Higher-order Case Research Articles

Backward error and perturbation bounds for high order Sylvester tensor equation

In this article, we investigate the backward error and perturbation bounds for the high order Sylvester tensor equation (STE). The bounds of the backward error and three types of upper bounds for the perturbed STE with or without dropping the second order terms are presented. The classic perturbation results for the Sylvester equation are extended to the high order case.

A generalization of Fueter's theorem in Dunkl–Clifford analysis

In this article, we first suggest an alternative approach to extend the original Fueter's theorem in Dunkl–Clifford analysis to a version of the higher order case. We then use this result to prove a generalized version of Fueter's theorem with an extra homogeneous Dunkl-monogenic polynomial P n (x 0, x ) in instead of the classical factor P n ( x ) in ℝ d .

Defect-based local error estimators for splitting methods, with application to Schrödinger equations, Part II. Higher-order methods for linear problems

In this work, defect-based local error estimators for higher-order exponential operator splitting methods are constructed and analyzed in the context of time-dependent linear Schrödinger equations. The technically involved procedure is carried out in detail for a general three-stage third-order splitting method and then extended to the higher-order case. Asymptotical correctness of the a posteriori local error estimator is proven under natural commutator bounds for the involved operators, and along the way the known (non)stiff order conditions and a priori convergence bounds are recovered. The theoretical error estimates for higher-order splitting methods are confirmed by numerical examples for a test problem of Schrödinger type. Further numerical experiments for a test problem of parabolic type complement the investigations.

A new DOA estimation approach using Volterra signal model

In this paper, we proposed a new direction-of-arrival estimation approach using Volterra signal model in spatial domain. The new technique basically uses additionally the second-order terms of Volterra series to produce augmented Volterra snapshots, the extension for higher-order case is straightforward. The resolution of the proposed method is high comparing with the standard multiple signal classification (MUSIC) algorithm and the 2q-MUSIC algorithm. Simulation results are presented to show the performance of the proposed technique.

High order recombination and an application to cubature on Wiener space

Particle methods are widely used because they can provide accurate descriptions of evolving measures. Recently it has become clear that by stepping outside the Monte Carlo paradigm these methods can be of higher order with effective and transparent error bounds. A weakness of particle methods (particularly in the higher order case) is the tendency for the number of particles to explode if the process is iterated and accuracy preserved. In this paper we identify a new approach that allows dynamic recombination in such methods and retains the high order accuracy by simplifying the support of the intermediate measures used in the iteration. We describe an algorithm that can be used to simplify the support of a discrete measure and give an application to the cubature on Wiener space method developed by Lyons and Victoir [Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 460 (2004) 169-198].

Fundamental Solutions for Cardinal Interpolation with Reduced Compact Support and Improved Attenuation

Riemenschneider and Shen have constructed compactly supported fundamental solutions for cardinal interpolation from integer shifts of B-splines of orders m and m + 1. In this article, we prove that the symbol functions of splines with order m and m + 2 are co-prime up to a power of z. This allows us to construct fundamental solutions using splines of order m, m + 1, and m + 2 that exhibit improved attenuation in low order cases, and reduced support for m ≥ 6. We present evidence that in higher order cases these new fundamental functions exhibit a 25% reduction in their support lengths compared to the fundamental solutions constructed from B-splines of orders m and m + 1.

Higher-order duality for a class of nondifferentiable multiobjective programming problems involving generalized type I and related functions

This paper extends the class of generalized type I functions introduced by Aghezzaf and Hachimi(2000) to the context of higher-order case and formulate a number of higher-order duals to a non-differentiable multi-objective programming problem and establishes higher-order duality results under the higher-order generalized type I functions introduced in the present paper. A special case that appears repeatedly in the literature is that the support function is the square root of a positive semi-definite quadratic form. This and other special cases can be readily generated from these results.

