Higher-order Case Research Articles

Solutions to generalized Sylvester matrix equation by Schur decomposition

This note deals with the problem of solving the generalized Sylvester matrix equation AV-EVF=BW, with F being an arbitrary matrix, and provides complete general parametric expressions for the matrices V and W satisfying this equation. The primary feature of this approach is that the matrix F is firstly transformed into triangular form by Schur decomposition and then unimodular transformation or singular value decomposition are employed. The results can be easily extended to second order case and high order case and can provide great convenience to the computation and analysis of the solutions to this class of equations, and can perform important functions in many analysis and design problems in control systems theory.

Modeling of Coupled Roll and Yaw Damping of a Floating Body in Waves

A mathematical model is described to investigate the damping moment of weakly nonlinear roll and yaw motions of a floating body in time domain under the action of sinusoidal waves. The mathematical formulation for added mass moment of inertia and damping is presented by approximating time-dependent coefficients and forcing moments when small distortion holds. Using perturbation technique, we obtain orderwise equations wherein the closed-form solution is obtained for zeroth-order case, and for higher-order cases we resort to numerical integration using Runge-Kutta method with adaptive step-size algorithm. In order to analyze the model result, we perform numerical experiment for a vessel of 19190 tons under the beam wave of 1 m height and frequency 0.74 rad/s. Closer inspection in damping analysis reveals that viscous effect becomes significant for roll damping; whereas for yaw damping, contribution from added mass variation becomes significant.

A weak discrete maximum principle for hp-FEM

In this paper, we prove a new discrete maximum principle (DMP) for the one-dimensional Poisson equation discretized by the hp-FEM. While the DMP for piecewise-linear elements is a classical result from the 1970s, no extensions to hp-FEM are available to the present day. Due to a negative result by Höhn and Mittelmann from 1981, related to quadratic Lagrange elements, it was long assumed that higher-order finite elements do not satisfy discrete maximum principles. In this paper we explain why it is not possible to make a straightforward extension of the classical DMP to the higher-order case, and we propose stronger assumptions on the right-hand side under which an extension is possible.

Operational methods and differential equations with applications to initial-value problems

The authors present and develop a general method of operational nature in order to explore and investigate such families of orthogonal polynomials and functions as the classical and generalized Hermite and Laguerre polynomials, as well as their higher-order and multi-index cases. They also consider several applications of these operational techniques to the solutions of a wide class of partial differential equations, including (for example) the heat, wave, and transport equations, some of which involve second-order Laguerre derivatives.

Higher order curvature flows on surfaces

We consider a sixth and an eighth order conformal flow on Riemannian surfaces, which arise as gradient flows for the Calabi energy with respect to a higher order metric. Motivated by a work of Struwe which unified the approach to the Hamilton-Ricci and Calabi flow, we extend the method to these higher order cases. Our results contain global existence and exponentially fast convergence to a constant scalar curvature metric. Uniform bounds on the conformal factor are obtained via the concentration-compactness result for conformal metrics. In the case of the sphere we use the idea of DeTurck's gauge flow to derive bounds up to conformal transformation. We prove exponential convergence by showing that the Calabi energy decreases exponentially fast. The problem of the non-trivial kernel in the evolution of the Calabi energy on the sphere is resolved by using Kazdan-Warner's identity.

Constructively Factoring Linear Partial Differential Operators in Two Variables

We study conditions under which a partial differential operator of arbitrary order $n$ in two variables or ordinary linear differential operator admits a factorization with a first-order factor on the left. The factorization process consists of solving, recursively, systems of linear equations, subject to certain differential compatibility conditions. In the generic case of partial differential operators one does not have to solve a differential equation. In special degenerate cases, such as ordinary differential, the problem is finally reduced to the solution of some Riccati equation(s). The conditions of factorization are given explicitly for second- and, and an outline is given for the higher-order case.

Degenerate nonlinear higher-order elliptic problems in domains with fine-grained boundary

We study the convergence of the solutions of the Dirichlet problem associated to a degenerate nonlinear higher-order elliptic equations in divergence form in variables domains, to a limit solution of the same type problem in a fixed domain, following the methods of the asymptotic expansion developed by Skrypnik [Methods for Analysis of Nonlinear Elliptic Boundary Value Problems, AMS, Providence, RI, 1994] modified to weighted higher-order case.

