Higher-order Case Research Articles

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

Process calculus based upon evaluation to committed form

An approach to the semantics of CCS-like communicating processes is proposed that is based upon evaluation of processes to input- or output-committed form, with no explicit mention of silent actions. This leads to a co-inductively defined notion of evaluation bisimilarity — a form of weak branching-time equivalence which is shown to be a congruence, even in the presence of summation. The relationship between this evaluation-based approach and the more traditional, labelled transition semantics is investigated. In particular, with some restriction on sums, CCS observation equivalence is characterised purely in terms of evaluation to committed form, and evaluation bisimilarity is characterised as a weak delay equivalence. These results are extended to the higher-order case, where evaluation bisimilarity coincides with Sangiorgi's weak context bisimilarity. An evaluation-based approach to π-calculus and the relationship with Milner and Sangiorgi's reduction-based notion of barbed bisimulation are also examined.

Extra Smoothness Requirements for Galerkin Methods for the Wave Equation

Rauch [SIAM J. Numer. Anal., 22 (1985), pp. 245–249] studied the convergence of a continuous-time Galerkin approximation of the second-order wave equation. He showed that using piecewise linear functions over a mesh of size h, the initial data must have three derivatives if the error is to be $O(h^2 )$. This corresponds to one “extra derivative– when contrasted with approximation theory or with results for parabolic differential equations. Rauch also suggested that some extra smoothness would be required for higher-order splines. Using similar techniques, sharp results are presented here which show how much extra smoothness is needed on the initial data for higher-order cases. For smoothest splines of order k on a uniform mesh, one needs $k + (\frac{1}{2})(\frac{k} {{(k - 1)}})$ derivatives to have convergence of order k; the extra smoothness is between $\tfrac{1} {2}$ and 1 derivative.

Inclusions and subtypes II: higher-order case

Inclusions and subtypes II: higher-order case

On the Poisson geometry of the Adler-Gel’fand-Dikii brackets

In this paper, we consider a symplectic leaf that goes through a singular point of the Adler-Gel’fand-Dikii Poisson bracket associated to SL(n,R). We find a finite-dimensional transverse section2 at the singular point and we prove that one can induce a Poisson structure on2 (the transverse structure) that is linearizable and equivalent to the Lie-Poisson structure on sl(n,R)*. This problem is closely related to finding normal forms for nth order scalar differential operators with periodic coefficient. We partially generalize a well-known result for Hill’s operators to the higher order case.

Interpretation of Lavrentiev Phenomenon by Relaxation: The Higher Order Case

We consider integral functionals of the calculus of variations of the form \[ F(u) = \int \limits _0^1 {f(x,u,u’, \ldots ,{u^{(n)}})dx} \] defined for $u \in {W^{n,\infty }}(0,1)$, and we show that the relaxed functional $F$ with respect to the weak $W_{{\text {loc}}}^{n,1}(0,1)$ convergence can be written as \[ \overline F (u) = \int \limits _0^1 {f(x,u,u’, \ldots ,{u^{(n)}})dx + L(u),} \] where the additional term $L(u)$, the Lavrentiev Gap, is explicitly identified in terms of $F$.

USES AND ABUSES OF FUZZINESS IN PHILOSOPHY: A SELECTIVE SURVEY OF HOW RECENT PHILOSOPHICAL LOGIC AND METAPHYSICS TREAT VAGUENESS

This is a selective summary of work done by philosophers on the concept of vagueness (or fuzziness) accompanied by a selective annotated bibliography organized topically. The main topics covered are: the sorites paradox, the semantic technique of supervaluations, the definition of first-order and higher-order borderline cases by means of a determinacy operator, arguments that the existence of vagueness precludes the coherence of observational predicates, such as color terms, and arguments (“The Problem of the Many”) that the existence of vagueness precludes the existence of ordinary things. Attempts to meet these last two arguments often encounter the problem of inappropriate precision, the problem of importing inappropriately high degrees of exactness into theoretical treatments of inexactness.

