G-invariance faible et isovariance en planification expérimentale

In this Note, we state a polynomial characterization of weakly invariant designs and show how to derive the construction of weakly invariant designs for the action of a compact group of matrices on an experimental domain, whose interior is not empty, from the construction of rotatable designs. As a consequence, it enables us to search for weakly invariant designs using techniques coming from computational commutative algebra and to benefit from the cumulated knowledge of rotatable designs which have been intensively studied since the seminal paper of Box and Hunter. To cite this article: F. Bertrand, C. R. Acad. Sci. Paris, Ser. I 347 (2009).     

Incorporating Angular Information into Parametric Models

The problem is considered of fitting a parametrically defined model in two or three dimensions to observed data, when angular information about the measured data points is available. Gauss-Newton methods based on correct separation of variables are developed. Some numerical results are included.     

Variance functions form-grouped cylindrically rotatable designs of type 3

Summary Formulae for variance of difference between two estimated responses are derived form-grouped first-order and 2-grouped second- and thirdorder cylindrically rotatable designs of type 3.     

Multiplicative designs I: The normal and reducible cases

Multiplicative designs, square designs with a rank one intersection pattern, are investigated. In particular we determine all designs for which some incidence matrix is reducible and we show if some incidence matrix is normal the design must be uniform with three families of exceptions. Some new designs are constructed.     

Oscillatory criteria for third-order nonlinear difference equation with impulses

In this paper, we study the oscillation of a third-order nonlinear difference equation with impulses. Some sufficient conditions for the oscillatory behavior of solutions of third-order impulsive nonlinear difference equation are obtained. Some known results in the literature are generalized and improved.     

Random arrays and functionals with multivariate rotational symmetries

Summary Our aim is to extend Schoenberg's classical theorem to higher dimensions, by establishing representations of arbitrary separately or jointly rotatable continuous linear random functionals in terms of multiple Wiener-Itô integrals and their tensor products. This leads to similar representations for separately or jointly rotatable arrays, and for separately or jointly exchangeable or spreadable random sheets.Research supported by NSF Grant DMS-9103050     

Symmetries, integrals and solutions of ordinary differential equations of maximal symmetry

Second- and third-order scalar ordinary differential equations of maximal symmetry in the traditional sense of point, respectively contact, symmetry are examined for the mappings they produce in solutions and fundamental first integrals. The properties of the ‘exceptional symmetries’, i.e. those not considered to be generic to scalar equations of maximal symmetry, can be recast into a form which is applicable to all such equations of maximal symmetry. Some properties of these symmetries are demonstrated.     

A single operational amplifier simulates general third-order linear systems

This paper presents a circuit configuration for simulating general third-order linear systems using a single operational amplifier. It discusses six alternative resistor-capacitor combinations available within the general network configuration to simulate a general form of third-order linear transfer functions. Attempt is made to keep the number of passive elements to a minimum. Restrictions on the systems-constants are worked out under which simulation is realisable with positive real values of the resistors and the capacitors; equations to calculate these values are set forth.     

Semiparametric analysis in double-sampling designs via empirical likelihood

Double-sampling designs are commonly used in real applications when it is infeasible to collect exact measurements on all variables of interest. Two samples, a primary sample on proxy measures and a validation subsample on exact measures, are available in these designs. We assume that the validation sample is drawn from the primary sample by the Bernoulli sampling with equal selection probability. An empirical likelihood based approach is proposed to estimate the parameters of interest. By allowing the number of constraints to grow as the sample size goes to infinity, the resulting maximum empirical likelihood estimator is asymptotically normal and its limiting variance-covariance matrix reaches the semiparametric efficiency bound. Moreover, the Wilks-type result of convergence to chi-squared distribution for the empirical likelihood ratio based test is established. Some simulation studies are carried out to assess the finite sample performances of the new approach.     

A comparison of some ODE solvers which require Jacobian evaluations

A variety of third-order ODE solvers which have a minimum configuration (i.e. minimum work per step) have been numerically tested and the results compared. They include implicit and explicit processes, and share the property that a Jacobian matrix must be evaluated at least once during the integration. Some of these processes have not been previously described in the literature.     

Oscillation criteria for certain third-order variable delay functional dynamic equations on time scales

The oscillation of certain third-order nonlinear variable delay functional dynamic equations with nonlinear neutral on time scales is discussed in this article. By using the generalized Riccati transformation and a lot of inequality techniques, some new oscillation criteria for the equations are established. Many of the results are new for the corresponding third-order difference equations and differential equations are as special cases. Some examples are given to illustrate the importance of our results.     

