Graded Codes

 Since S.D. Berman showed in 1967 that cyclic codes and Reed Muller codes can be studied as ideals in a group algebra KG (over a finite field K and G is considered, in each case, a finite cyclic group and a 2-group respectively), several authors have investigated these codes. It has been observed, that the presence of additional algebraic structure make their study more effective. Following these ideas, we consider codes which are (graded) left ideals of a graded ring R or more generally, codes as (graded) left R-modules. In this paper, a linear code is graded if it is a graded R-module for some multiplicative group G and a G-graded K-algebra R. We will show that some important properties of graded codes can be obtained from their homogeneous components and then we generalize some results about codes as ideals in group algebras. In particular, we study cocyclic codes as ideals in a twisted group algebra, and more generally, in a crossed product. Received: May 22, 1999; revised version: August 20, 2001     

鲁棒线性最优化的若干扩展

本文基于凸锥理论对鲁棒线性最优化作了若干拓展。本文的拓展分为三部分。首先我们放松了对不确定集的限制,把鲁棒线性最优化拓展到凸锥和子空间平移的交的不确定集的情形。其次我们考虑了由凸不等式定义的不确定集的鲁棒线性最优化。再次,我们把鲁棒线性最优化拓展到了包含系数不确定性和解的实现误差的情形。对某些特殊的情形,我们导出了鲁棒线性最优化的确定性等价问题。     

Positively equicontinuous flows

Let G be a positively equicontinuous flow of homeomorphisms of a locally compact metric space E. We show how the dynamics of such a flow are rich. We study when regularly almost periodic elements in G are periodic and we describe orbits and their limit sets. In particular, we show that the limit set L(G?) of G is a closed subset on which G is equicontinuous and that if G moreover has closed orbits, then any w-limit set is a periodic orbit.     

Ergodic measure-expansive diffeomorphisms

In this paper, we consider the set of diffeomorphisms which are measure-expansive for any ergodic measure, and study the set from the viewpoint of geometric theory of dynamical systems. It is proved that (1) there exists a non-empty C¹-open set of robustly non-hyperbolic and transitive diffeomorphisms such that each element of the set is measure-expansive for any ergodic measure, and that (2) C¹-generically, a diffeomorphism is measure-expansive for any generic ergodic measure.     

On Systems of Linear Equations with Nonnegative Coefficients

We consider a system of linear equations with positive coefficients, where the entries of the nonnegative irreducible coefficient matrix depend on a parameter vector. We say that the parameter vector is feasible if there exists a positive solution to this system. A set of all feasible parameter vectors is called the feasibility set. If all the positive entries are log-convex functions, the paper shows that the associated Perron root is log-convex on the parameter set and the l¹-norm of the solution is log-convex on the feasibility set. These results imply that the feasibility set is a convex set regardless whether the l¹-norm of the solution is bounded by some positive real number or not. Finally, we show important applications of these results to wireless communication networks and prove some other interesting results for this special case.This work was partly supported by the German Federal Ministry of Education and Research (BMBF) under grant BU150This work was supported by the German Research Foundation (DFG) under grant BO1734/1-2     

Abundance of elliptic dynamics on conservative three-flows

We consider a compact three-dimensional boundaryless Riemannian manifold M and the set of divergence-free (or zero divergence) vector fields without singularities, then we prove that this set has a C ¹-residual (dense G _δ) such that any vector field inside it is Anosov or else its elliptical orbits are dense in the manifold M. This is the flow-setting counterpart of Newhouse's Theorem 1.3 (S. Newhouse, Quasi-elliptic periodic points in conservative dynamical systems, Am. J. Math. 99 (1977), pp. 1061–1087). Our result follows from two theorems, the first one says that if Λ is a hyperbolic invariant set for some class C ¹ zero divergence vector field X on M, then either X is Anosov, or else Λ has empty interior. The second one says that, if X is not Anosov, then for any open set U ? M there exists Y arbitrarily close to X such that Y ^t has an elliptical closed orbit through U.     

