environmental constraint. Fourteen participants performed a breaststroke-swimming task that required them to develop a specific biomechanically expert pattern and in turn provide the basis for a suitable task vehicle to study the functional role of movement variability. Inter-limb coordination was defined based on the computation of continuous relative phase between elbow and knee oscillators. Unsupervised cluster analysis on arm–leg coordination revealed the existence of different patterns of coordination when participants achieved the same task goal under different levels of environmental constraints (i.e. different amounts of forward resistances). In addition, clusters differed in terms of higher order derivatives (e.g., joint angular velocity, joint amplitude), suggesting an effective role for degeneracy in learning by allowing the exploration of the key relationships between motor organization and interacting constraints. There is evidence to suggest that neurobiological degeneracy supports the potential for motor re-organization to enhance motor learning.

High-order spline filter: Design and application to surface metrology

The differences between the transmission characteristics of the standard cubic spline filter and the Gaussian filter lead to different evaluation results even for the same profile. There is indeed an adverse impact on the comparison of measurement results and the applications of the related international standards. A novel high-order spline filter is proposed to resolve this practical problem of approximating the Gaussian filtering characteristic. The design of the new filter is based on an improved variational approach by adding the high order derivative terms to the bending energy part whose structural parameters are determined by the aid of the universal Taylor series, so as to realize the convergence to the function of the Gaussian filter. In addition, a cascade algorithm in terms of the low-order filters is also developed in order to ensure stable performance of the high-order filter. Its effectiveness and application were verified by the experiments.

Fundamentals of higher order stochastic equations

Stochastic processes that correspond to equations of arbitrary differential order are obtained. We show that stochastic paths in a complex extension of the original phase-space allow an implementation of such higher-order derivative terms. The resulting stochastic process is equivalent to the original partial differential equation in the sense of having equivalent analytic moments. However, the correspondence has unusual properties. Only the analytic moments are convergent, while non-analytic moments such as the complex moduli are non-convergent. These results unify previous approaches that transform higher-derivative equations into probabilistic stochastic equations. Larger ensembles are required as time-steps are reduced, giving these equations unusual convergence properties. This type of process is relevant to the question of how to obtain stochastic quantum simulations using the Wigner representation.

Parameter Fitted Scheme for Singularly Perturbed Delay Differential Equations

In this paper, we presented a parameter fitted scheme to solve singularly perturbed delay differential equations of second order with left and right boundary. In this technique, approximating the term containing negative shift by Taylor series, we modified the singularly perturbed delay differential equations. We introduced a fitting parameter on the highest order derivative term of the modified problem. The fitting parameter is to be determined from the scheme using the theory of singular Perturbation. Finally, we obtained a three term recurrence relation that can be solved using Thomas algorithm. The applicability of the method is tested by considering four linear problems (two problems on left layer and two problems on right layer). It is observed that when the delay parameter is smaller than the perturbation parameter, the layer behavior is maintained.

An approximation of anisotropic metrics from higher order interpolation error for triangular mesh adaptation

We propose an efficient algorithm for the numerical approximation of metrics, used for anisotropic mesh adaptation on triangular meshes with finite element computations. We derive the metrics from interpolation error estimates expressed in terms of higher order derivatives, for the Pk-Lagrange finite element, k>1. Numerical examples of mesh adaptation done using metrics computed with our Algorithm, and derived from higher order derivatives as error estimates, show that we obtain the right directions of anisotropy.

Krylov implicit integration factor WENO methods for semilinear and fully nonlinear advection–diffusion–reaction equations