Interpretation of Lavrentiev phenomenon by relaxation: the higher order case

We consider integral functionals of the calculus of variations of the form \[ F ( u ) = ∫ 0 1 f ( x , u , u ′ , … , u ( n ) ) d x F(u) = \int \limits _0^1 {f(x,u,u’, \ldots ,{u^{(n)}})dx} \] defined for u ∈ W n , ∞ ( 0 , 1 ) u \in {W^{n,\infty }}(0,1) , and we show that the relaxed functional F F with respect to the weak W loc n , 1 ( 0 , 1 ) W_{{\text {loc}}}^{n,1}(0,1) convergence can be written as \[ F ¯ ( u ) = ∫ 0 1 f ( x , u , u ′ , … , u ( n ) ) d x + L ( u ) , \overline F (u) = \int \limits _0^1 {f(x,u,u’, \ldots ,{u^{(n)}})dx + L(u),} \] where the additional term L ( u ) L(u) , the Lavrentiev Gap, is explicitly identified in terms of F F .

Toward formal development of programs from algebraic specifications: Parameterisation revisited

Parameterisation is an important mechanism for structuring programs and specifications into modular units. The interplay between parameterisation (of programs and of specifications) and specification (of parameterised and of non-parameterised programs) is analysed, exposing important semantic and methodological differences between specifications of parameterised programs and parameterised specifications. The extension of parameterisation mechanisms to the higher-order case is considered, both for parameterised programs and parameterised specifications, and the methodological consequences of such an extension are explored. A specification formalism with parameterisation of an arbitrary order is presented. Its denotational-style semantics is accompanied by an inference system for proving that an object satisfies a specification. The formalism includes the basic specification-building operations of the ASL specification language and is institution independent.

Fixed points and frontiers: a new perspective

AbstractAbstract interpretation is the collective name for a family of semantics-based techniques for compile-time analysis of programs. One of the most costly operations in automating such analyses is the computation of fixed points. The frontiers algorithm is an elegant method, invented by Chris Clack and Simon Peyton Jones, which addresses this issue.In this article we present a new approach to the frontiers algorithm based on the insight that frontiers represent upper and lower subsets of a function's argument domain. This insight leads to a new formulation of the frontiers algorithm for higher-order functions, which is considerably more concise than previous versions.We go on to argue that for many functions, especially in the higher-order case, finding fixed points is an intractable problem unless the sizes of the abstract domains are reduced. We show how the semantic machinery of abstract interpretation allows us to place upper and lower bounds on the values of fixed points in large lattices by working within smaller ones.

One universally efficient estimation of the first-order autoregressive parameter and universal data compression

A universal nearly efficient estimator is proposed for the first-order autoregressive (AR) model where the probability distribution of the driving noise is unknown. It is shown that universal estimators for the AR model can be derived from universal data compression algorithms and universal tests for randomness. In other words, estimators derived appropriately from efficient universal codes can be expected to inherit good estimation performance under some conditions. The proposed estimator has a simple information-theoretic interpretation related to universal coding, which can be easily generalized to the higher-order case and to other parametric models, e.g. the one-sample location model, the two-sample location model, and the linear regression model. >

Higher-order differential equations and higher-order lagrangian mechanics

The study of higher-order mechanics, by various geometrical methods, in the framework of the theory of higher-order tangent bundles or jet spaces, has been undertaken by a number of authors recently: for example, Tulczyjew [16, 17], Rodrigues [14, 15] de León [8], Krupka and Musilova [11, and references therein]. In this article we wish to complement these studies by approaching the subject from a new point of view, one which we developed for second-order differential equation fields and first-order Lagrangian mechanics in [19]. In particular, our aim is to show that many of the results we obtained there may be extended to the higher-order case.