A precise computation of stress intensity factor on the front of a convex planar crack

In this paper, we consider a general integral expression for mode I stress intensity factor along the fronts of convex planar cracks. For this integral approximation, we develop a simple numerical quadrature formula on every convex set Ω which allows a precise estimation of the error. This permits the use of extrapolation techniques for the accurate computation of the integral. Copyright © 2002 John Wiley & Sons, Ltd.     

On the reformulation of topology optimization problems as linear or convex quadratic mixed 0–1 programs

We consider equivalent reformulations of nonlinear mixed 0–1 optimization problems arising from a broad range of recent applications of topology optimization for the design of continuum structures and composite materials. We show that the considered problems can equivalently be cast as either linear or convex quadratic mixed 0–1 programs. The reformulations provide new insight into the structure of the problems and may provide a foundation for the development of new methods and heuristics for solving topology optimization problems. The applications considered are maximum stiffness design of structures subjected to static or periodic loads, design of composite materials with prescribed homogenized properties using the inverse homogenization approach, optimization of fluids in Stokes flow, design of band gap structures, and multi-physics problems involving coupled steady-state heat conduction and linear elasticity. Several numerical examples of maximum stiffness design of truss structures are presented. The research is funded by the Danish Natural Science Research Council and the Danish Research Council for Technology and Production Sciences.     

On the concepts of admissibility and coherence

In the present paper it will be argued that if a parameter value assigns zero probability to an open set containing the actual response, then that parameter value should be excluded from the parameter space. When this is done, the model becomes a restricted model, and sampling theory inferences should be focused on the sampling distribution of an hypothetical independent response from the restricted model. The above situation may arise when probability densities vanish outside a compact set. This phenomenon arises frequently in the real world, but it is usually ignored for reasons of mathematical convenience. Although many statistical procedures remain substantially the same when we consider this restricted model, admissibility propertics may be drastically changed, and inferences which are known to be inadmissible may turn out to be really admissible. Thus, Stein's phenomenon concerning the inadmissibility of the sample mean as an estimator of the population mean of ap-variate normal distribution whenp≥3may be explained by the fact that the distribution has a compact support but this has been ignored by reasons of mathematical convenience. The introduction of a restricted model is also important in the study of coherence. Thus, it will be shown that Brunk's theory of countably additive coherence, which admits the use of countably additive improper priors, can be improved with the introduction of a restricted model. Thus, the theory will be unified because it will be proved that the posterior is really coherent if and only if it is a Bayes posterior, and it will be simplified because it will not be required that the prior be minimally compatible with the model.     

Vibrations analysis of flexible multibody systems

This paper is the continuation of a previous work (Pascal, 1988a) in which we consider the dynamical analysis of flexible space vehicles modelled by a chain of rigid and elastic bodies with tree structure. The multibody system consists of n + 1 bodies (S$iiei:) (i = 0, 1,…n) interconnected by n hinges l_a (a = 1,…n). The only external forces and torques are exerted on the first body which is assumed to be rigid. On each individual flexible appendage (Si), the only external forces and torques are those introduced by the hinges. Assuming that the multibody system undergoes small vibrations around an equilibrium position, we define in the frequency domain the linear transformation giving the resultant forces and torques on the boundaries of each flexible appendage (Si) in terms of the displacements of these boundaries. The motion of each flexible appendage is represented by a set of component modes, which are the modes obtained when the appendage vibrates independently with respect to the other parts of the whole system. Two different sets of vibration modes are used to give an expansion of the transfer function between forces and torques exerted on the boundaries of the body (Si) and displacements of these boundaries     

Procedures to investigate injectivity of polynomial maps and to compute the inverse

In this paper a polynomial map from C ⁿ to C ^m is studied in order to investigate if it is injective out of a set of measure zero. We propose a procedure, based on truncated Gröbner basis computations, which when successful, allows to reduce the problem to an easier map, and so gives a speed-up of the general algorithms using Gröbner basis techniques. Moreover, for the special case of a polynomial map from C ⁿ to C ⁿ where the polynomials are at most quadratic, we propose two criteria for non-injectivity based on the structure of the Jacobian matrix and requiring only basic symbolic computations.     