Garch-Type Models and Performance of Information Criteria

GARCH model has gained popularity during the last two decades, because of their ability to capture non-linear dynamics in the real life data which we often observe especially in financial markets. This paper discuss four common information criteria (AIC, AICc, BIC and HQ) and their ability of correct selection in the presence of GARCH effect, based on their probability of correct selection as a measure of performance. The investigation has been performed using Monte Carlo simulation of conditional variance GARCH processes with six different kinds of DGPs including; GARCH(p, q), p = 1, 2 and q = 1, 2, GARCH(1, 1)-Leverage, and GARCH(1, 1)-Spillover. All these models are further simulated with different parameter combinations to study the possible effect of different volatility structures on these information criteria. We notice an impact from the volatility structure of time series on the performance of these criteria. Specially for higher order GARCH processes, the BIC and HQ criterion select the lower order process at small sample size. Furthermore, we observe different behavior of information criteria for changing the parameter set in higher order case. Moreover, the influence of sample size, having an impact on the the performance of these criteria towards correct selection, is observed.

A comparative study of approximate symmetry and approximate homotopy symmetry to a class of perturbed nonlinear wave equations

A comparative study of approximate symmetry and approximate homotopy symmetry to a class of perturbed nonlinear wave equations is performed. First, complete infinite-order approximate symmetry classification of the equation is obtained by means of the method originated by Fushchich and Shtelen. An optimal system of one-dimensional subalgebras is derived and used to construct general formulas of approximate symmetry reductions and similarity solutions. Second, we study approximate homotopy symmetry of the equation and construct connections between the two symmetry methods for the first-order and higher-order cases, respectively. The series solutions derived by the two methods are compared.

Periodic Solutions and Invariant Manifolds for an Even-Order Differential Equation with Power Nonlinearity

The PDE Δmu = u|u|p–1, with \({m\in\mathbb{N}}\) and \({p\in(1,\infty)}\), serves as a paradigm for a large class of higher-order elliptic equations with power-like nonlinearities. In studying radially symmetric solutions of this PDE and their asymptotic behavior, it is of critical importance to understand the dynamics of the associated ODE, u(2m) = u|u|p–1, which is the subject of the present paper. Most solutions of the ODE blow up in finite time, diverging to either ∞ or −∞. In the phase space \({\mathbb{R}^{2m}}\), the orbits of these two types of solutions are separated by a (2m−1)-dimensional manifold \({\fancyscript{M}}\), which is unordered and homeomorphic to each of the coordinate hyperplanes under orthogonal projection. Solutions with orbits on \({\fancyscript{M}}\) do not blow up or, else, are unbounded from above and from below (oscillatory blow-up). In the second-order case, \({\fancyscript{M}}\) coincides with the stable manifold of the trivial equilibrium. In the fourth-order case, which is our main focus, \({\fancyscript{M}}\) contains a two-dimensional manifold \({\fancyscript{M}_0}\) of periodic orbits. There exists a unique periodic solution \({\bar u}\) with \({\bar u(0)=1}\) and \({\bar u'(0)=0}\), which shares most of the symmetry properties of the common cosine; all nontrivial periodic solutions are obtained from \({\bar u}\) via scaling and phase-shifting. Every solution with orbit on \({\fancyscript{M}\setminus\fancyscript{M}_0}\) converges to either the trivial equilibrium or one of the nontrivial periodic orbits; oscillatory blow-up does not occur. In higher-order cases, the structure of the manifold \({\fancyscript{M}}\) remains largely open.

Universal recursive formulae for Q-curvatures

We formulate and discuss two conjectures concerning recursive formulae for Branson's Q-curvatures. The proposed formulae describe all Q-curvatures on manifolds of all even dimensions in terms of respective lower order Q-curvatures and lower order GJMS-operators. They are universal in the dimension of the underlying space. The recursive formulae are generated by an algorithm which rests on the theory of residue families of [Juhl, Progr. Math. 275, 2009]. We attempt to resolve the algorithm by formulating a conjectural description of the coe‰cients in the recursive formulae in terms of interpolation polynomials associated to compositions of natural numbers. We prove that the conjectures cover Q4 and Q6 for general metrics, and Q8 for conformally flat metrics. The result for Q8 is proved here for the first time. Moreover, we display explicit (conjectural) formulae for Q-curvatures of order up to 16, and test high order cases for round spheres and Einstein metrics.