Vagueness as closeness

This paper presents and defends a definition of vagueness, compares it favourably with alternative definitions, and draws out some consequences of accepting this definition for the project of offering a substantive theory of vagueness. The definition is roughly this: a predicate ‘F’ is vague just in case for any objects a and b, if a and b are very close in respects relevant to the possession of F, then ‘Fa’ and ‘Fb’ are very close in respect of truth. The definition is extended to cover vagueness of many-place predicates, of properties and relations, and of objects. Some of the most important advantages of the definition are that it captures the intuitions which motivate the thought that vague predicates are tolerant, without leading to contradiction, and that it yields a clear understanding of the relationships between higher-order vagueness, sorites susceptibility, blurred boundaries, and borderline cases. The most notable consequence of the definition is that the correct theory of vagueness must countenance degrees of truth.

On Dependency Pair Method for Proving Termination of Higher-Order Rewrite Systems

This paper explores how to extend the dependency pair technique for proving termination of higher-order rewrite systems. In the first order case, the termination of term rewriting systems are proved by showing the non-existence of an infinite R-chain of the dependency pairs. However, the termination and the non-existence of an infinite R-chain do not coincide in the higher-order case. We introduce a new notion of dependency forest that characterize infinite reductions and infinite R-chains, and show that the termination property of higher-order rewrite systems R can be checked by showing the non-existence of an infinite R-chain, if R is strongly linear or non-nested.

RESOLUTION OF SINGULARITIES FOR A FAMILY OF THIRD-ORDER DIFFERENTIAL SYSTEMS WITH SMALL MEROMORPHIC SOLUTION SPACES

Comparing the resolution of singularities for differential equations of Painleve type, there are important differences between the second-order Painleve equations and those of higher order. Unlike the second-order case, in higherorder cases there may exist some meromorphic solution spaces with codimension 2. In this paper, we will give an explicit global resolution of singularities for a 3-parameter family of third-order differential systems with meromorphic solution spaces of codimension 2.

Preconditioning the Mortar Method by Substructuring: The High Order Case.

AbstractWe analyze a class of preconditioners for the mortar method, based on substructuring. After splitting in a suitable way the degrees of freedom in interior, edge and vertex, we study the performance of a block Jacobi type preconditioner for which the condition number of the preconditioned matrix only grows polylogarithmically. Unlike the previous work by Achdou, Maday and Widlund [1], which is restricted to the case of first order finite element, this paper relies on abstract assumptions and therefore applies to finite element of any order. Moreover, the use of a suitable coarse preconditioner (whose effect we analyze) makes this technique more efficient. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Towards Safe and Effective High-Order Legendre Transforms with Applications to FFTs for the 2-sphere

This paper proposes and investigates a method for reducing potential numerical instability in the fast spherical harmonic transform algorithms proposed previously in [5,11]. The key objective of this study is a numerically reliable fast algorithm for computing the discrete Legendre transform (DLT); that is, the projection of sampled data onto the associated Legendre functions within a specified range of degrees. A simple divide-and-conquer approach derives from a factorization of high-degree Legendre functions into Legendre functions of lower-degree, exploiting the fact that the complexity of projection onto Legendre functions decreases with decreasing degree. Combining the resulting fast DLT algorithms for each relevant order of associated Legendre function results in an O(Nlog 2N) algorithm for computing the spherical harmonic expansion of a function sampled at N points on the sphere. While fast DLT algorithms of this form are exact in exact arithmetic, actual (finite precision) implementations of the earlier variants display instabilities which generally grow with the the order of the associated Legendre function in the transform. Here we return to the basic algorithm, present a slight modification of the general schema and examine the error mechanism for the higher-order cases. This study suggests a new approach to the high-order DLTs achieved by substitution of a simple alternative factorization of the associated Legendre functions in place of that used previously. This technique improves stability significantly for a wide range of useful problem sizes and may be used with any of the variants of the basic algorithm previously proposed. We present a description of the use of the new Legendre decomposition in the basic fast algorithm along with numerical experiments demonstrating the large advances in stability and efficiency of the new approach over the previous results.