From Fourier Expansions to Arithmetic-Haar Expressions on Quaternion Groups

Arithmetic expressions for switching functions are introduced through the replacement of Boolean operations with arithmetic equivalents. In this setting, they can be regarded as the integer counterpart of Reed-Muller expressions for switching functions. However, arithmetic expressions can be interpreted as series expansions in the space of complex valued functions on finite dyadic groups in terms of a particular set of basic functions. In this case, arithmetic expressions can be derived from the Walsh series expansions, which are the Fourier expansions on finite dyadic groups. In this paper, we extend the arithmetic expressions to non-Abelian groups by the example of quaternion groups. Similar to the case of finite dyadic groups, the arithmetic expressions on quaternion groups are derived from the Fourier expansions. Attempts are done to get the related transform matrices with a structure similar to that of the Haar transform matrices, which ensures efficiency of computation of arithmetic coefficients. Received: October 5, 1999; revised version: June 14, 2000     

Optimal sensor placement for joint parameter and state estimation problems in large-scale dynamical systems with applications to thermo-mechanics

We consider large-scale dynamical systems in which both the initial state and some parameters are unknown. These unknown quantities must be estimated from partial state observations over a time window. A data assimilation framework is applied for this purpose. Specifically, we focus on large-scale linear systems with multiplicative parameter-state coupling as they arise in the discretization of parametric linear time-dependent partial differential equations. Another feature of our work is the presence of a quantity of interest different from the unknown parameters, which is to be estimated based on the available data. In this setting, we employ a simplicial decomposition algorithm for an optimal sensor placement and set forth formulae for the efficient evaluation of all required quantities. As a guiding example, we consider a thermo-mechanical PDE system with the temperature constituting the system state and the induced displacement at a certain reference point as the quantity of interest.     

Single-period assortment and stock-level decisions for dual sales channels with capacity limits and uncertain demand

This paper addresses a dual channel, clicks-and-mortar retailer's problem of determining which among a set of products with seasonal demand will occupy limited retail shelf space, which products will be offered via an online channel, and which items will be available through both channels. Using a consumer choice model in which the set of products offered influences each product's demand in each channel, we consider stocking and price decisions under uncertain demand in a single-period setting with a constraint on the probability of stocking out. The resulting model is a large-scale, chance-constrained, two-stage stochastic programme. We propose a sample average approximation (SAA) method that permits quickly arriving at near-optimal solutions for this complex problem class. We also exercise the proposed model to gain insights on the problem's key tradeoffs and properties of optimal solutions.     

Fracture and fatigue study of unidirectional glass/epoxy laminate under different mode of loading

Interlaminar fracture is the dominant failure mechanism in most advanced composite materials. The delaminating behaviour of materials is quantified in terms of the strain energy release rate G. In this paper, the experimental measurements of the fatigue delaminating growth for some combinations of energy release rate mode ratio have been carried out on unidirectional glass/epoxy laminates. On this base the constants in the Paris equation have been determined for each G_II/G_T considered modal ratio. The fatigue threshold strain energy release rate Δ G_Tth , below which delaminating doesn't occur, were measured. Three type specimens were tested, namely: double cantilever beam (DCB), end‐loaded split (ELS) and mixed‐mode bending (MMB) under mode I, mode II and mixed‐mode (I + II) loading, respectively. Scanning electron microscopy techniques were used to identify the fatigue delamination growth mechanisms and to define the differences between the various modes of fracture.     