Implicit integration factor (IIF) methods are originally a class of efficient “exactly linear part” time discretization methods for solving time-dependent partial differential equations (PDEs) with linear high order terms and stiff lower order nonlinear terms. For complex systems (e.g. advection–diffusion–reaction (ADR) systems), the highest order derivative term can be nonlinear, and nonlinear nonstiff terms and nonlinear stiff terms are often mixed together. High order weighted essentially non-oscillatory (WENO) methods are often used to discretize the hyperbolic part in ADR systems. There are two open problems on IIF methods for solving ADR systems: (1) how to obtain higher than the second order global time discretization accuracy; (2) how to design IIF methods for solving fully nonlinear PDEs, i.e., the highest order terms are nonlinear. In this paper, we solve these two problems by developing new Krylov IIF-WENO methods to deal with both semilinear and fully nonlinear advection–diffusion–reaction equations. The methods can be designed for arbitrary order of accuracy. The stiffness of the system is resolved well and the methods are stable by using time step sizes which are just determined by the nonstiff hyperbolic part of the system. Large time step size computations are obtained. We analyze the stability and truncation errors of the schemes. Numerical examples of both scalar equations and systems in two and three spatial dimensions are shown to demonstrate the accuracy, efficiency and robustness of the methods.

Higher-order optimality conditions for proper efficiency in nonsmooth vector optimization using radial sets and radial derivatives

We establish both necessary and sufficient optimality conditions of higher orders for various kinds of proper solutions to nonsmooth vector optimization in terms of higher-order radial sets and radial derivatives. These conditions are for global solutions and do not require continuity and convexity assumptions. Examples are provided to show advantages of the results over existing ones in a number of cases.

Restriction estimates via the derivatives of the heat semigroup and connection with dispersive estimates

We consider an abstract non-negative self-adjoint operator H on an L2- space. We derive a characterization for the restriction estimate lldEH(λ)/dλll Lp→Lp ≤ Cλ d/2 (1/p − 1/p')−1 (involving the Radon-Nikodym derivative of the spectral measure) in terms of higher order derivatives of the semigroup e−tH. We provide an alternative proof of a result in [1] which asserts that dispersive estimates imply restriction estimates. We also prove Lp − Lp' estimates for the derivatives of the spectral resolution of H.

High frequency limit for gravitational perturbations of cosmological models in modified gravity theories

In general relativity, it has been shown that the effective gravitational stress-energy tensor for short-wavelength metric perturbations acts just like that for a radiation fluid, and thus, in particular, cannot provide any effects that mimic dark energy. However, it is far from obvious if this property of the effective gravitational stress-energy tensor is a specific nature held only in the Einstein gravity, or holds also in other theories of gravity. In particular, when considering modified gravity theories that involve higher order derivative terms, one may expect to have some non-negligible effects arising from higher order derivatives of short-wavelength perturbations. In this paper, we argue this is not the case at least in the cosmological context. We show that when the background, or coarse-grained metric averaged over several wavelengths has FLRW symmetry, the effective gravitational stress-energy tensor for metric perturbations of a cosmological model in a simple class of f(R) gravity theories, as well as that obtained in the corresponding scalar-tensor theory, takes a similar form to that in general relativity and is in fact traceless, hence acting again like a radiation fluid.

Yang-Mills-Higgs models with higher order derivatives

Models with higher order derivative terms in the kinetic energy appear not only as effective theories, they can be considered as elementary, renormalizable models in their own right. The extension of Higgs mechanism is discussed for Yang-Mills-Higgs models where the kinetic energy of the scalar field contains higher order derivatives. The connection term of the covariant derivatives generates a local potential for the gauge field in the phase with spontaneously broken global gauge symmetry which may break Lorentz symmetry in the vacuum. Relativistic, massless gauge bosons spectrum is found for small fluctuations where the only remnant of the breakdown of the Lorentz symmetry is a shift of the energy induced by the expectation value of the temporal component of the gauge field, reminiscent of a chemical potential.

Higher derivative extension of 6D chiral gauged supergravity

Abstract Six-dimensional (1, 0) supersymmetric gauged Einstein-Maxwell supergravity is extended by the inclusion of a supersymmetric Riemann tensor squared invariant. Both the original model as well as the Riemann tensor squared invariant are formulated off-shell and consequently the total action is off-shell invariant without modification of the supersymmetry transformation rules. In this formulation, superconformal techniques, in which the dilaton Weyl multiplet plays a crucial role, are used. It is found that the gauging of the U(1) R-symmetry in the presence of the higher-order derivative terms does not modify the positive exponential in the dilaton potential. Moreover, the supersymmetric Minkowski4 × S 2 compactification of the original model, without the higher-order derivatives, is remarkably left intact. It is shown that the model also admits non-supersymmetric vacuum solutions that are direct product spaces involving de Sitter spacetimes and negative curvature internal spaces.

