Generalized sufficient conditions for modular termination of rewriting

Modular properties of term rewriting systems, i.e. properties which are preserved under disjoint unions, have attracted an increasing attention within the last few years. Whereas confluence is modular this does not hold true in general for termination. By means of a careful analysis of potential counterexamples we prove the following abstract result. Whenever the disjoint union _{1 2} of two (finitely branching) terminating term rewriting systems ₁, ₂ is non-terminating, then one of the systems, say ₁, enjoys an interesting (undecidable) property, namely it is not termination preserving under non-deterministic collapses, i.e. ₁ {itG(x, y) x, G(x, y) y} is non-terminating, and the other system ₂ is collapsing, i.e. contains a rule with a variable right hand side. This result generalizes known sufficient criteria for modular termination of rewriting and provides the basis for a couple of derived modularity results. Furthermore, we prove that the minimal rank of potential counterexamples in disjoint unions may be arbitrarily high which shows that interaction of systems in such disjoint unions may be very subtle. Finally, extensions and generalizations of our main results in various directions are discussed. In particular, we show how to generalize the main results to some restricted form of non-disjoint combinations of term rewriting systems, namely for combined systems with shared constructors.This research was supported by the Deutsche Forschungsgemeinschaft, SFB 314 (D4-Projekt).     

Necessary and sufficient conditions for S-lemma and nonconvex quadratic optimization

The celebrated S-lemma establishes a powerful equivalent condition for the nonnegativity of a quadratic function over a single quadratic inequality. However, this lemma fails without the technical condition, known as the Slater condition. In this paper, we first show that the Slater condition is indeed necessary for the S-lemma and then establishes a regularized form of the S-lemma in the absence of the Slater condition. Consequently, we present characterizations of global optimality and the Lagrangian duality for quadratic optimization problems with a single quadratic constraint. Our method of proof makes use of Brickman’s theorem and conjugate analysis, exploiting the hidden link between the convexity and the S-lemma.     

On enforcement of corner conditions in triangular differential quadrature

This paper is concerned with the enforcement of corner conditions in the recently proposed triangular differential quadrature method. The sensitivity of solution to some corner conditions is exemplified and a reduced quadrature technique is introduced to overcome the sensitivity. Numerical examples in the context of bending and vibration of Mindlin plates are studied to validate the proposed reduced quadrature technique. It is shown that the technique is effective to cope with sensitivity of solution to corner conditions. The effect of the order of the reduced quadrature on accuracy of solution is also studied. It is found that satisfactory convergence is usually achieved when the order of reduced quadrature employed at a corner is larger than one half of the order of the triangular differential quadrature approximation in the entire triangle but two orders less than the order of the triangular differential quadrature approximation.     

A characterization and sufficient conditions for the total time on test transform order

The total time on test transform has been used in reliability to compare two random lifetimes. Despite that it has been used widely in this context, it is not very clear in which sense the random variables are being compared. In this paper we provide a characterization which reveals the true nature of this comparison. Given that this comparison is of particular interest when the stochastic order does not hold, we also provide sufficient conditions for the comparisons of the total time on test transforms, when the stochastic order does not hold. Applications to the comparison of several parametric families of distributions are provided.     

Several sufficient conditions for a map and a semi-flow to be ergodically sensitive

In this paper, we are concerned solely with the ergodic sensitivity for maps (resp. semi-flows), where these maps and semi-flows may not be continuous (in topology). A few new sufficient conditions under which a map (resp. a semi-flow) on a metric space is ergodically sensitive are presented, where such maps and semi-flows may not be continuous (in topology). In particular, we prove that the topologically strong ergodicity of a measure-preserving map (resp. a measure-preserving semi-flow) on a metric probability space with a fully supported measure implies its ergodical sensitivity.     

On the treatment of time-dependent boundary conditions in splitting methods for parabolic differential equations

Splitting methods for time-dependent partial differential equations usually exhibit a drop in accuracy if boundary conditions become time-dependent. This phenomenon is investigated for a class of splitting methods for two-space dimensional parabolic partial differential equations. A boundary-value correction discussed in a paper by Fairweather and Mitchell for the Laplace equation with Dirichlet conditions, is generalized for a wide class of initial boundary-value problems. A numerical comparison is made for the ADI method of Peaceman-Rachford and the LOD method of Yanenko applied to problems with Dirichlet boundary conditions and non-Dirichlet boundary conditions.     