ROS3P—An Accurate Third-Order Rosenbrock Solver Designed for Parabolic Problems

In this note we present a new Rosenbrock solver which is third-order accurate for nonlinear parabolic problems. Since Rosenbrock methods suffer from order reduction when they are applied to partial differential equations, additional order conditions have to be satisfied. Although these conditions have been known for a longer time, from the practical point of view only little has been done to construct new methods. Steinebach modified the well-known solver RODAS of Hairer and Wanner to preserve its classical order four for special problem classes including linear parabolic equations. His solver RODASP, however, drops down to order three for nonlinear parabolic problems. Our motivation here was to derive an efficient third-order Rosenbrock solver for the nonlinear situation. Such a method exists with three stages and two function evaluations only. A comparison with other third-order methods shows the substantial potential of our new method.This revised version was published online in October 2005 with corrections to the Cover Date.     

Shape optimization by the homogenization method

Summary. In the context of shape optimization, we seek minimizers of the sum of the elastic compliance and of the weight of a solid structure under specified loading. This problem is known not to be well-posed, and a relaxed formulation is introduced. Its effect is to allow for microperforated composites as admissible designs. In a two-dimensional setting the relaxed formulation was obtained in [6] with the help of the theory of homogenization and optimal bounds for composite materials. We generalize the result to the three dimensional case. Our contribution is twofold; first, we prove a relaxation theorem, valid in any dimensions; secondly, we introduce a new numerical algorithm for computing optimal designs, complemented with a penalization technique which permits to remove composite designs in the final shape. Since it places no assumption on the number of holes cut within the domain, it can be seen as a topology optimization algorithm. Numerical results are presented for various two and three dimensional problems. Received July 4, 1995     

An empirical likelihood approach to data analysis under two-stage sampling designs

A new empirical likelihood approach is developed to analyze data from two-stage sampling designs, in which a primary sample of rough or proxy measures for the variables of interest and a validation subsample of exact information are available. The validation sample is assumed to be a simple random subsample from the primary one. The proposed empirical likelihood approach is capable of utilizing all the information from both the specific models and the two available samples flexibly. It maintains some nice features of the empirical likelihood method and improves the asymptotic efficiency of the existing inferential procedures. The asymptotic properties are derived for the new approach. Some numerical studies are carried out to assess the finite sample performance.     

On construction of a fourth order rotatable design with three factors

The necessary and sufficient conditions for a set of points to form a rotatable design of order four have been given by Patel and Arap Koske [1], In this paper, a set of 98 points for three factors which satisfies these conditions is given.     

Nonlinear Dispersive Systems: Theory and Examples

Some recent results in the asymptotic theory of differential equations are applied to certain third-order scalar and vector boundary-value problems that model various nonlinear physicochemical and dispersive wave phenomena. The key to our approach is the reduction of the third-order problems to asymptotically equivalent second-order ones which are more amenable to analysis. Many examples are discussed, including the problem of solitary-wave solutions of generalized Korteweg-de Vries equations and coupled systems of such equations.     

EFFICIENT AND ACCURATE CHEBYSHEV DUAL-PETROV-GALERKIN METHODS FOR ODD-ORDER DIFFERENTIAL EQUATIONS

Efficient and accurate Chebyshev dual-Petrov-Galerkin methods for solving first-order equation,third-order equation,third-order KdV equation and fifth-order Kawahara equa-tion are proposed.Some Sobolev bi-orthogonal basis functions are constructed which lead to the diagonalization of discrete systems.Accordingly,both the exact solutions and the approximate solutions are expanded as an infinite and truncated Fourier-like series,respec-tively.Numerical experiments illustrate the effectiveness of the suggested approaches.     

Efficient space-filling and non-collapsing sequential design strategies for simulation-based modeling

Simulated computer experiments have become a viable cost-effective alternative for controlled real-life experiments. However, the simulation of complex systems with multiple input and output parameters can be a very time-consuming process. Many of these high-fidelity simulators need minutes, hours or even days to perform one simulation. The goal of global surrogate modeling is to create an approximation model that mimics the original simulator, based on a limited number of expensive simulations, but can be evaluated much faster. The set of simulations performed to create this model is called the experimental design. Traditionally, one-shot designs such as the Latin hypercube and factorial design are used, and all simulations are performed before the first model is built. In order to reduce the number of simulations needed to achieve the desired accuracy, sequential design methods can be employed. These methods generate the samples for the experimental design one by one, without knowing the total number of samples in advance. In this paper, the authors perform an extensive study of new and state-of-the-art space-filling sequential design methods. It is shown that the new sequential methods proposed in this paper produce results comparable to the best one-shot experimental designs available right now.     

Compacton,peakon and folded localized excitations for the (2 + 1)-dimensional Broer–Kaup system

In high dimensions there are abundant coherent soliton excitations. From the variable separation solutions for the (2 + 1)-dimensional Broer–Kaup system, three kinds of new localized excitations in this system are obtained. Some interesting novel features of these structures are revealed.     

Asymptotic properties of third-order half-linear neutral dynamic equations with mixed arguments

This work is concerned with the asymptotic behavior of a class of third-order half-linear neutral dynamic equations with mixed arguments on a time scale. Some new criteria and an example are presented.