On higher order Bragg resonance of water waves by bottom corrugations

The exact theory of linearized water waves in a channel of indefinite length with bottom corrugations of finite amplitude (Howard & Yu, J. Fluid Mech., vol. 593, 2007, pp. 209–234) is extended to study the higher order Bragg resonances of water waves occurring when the corrugation wavelength is close to an integer multiple of half a water wavelength. The resonance tongues (ranges of water-wave frequencies) are given for these higher order cases. Within a resonance tongue, the wave amplitude exhibits slow exponential modulation over the corrugations, and slow sinusoidal modulation occurs outside it. The spatial rate of wave amplitude modulation is analysed, showing its quantitative dependence on the corrugation height, water-wave frequency and water depth. The effects of these higher order Bragg resonances are illustrated using the normal modes of a rectangular tank.

Reply to “Comments on 'Broadband Matching Limitations for Higher Order Spherical Modes'”

The authors are grateful to Prof. Kogan for calling to their attention the work in (Kogan, 1980), of which they were unaware. The approach he describes may have an advantage over their approach, since it appears that it requires less numerical work to formulate the set of simultaneous equations to be solved. They also thank Prof. Kogan for pointing out the need to select the proper branch in the TMoi case; in the higher order cases, we believe that the optimization should, in principle, find the minimum solution to Fano's equations.

Coq formalization of the higher-order recursive path ordering

The recursive path ordering (RPO) is a well-known reduction ordering introduced by Dershowitz, that is useful for proving termination of term rewriting systems (TRSs). Jouannaud and Rubio generalized this ordering to the higher-order case thus creating the higher-order recursive path ordering (HORPO). They proved that this ordering can be used for proving termination of higher-order TRSs, which essentially comes down to proving well-foundedness of HORPO. This result entails well-foundedness of RPO and termination of simply typed lambda calculus (as the β-reduction relation is included in HORPO). This paper describes our undertaking of providing a complete, axiom-free, fully constructive formalization of those results in the proof assistant Coq. The formalization can be divided into three parts: finite multisets and two variants of multiset extensions of a relation, simply typed lambda calculus with termination of β-reduction as the main result, HORPO with a proof of its well-foundedness; also decidability of HORPO has been proved and, due to its constructive nature, a certified algorithm to verify whether two terms can be oriented with HORPO can be extracted from the proof.

Higher-order term indexing using substitution trees

We present a higher-order term indexing strategy based on substitution trees for simply typed lambda-terms. There are mainly two problems in adapting first-order indexing techniques. First, many operations used in building an efficient term index and retrieving a set of candidate terms from a large collection are undecidable in general for higher-order terms. Second, the scoping of variables and binders in the higher-order case presents challenges. The approach taken in this article is to reduce the problem to indexing linear higher-order patterns, a decidable fragment of higher-order terms, and delay solving terms outside of this fragment. We present insertion of terms into the index based on computing the most specific linear generalization of two linear higher-order patterns, and retrieval based on matching two linear higher-order patterns. Our theoretical framework maintains that terms are in βη-normal form, thereby eliminating the need to renormalize and raise terms during insertion and retrieval. Finally, we prove correctness of our presented algorithms. This indexing structure is implemented as part of the Twelf system to speed up the execution of the tabled higher-logic programming interpreter.

Accuracy preserving limiter for the high-order accurate solution of the Euler equations

Higher-order finite-volume methods have been shown to be more efficient than second-order methods. However, no consensus has been reached on how to eliminate the oscillations caused by solution discontinuities. Essentially non-oscillatory (ENO) schemes provide a solution but are computationally expensive to implement and may not converge well for steady-state problems. This work studies the extension of limiters used for second-order methods to the higher-order case. Requirements for accuracy and efficient convergence are discussed. A new limiting procedure is proposed. Ringleb’s flow problem is used to demonstrate that nearly nominal orders of accuracy for schemes up to fourth-order can be achieved in smooth regions using the new limiter. Results for the fourth-order accurate solution of transonic flow demonstrates good convergence properties and significant qualitative improvement of the solution relative the second-order method. The new limiter can also be successfully applied to reduce the dissipation of second-order schemes with minimal sacrifices in convergence properties relative to existing approaches.