Finite Element Analysis for Mass-Lumped Three-Step Taylor Galerkin Method for Time Dependent Singularly Perturbed Problems with Exponentially Fitted Splines

The motive of the current study is to derive pointwise error estimates for the three-step Taylor Galerkin finite element method for singularly perturbed problems. Pointwise error estimates have not been derived so far for the said method in the finite element framework. Singularly perturbed problems represent a class of problems containing a very sharp boundary layer in their solution. A small parameter called singular perturbation parameter is multiplied with the highest order derivative terms. When this parameter becomes smaller and smaller, a boundary layer occurs and the solution changes very abruptly in a very small portion of the domain. Because of this sudden change in the nature of the solution, it becomes very difficult for the numerical methods to capture the solution accurately specially in the boundary layer region. In the present study finite element analysis has been carried out for such one-dimensional singularly perturbed time dependent convection-diffusion equations. Exponentially fitted splines have been used for the three-step Taylor Galerkin finite element method to converge. Pointwise error estimates have been derived for the method and it is shown that the method is conditionally convergent of first order accurate in space and third order accurate in time. Numerical results have been presented for both the linear and nonlinear problems.

U(1) symmetry and elimination of spin-0 gravitons in Horava-Lifshitz gravity without the projectability condition

In this paper, we show that the spin-0 gravitons appearing in Horava-Lifshitz gravity without the projectability condition can be eliminated by extending the gauge symmetries of the foliation-preserving diffeomorphisms to include a local U(1) symmetry. As a result, the problems of stability, ghost, strong coupling, and different speeds in the gravitational sector are automatically resolved. In addition, with the detailed balance condition softly breaking, the number of independent coupling constants can be significantly reduced (from more than 70 down to 15), while the theory is still UV complete and possesses a healthy IR limit, whereby the prediction powers of the theory are considerably improved. The strong coupling problem in the matter sector can be cured by introducing an energy scale $M_{*}$, so that $M_{*} < \Lambda_{\omega}$, where $M_{*}$ denotes the suppression energy of high order derivative terms, and $\Lambda_{\omega}$ the would-be strong coupling energy scale.

Strong coupling in nonrelativistic general covariant theory of gravity

We study the strong coupling problem in the Horava-Melby-Thompson setup of the Horava-Lifshitz gravity with an arbitrary coupling constant $\lambda$, generalized recently by da Silva, where $\lambda$ describes the deviation of the theory in the infrared from general relativity that has $\lambda_{GR} = 1$. We find that a scalar field in the Minkowski background becomes strong coupling for processes with energy higher than $\Lambda_{\omega} [\equiv (M_{pl}/c_1)^{3/2} M_{pl}|\lambda - 1|^{5/4}]$, where generically $c_1 \ll M_{pl}$. However, this problem can be cured by introducing a new energy scale $M_{*}$, so that $M_{*} < \Lambda_{\omega}$, where $M_{*}$ denotes the suppression energy of high order derivative terms of the theory.

A numerical study of singularly perturbed generalized Burgers–Huxley equation using three-step Taylor–Galerkin method

In the present work, a numerical study has been carried out for the singularly perturbed generalized Burgers–Huxley equation using a three-step Taylor–Galerkin finite element method. A Burgers–Huxley equation represents the traveling wave phenomena. In singular perturbed problems, a very small positive parameter, ϵ , called the singular perturbation parameter is multiplied with the highest order derivative term. As this parameter tends towards zero, the problem exhibits boundary layers. The traditional methods fail to capture the boundary layers when ϵ becomes very small. In this paper a three-step Taylor–Galerkin finite element method is used to capture the boundary layers. The method is third-order accurate and has inbuilt upwinding. Stability analysis has been carried out and the numerical results show that the method is efficient in capturing the boundary layers.