Hardy inequalities with optimal constants and remainder terms

We show that the classical Hardy inequalities with optimal constants in the Sobolev spaces $W_0^{1,p}$ and in higher-order Sobolev spaces on a bounded domain $\Omega \subset \mathbb {R}^n$ can be refined by adding remainder terms which involve $L^p$ norms. In the higher-order case further $L^p$ norms with lower-order singular weights arise. The case $1<p<2$ being more involved requires a different technique and is developed only in the space $W_0^{1,p}$.

A Schwarz Preconditioner for a Hybridized Mixed Method

AbstractIn this paper, we provide a Schwarz preconditioner for the hybridized versions of the Raviart-Thomas and Brezzi-Douglas-Marini mixed methods. The preconditioner is for the linear equation for Lagrange multipliers arrived at by eliminating the flux as well as the primal variable. We also prove a condition number estimate for this equation when no preconditioner is used. Although preconditioners for the lowest-order case of the Raviart-Thomas method have been constructed previously by exploiting its connection with a nonconforming method, our approach is different in that we use a new variational characterization of the Lagrange multiplier equation. This allows us to precondition even the higher-order cases of these methods.

On uniqueness for a higher-order Cauchy problem of mathematical physics

Considerable attention is currently being devoted to Cauchy Problems of Mathematical Physics in view of their increasing importance for the solution of applied problems. Many Cauchy problems in mathematical physics (like elasticity or Maxwell's equations) arises in connection with the process described by partial differential equations. If one cannot guarantee the uniqueness of a solution to these problems, then the problem does not make sense and there is no hope of handling them. It is shown here that for quite a complicated higher-order case one can achieved uniqueness under physically reasonable assumptions.

Understanding the destruction ofnth-order quantum coherence in terms of multipath interference

In this paper we depict the high order quantum coherence of a boson system by using the multi-particle wave amplitude, whose norm square is just the high order correlation function. This multi-time amplitude can be shown to be a superposition of several "multi-particle paths". When the environment or a apparatus entangles with them to form a generalized "which-way" measurement for many particle system, the quantum decoherence happens in the high order case dynamically. An explicit illustration is also given with an intracavity system of two modes interacting with a moving mirror.

On the Number of Zeros of Nonoscillatory Solutions to Higher-Order Linear Ordinary Differential Equations

This paper is concerned with the problem of counting the number of zeros of bounded nonoscillatory solutions of higher-order linear ordinary differential equations involving a parameter. It is shown that a recent result for the second-order case is still valid for the higher-order case.

The delay approximation of discrete control in higher-order systems

Physical machinery of high mathematical order operating in industry increasingly nowadays requires the replacement of early analog controllers by imbedded digital systems. Using a 7 th-order longwall shearer as the motivating case-study, the paper examines the possibility of easing the digital control system design process by substituting a continuous transport-delay for the digital sample-hold operation. Based on first- and second-order models, this is shown to produce a design procedure that is pessimistic (i.e., safe) in the main and to a practically acceptable degree. Similar response speeds are generated by the discrete- and delay-models. By simulation, the technique is shown to carry over well to the higher-order case study aforementioned. This provides confidence in the delay-for-sampler substitution, the advantages of which are that the need to generate multiple z-transform models for a range of sampling intervals and the difficulties of interpreting z-plane plots (compared to those of familiar Nyquist diagrams and root loci) are avoided.

Scalable context-sensitive flow analysis using instantiation constraints

This paper shows that a type graph (obtained via polymorphic type inference) harbors explicit directional flow paths between functions. These flow paths arise from the instantiations of polymorphic types and correspond to call-return sequences in first-order programs. We show that flow information can be computed efficiently while considering only paths with well matched call-return sequences, even in the higher-order case. Furthermore, we present a practical algorithm for inferring type instantiation graphs and provide empirical evidence to the scalability of the presented techniques by applying them in the context of points-to analysis for C programs.

A second order scheme for the navier-stokes equations: stability and convergence *

In this paper, a fully discretized projection method is introduced. It contains a parameter operator. Depending on this operator, we can obtain a first-order scheme, which is appropriate for theoretical analysis, and a second-order scheme, which is more suitable for actual computations. In this method, the boundary conditions of the intermediate velocity field and pressure are not needed. We give the proof of the stability and convergence for the first-order case. For the higher order cases, the proof were different, and we will present it elsewhere. In a forthcoming article [7], we apply this scheme to the driven-cavity problem and compare it with other schemes