The solution of the recurrence relation f n (t)=a n (t) f n−1(t)−b n (t)(∂/∂t) f n−1(t)

Many linear recurrence relations for combinatorial numbers depending on two indices – like, e.g. the Stirling numbers – can be transformed into a sequence of linear differential equations (of first order) for the corresponding generating functions. In this paper, the most general sequence of such differential equations is considered where the coefficient functions are assumed to be analytic. Using an Ansatz of factoring the sought-for solution into the product of two functions each satisfying a particular associated differential equation, an explicit solution is derived. Some concrete examples are treated. Furthermore, first results for sequences of linear differential equations of second-order are presented and the difficulties of treating the higher order case within the above mentioned method are discussed.

SIMPLEST NORMAL FORMS FOR PLANAR SYSTEMS ON EQUILIBRIUM MANIFOLDS

Equilibrium manifold is a manifold that consists of equilibrium points. Planar systems with one-dimensional equilibrium manifolds are considered in this paper. First, for such planar systems, a unified equation with the horizontal axis as the equilibrium curve is formulated. Then, according to the corresponding linearized systems, different cases are discussed: For the nondegenerate case, the simplest normal form of a system with simplified Bogdanov–Takens singularities is obtained; for the general first-order degenerative case, the simplest normal forms are completely characterized; finally, for the general higher-order degenerative case, deduction of the simplest normal form is illustrated.

Fourier–Bessel-type series: The fourth-order case

The structured higher-order Bessel-type linear ordinary differential equations were first discovered in 1994. There is a denumerable infinity of these higher-order equations, all of then of even-order. These differential equations possess many of the properties of the classical second-order Bessel differential equation, but these higher-order cases bring remarkable new analytic structures. In many ways it is sufficient to study the properties of the fourth-order Bessel-type differential equation to be able to assess the corresponding properties of the sixth-and higher-order cases. This paper follows a number of earlier papers devoted to the study of the fourth-order case. These publications show the connections between the special function properties of solutions of the differential equation, and the properties of linear differential operators generated by the associated linear differential expression in certain weighted Lebesgue, and Lebesgue–Stieltjes function spaces. To follow the earlier papers on the study of the fourth-order Bessel-type differential equation, this present paper determines the form of the Fourier–Bessel-type series which best extends the classical theory of the second-order Fourier–Bessel series. In fact the Fourier–Bessel-type series are based on a new orthogonal system in terms of the regular eigensolutions of the fourth-order Bessel-type equation. The corresponding eigenvalues are obtained by restricting the spectral parameter to the zeros of an analytic function arising already in the Dini boundary conditions.

Universal Piecewise Linear Prediction Via Context Trees

This paper considers the problem of piecewise linear prediction from a competitive algorithm approach. In prior work, prediction algorithms have been developed that are universal with respect to the class of all linear predictors, such that they perform nearly as well, in terms of total squared prediction error, as the best linear predictor that is able to observe the entire sequence in advance. In this paper, we introduce the use of a tree, to compete against a doubly exponential number of piecewise linear (affine) models. We use the context tree to achieve the total squared prediction error performance of the best piecewise linear model that can choose both its partitioning of the regressor space and its real-valued prediction parameters within each region of the partition, based on observing the entire sequence in advance, uniformly, for every bounded individual sequence. This performance is achieved with a prediction algorithm whose complexity is only linear in the depth of the context tree per prediction. Upper bounds on the regret with respect to the best piecewise linear predictor are given for both the scalar and higher order case, and lower bounds on the regret are given for the scalar case. An explicit algorithmic description and examples demonstrating the performance of the algorithm are given.