Necessary and sufficient invariance conditions in mismatched uncertain variable structure systems

Abstract

This paper investigates new invariance conditions in mismatched uncertain variable structure systems associated with a new sliding mode control, at the same time, retaining the benefits achieved in conventional variable structure systems design, namely, fast response, good robustness, and stability. Based on new equivalent state idea and two sets of switching surfaces, necessary and sufficient invariance conditions are derived such that matched and mismatched uncertainties completely vanish from the sliding mode dynamics. In terms of linear matrix inequalities, we give explicit formulas of linear switching surfaces to guarantee that the system in the new sliding mode is quadratically stable. Additionally, we give a control law to perform the new sliding mode. Finally, a numerical example is given to illustrate the effectiveness of the proposed approach.     

On the algebraic characterization of a Mueller matrix in polarization optics. II. Necessary and sufficient conditions for Jones-derived Mueller matrices

Abstract

We show that every Mueller matrix, that is a real 4 × 4 matrix M which transforms Stokes vectors into Stokes vectors, may be factored as M = L ₂ KL ₁ where L ₁ and L ₂ are orthochronous proper Lorentz matrices and K is a canonical Mueller matrix having only two different forms, namely a diagonal form for type-I Mueller matrices and a non-diagonal form (with only one non-zero off-diagonal element) for type-II Mueller matrices. Using the general forms of Mueller matrices so derived, we then obtain the necessary and sufficient conditions for a Mueller matrix M to be Jones derived. These conditions for Jones derivability, unlike the Cloude conditions which are expressed in terms of the eigenvalues of the Hermitian coherency matrix T associated with M, characterize a Jones-derived matrix M through the G eigenvalues and G eigenvectors of the real symmetric N matrix N = [Mtilde]GM associated with M. Appending the passivity conditions for a Mueller matrix onto these Jones-derivability conditions, we then arrive at an algebraic identification of the physically important class of passive Jones-derived Mueller matrices.     

Some necessary conditions for a maximal solution to a family of elliptic equations

We established some necessary conditions that a class of non-local variational problems has a maximal element over a preassigned set, _p, of functions. The exact nature of the set _p is not specified; this depends—in part—on the physical process being modeled. However we do impose some very general conditions on a maximizing element so that our computations may be carried out.

The class of problems we consider occurs in various engineering problems. In particular we apply our results to a class of problems similar to that encountered in the determination of optimum two-dimensional bearing profiles for Hydrodynamic Lubrication Theory.     

Phase anomalies in fourier-transform spectrometers: an absorbing beam splitter is neither sufficient nor necessary

Recently it has been a topic of some discussion that the phase associated with part of the self-emission of a Fourier-transform spectrometer may differ neither by 0 nor by pi rad from the phase of an external target if there is absorption in the beam splitter. The conventional interpretation of this has been to separate the self-emission into three terms: instrument emission from the input port, in phase with the external target; emission from the secondary input port, pi rad from the target; and emission from an absorbing beam splitter with an anomalous phase (neither 0 nor pi rad with respect to the target). There is another necessary condition that has not received much attention, that the instrument not be isothermal. For polarized radiation there is an additional condition that suppresses the anomalous phase, and for unpolarized radiation there is a way to produce an anomalous phase without beam-splitter absorption.     

Instrument for flatness checks on large workpieces

Conclusion The IS41M provides high accuracy in checking large surfaces for planarity deviation, and also during equipment installation. It contains no components requiring high-precision manufacture. The instrument is convenient to use and can be prepared for operation in not more than 10–15 min. The data do not need complicated processing and provide direct information on the deviation from planarity at the relevant point.Components of a machine can be set up in a single plane under direct control of the instrument, and the surface can be oriented at any angle in space.The instrument is suitable for use in the laboratory and under plant conditions and even in the open air, provided there is no precipitation.Translated from Izmeritel'naya Tekhnika, No. 8, pp. 48–49, August, 1978.     

Effect of growth conditions on the flatness of micro-crystal terraces of CdTe thin film

An atomic force microscopy study of the surface roughness of CdTe thin films growth by the close-space vapor transport technique on Corning glass substrate under different growth conditions (substrate temperature, Cd overpressure, and annealing) is presented. The roughness measurements show that under certain growth conditions the surface of the micro-crystals is flatter—on the micrometer scale—than in the case of CdTe layers grown by molecular beam epitaxy (MBE) on a CdTe substrate. ©1999 Kluwer Academic Publishers     

动态设定型板形板厚自动控制系统

文章介绍了应用解析板形方程推出板形最佳轧制规程和板形板厚协调控制新方法。该方法的主要特点：系统中采用了静、动态负荷分配,动态设定厚度自动控制系统（DAGC）完成了板形板厚的闭环控制,在算法中采用了贝尔曼动态规划。新方法将会改变目前轧机设计和控制思想：设计合理刚度,强调调度计划、配置轧辊凸度和优化轧制规程等方面的作用,实现信息控制。