The MITC9 <Emphasis Type="Italic">shell</Emphasis> element in <Emphasis Type="Italic">plate</Emphasis> bending: mathematical analysis of a simplified case

We consider the 9-node shell element referred to as the MITC9 shell element in plate bending solutions and present a simplified mathematical analysis. The element uses bi-quadratic interpolations of the rotations and transverse displacement, and the “rotated Raviart-Thomas” interpolations for the transverse shear stresses. A rigorous mathematical analysis of the element is still lacking, even for the simplified case of plate solutions (that is, flat shells), although the numerical evidence suggests a good and reliable behavior. Here we start such an analysis by considering a very simple particular case; namely, a rectangular plate, clamped all around the boundary, and solved with a uniform decomposition. Moreover, we consider only the so-called limit case, corresponding to the limit equations that are obtained for the thickness t going to zero. While the mathematical analysis of the limit case is simpler, such analysis, in general, gives an excellent indication of whether shear locking is present in the real case t > 0. We detail that the element in the setting considered shows indeed optimal behavior.     

Testing fast aspheric convex surfaces with a linear array of sources

We describe a noncontact test procedure with which to obtain the shapes of fast convex surfaces. For this, an array of sources is positioned in a straight line and separated in such a way that the image by reflection on the surface consists of a set of equally spaced bright spots. By rotating the surface, we test different meridians such that, after 360 degrees, the entire surface is measured. We present the source array design and the surface evaluation algorithm. We found that, to reduce numerical error in the evaluation of the shape of the surface, a numerical integration must be performed by a method that uses parabolic arcs instead of the traditional method that uses trapezoids. Through some numerical simulations we analyzed the accuracy of the method by introducing random displacements into the measured data. We found that to measure the quality of the surface with accuracy better than 5 microm, we have to measure the coordinates of the centroids on the image plane with an accuracy better than 0.5 pixel, and we to have measure the positions of the linear sources with an accuracy better than 0.5 mm. Experimental results for the testing of a carbon fiber convex sphere of 383.6-mm diameter (f/0.398) are shown.     

Convergence of Hybrid Space Mapping Algorithms

The space mapping technique is intended for optimization of engineering models which involve very expensive function evaluations. It may be considered a preprocessing method which often provides a very efficient initial phase of an optimization procedure. However, the ultimate rate of convergence may be poor, or the method may even fail to converge to a stationary point.We consider a convex combination of the space mapping technique with a classical optimization technique. The function to be optimized has the form H &#x25CB; f where H : R ^m &#x2192; R is convex and f : R ⁿ &#x2192; R ^m is smooth. Experience indicates that the combined method maintains the initial efficiency of the space mapping technique. We prove that the global convergence property of the classical technique is also maintained: The combined method provides convergence to the set of stationary points of H &#x25CB; f.     

On smooth relaxations of obstacle sets

We consider the problem of avoiding obstacle sets described by finitely many smooth convex inequality constraints, as it frequently occurs in, for example, trajectory or location planning. We present and discuss a general method to relax such sets by the upper level set of a single smooth convex function, covering different smoothing approaches like hyperbolic and entropic smoothing. Based on error bounds and Lipschitz continuity, special attention is paid to the computations of the maximal geometric approximation error and of a guaranteed safety margin. Our results thus allow to safely avoid the obstacle by obeying a single nonconvex smooth constraint. Numerical results indicate that our technique gives rise to smoothing methods which perform well even for smoothing parameters very close to zero.     

Robust mixed H 2/H ∞ controller for uncertain neutral state‐input delayed systems via LMI optimization approach

Abstract

In this paper, the problem of designing a robust mixed H ₂/H _∞ controller for a class of uncertain neutral state‐input delays system is considered. Based on Lyapunov‐Krasovskii functional theory, a delay‐dependent criterion is derived for the existence of a desired mixed H ₂/H _∞ controller, which can be easily constructed by certain feasible linear matrix inequalities (LMIs). Furthermore, a convex optimization problem is formulated to solve for a robust mixed H ₂/H _∞ controller which achieves the minimization of an upper bound of the closed‐loop H ₂ perforance measure.