Discretized likelihood methods—asymptotic properties of discretized likelihood estimators (DLE's)

Suppose thatX1,X2, ...,X n , ... is a sequence of i.i.d. random variables with a densityf(x, θ). Letc n be a maximum order of consistency. We consider a solution $$\hat \theta _n $$ of the discretized likelihood equation $$\sum\limits_{i = 1}^n {\log f(X_i ,\hat \theta _n + rc_n^{ - 1} ) - } \sum\limits_{i = 1}^n {\log f(X_i ,\hat \theta _n ) = a_n (\hat \theta _n ,r)} $$ wherea n (θ,r) is chosen so that $$\hat \theta _n $$ is asymptotically median unbiased (AMU). Then the solution $$\hat \theta _n $$ is called a discretized likelihood estimator (DLE). In this paper it is shown in comparison with DLE that a maximum likelihood estimator (MLE) is second order asymptotically efficient but not third order asymptotically efficient in the regular case. Further it is seen that the asymptotic efficiency (including higher order cases) may be systematically discussed by the discretized likelihood methods.

Nonlinear effects upon waves near caustics

According to linear theory the wave intensity of a slowly varying wave train becomes particularly large near caustics. In this paper it is shown how the waves are modified when the wave intensity is sufficient for nonlinear effects to begin to be important. Two types of near-linear caustics can arise in which nonlinearity either tends to advance or to retard the reflexion of waves from the caustic. General examples are given in terms of one-dimensional wave propagation, and of propagation in a uniform medium. Detailed consideration is given to a particular example: small-amplitude water waves on deep currents. This helps to provide an interpretative framework for the large-amplitude results presented in the companion paper (Peregrine & Thomas 1979). For the more exceptional case of triple roots, or cusped caustics, the increase in wave intensity is even more dramatic. In three appendices the analysis for caustics is extended to some higher-order cases.

A Complete Mechanization of Second-Order Type Theory

A generalization of the resolution method for higher order logic is presented. The languages acceptable for the method are phrased in a theory of types of order w (all finite types)—including the λ-operator, propositional functors, and quantifiers. The resolution method is, of course, a machine-oriented theorem search procedure based on refutation. In order to make this method suitable for higher order logic, it was necessary to overcome two sorts of difficulties. The first is that the unifying substitution procedure—an essential feature of the classic first-order resolution—must be generalized (it is noted that for the higher order unification the proper notion of substitution will include λ-normalization). A general unification algorithm is produced and proved to be complete for second-order languages. The second difficulty arises because in higher order languages, semantic intent is essentially more “interwoven” in formulas than in first-order languages. Whereas quantifiers could be eliminated immediately in first-order resolution, their elimination must be deferred in the higher order case. The generalized resolution procedure which the author produces thus incorporates quantifier elimination along with the familiar features of unification and tautological reduction. It is established that the author's generalized resolution procedure is complete with respect to a natural notion of validity based on Henkin's general validity for type theory. Finally, there are presented examples of the application of the method to number theory and set theory.

Sub-optimal Pole Locations of Pulse Networks†

The main requirement of any pulse transmission network is small rise time (10-90%) with little or no overshoot for step inputs. In this paper an attempt is made to find the polo locations of third, fourth, fifth and sixth-order ail-pole transfer functions to give the least rise time. In the third-order case, the optimum pole locations, subject to the step response being monotonic. are found, while in the higher-order cases the pole locations along the Butterworth angles giving the least rise time are obtained.

Analysis and Synthesis Techniques of Oriented Communication Nets

An oriented communication net is a communication network in which channel capacities between pairs of terminals are not symmetrical. Such a system can be represented by an oriented graph. The concept of a minimum-valued cut is used to determine the terminal-capacity matrix which gives the maximum possible communication between any ordered pair of terminals. Necessary conditions for a terminal capacity matrix to be realizable as an oriented communication network are obtained. These conditions are shown to be sufficient when the order of the terminal capacity matrix is three or less. They are not sufficient for higher-order cases in general. Synthesis techniques based on matrix partition and matrix addition are used to realize any three-by-three terminal capacity matrix with extensions to some higher-order cases.