A note on weighing designs

Summary Banerjee [1], [2] has shown that the arrangements afforded by a Balanced Incomplete Block Design can be used as an efficient spring balance design. Such designs suffer from one drawback viz., there are only a few or no degrees of freedom left for the estimation of error-variance,σ ². To overcome this difficulty, it has been suggested that the whole design may be repeated a certain number of times to get an estimate of the error variance. In the present note an attempt has been made to give an alternative design where there is no necessity of such repetition. It has been also shown that these designs give a lesser variance of the estimated weights than the repeated design. Institute of Agricultural Research Statistics     

Construction of uniform designs without replications

A uniform design scatters its design points evenly on the experimental domain according to some discrepancy measure. In this paper all the design points of a full factorial design can be split into two subdesigns. One is called the complementary design of the other. The complementary design theories of characterizing one design through the other under the four commonly used discrepancy measures are investigated. Based on these complementary design theories, some general rules for searching uniform designs through their complementary designs are proposed. An efficient method to check if a design has repeated points is introduced and a modified threshold-accepting algorithm is proposed to search uniform or nearly uniform designs without replications. The new algorithm is shown to be more efficient by comparing with other existing methods. Many new uniform or nearly uniform designs without replications are tabulated and compared.     

Some observations on repeated spring balance weighing designs

Summary Dey [3] has suggested a spring balance weighing design in preference to “repeated designs”, and later, Kulshreshtha and Dey [5] have suggested yet one more weighing design which, they say, would be preferred to “repeated designs” and to those suggested in [3], provided one is interested in estimating the weights of some of the objects with increased precision at the cost of precision for others. It has been shown here that, while the above findings may be true in some situations, one might, in a given problem, prefer “repeated designs” to those suggested in [3] and [5]. NSF Grant No. GP-28312 and GP-36562.     

Discontinuous solutions to hyperbolic systems under operator splitting

Two-dimensional systems of linear hyperbolic equations are studied with regard to their behavior under a solution strategy that in alternate time-steps exactly solves the component one-dimensional operators. The initial data is a step function across an oblique discontinuity. The manner in which this discontinuity breaks up under repeated applications of the split operator is analyzed, and it is shown that the split solution will fail to match the true solution in any case where the two operators do not share all their eigenvectors. The special case of the fluid flow equations is analyzed in more detail, and it is shown that arbitrary initial data gives rise to “pseudo acoustic waves” and a nonphysical stationary wave. The implications of these findings for the design of high-resolution computing schemes are discussed.     

Iterative computation of derivatives of repeated eigenvalues and the corresponding eigenvectors

Recently the authors proposed a simultaneous iteration algorithm for the computation of the partial derivatives of repeated eigenvalues and the corresponding eigenvectors of matrices depending on several real variables. This paper analyses the properties of that algorithm and extends it in several ways. The previous requirement that the repeated eigenvalue be dominant is relaxed, and the new generalized algorithm given here allows the simultaneous treatment of simple and repeated eigenvalues. Methods for accelerating convergence are examined. Numerical results support our theoretical analysis. Copyright © 2000 John Wiley & Sons, Ltd.     

Augmented line sampling for approximation of failure probability function in reliability-based analysis

An efficient approach, called augmented line sampling, is proposed to locally evaluate the failure probability function (FPF) in structural reliability-based design by using only one reliability analysis run of line sampling. The novelty of this approach is that it re-uses the information of a single line sampling analysis to construct the FPF estimation, repeated evaluations of the failure probabilities can be avoided. It is shown that, when design parameters are the distribution parameters of basic random variables, the desired information about FPF can be extracted through a single implementation of line sampling. Line sampling is a highly efficient and widely used reliability analysis method. The proposed method extends the traditional line sampling for the failure probability estimation to the evaluation of the FPF which is a challenge task. The required computational effort is neither relatively sensitive to the number of uncertain parameters, nor grows with the number of design parameters. Numerical examples are given to show the advantages of the approach.     

A station strategy to deter backoff attacks in IEEE 802.11 LANs

The IEEE 802.11 MAC protocol now prevailing in wireless LANs is vulnerable to selfish backoff attacks consisting in selection of short backoff times in the constituent CSMA/CA procedure. Administrative prevention of such attacks fails in ad hoc configurations, where stations' behavior cannot be mandated. In this paper we take an incentive-oriented game-theoretic approach whereby stations are allowed to maximize their payoffs (achieved success rates). Using a fairly accurate performance model we show that a noncooperative CSMA/CA game then arises with a payoff structure characteristic of a Prisoners' Dilemma. For a repeated CSMA/CA game, a novel SPELL strategy is proposed and shown to yield to simple algorithmic design. Assuming that the stations are rational players and wish to maximize a mean-value-type long-term utility, SPELL is further shown to deter a single attacker by providing a disincentive to deviate from SPELL.     

Piecewise polynomial Taylor methods for initial value problems

A class of explicit Taylor-type methods for numerically solving first-order ordinary differential equations is presented. The basic idea is that of generating a piecewise polynomial approximating function, with a given order of differentiability, by repeated Taylor expansion. Sharp error bounds for the approximation and its derivatives are given along with a stability analysis.This work was supported by the United States Atomic Energy Commission.     

Persymmetric determinants

The object of this paper is to develop some of the results in the author's joint paper with Dale [2] concerning the derivatives of persymmetric determinants whose elements are Appell functions.Four new double-sum identities are presented which are valid for arbitrary persymmetric determinants. Two of these identities are applied to give direct proofs of two results in [2], A simple formula is given for the derivative of a Turanian of order n with Appell polynomial elements and the result is applied repeatedly to show that its degree is far lower than expected. It is shown that one particular determinant has simple derivatives of all orders and that its degree too is far lower than expected. The formula for the derivative of (first) cofactors is shown to be extensible in a simple manner to the derivatives of second cofactors.     

Flux-splitting schemes for parabolic equations with mixed derivatives

Difference schemes of required quality are often difficult to construct as applied to boundary value problems for parabolic equations with mixed derivatives. Specifically, difficulties arise in the design of monotone difference schemes and unconditionally stable locally one-dimensional splitting schemes. In parabolic problems, certain opportunities are offered by restating the problem in question so that the quantities to be determined are fluxes (directional derivatives). The original problem is then rewritten as a boundary value one for a system of equations in flux variables. Weighted schemes for parabolic equations in flux coordinates are examined. Unconditionally stable locally one-dimensional flux schemes that are first- and second-order accurate in time are constructed for a parabolic equation without mixed derivatives. A feature of systems in flux variables for equations with mixed derivatives is that the terms with time derivatives are coupled with each other.     

On partial derivatives of multivariate Bernstein polynomials

It is shown that Bernstein polynomials for a multivariate function converge to this function along with partial derivatives provided that the latter derivatives exist and are continuous. This result may be useful in some issues of stochastic calculus.     

IPA Derivatives for Make-to-Stock Production-Inventory Systems with Backorders

A single-stage Make-to-Stock (MTS) production-inventory system consists of a production facility coupled to an inventory facility, and is subject to a policy that aims to maintain a prescribed inventory level (called base stock) by modulating production capacity. This paper considers a class of single-stage, single-product MTS systems with backorders, driven by random demand and production capacity, and subject to a continuous-review base-stock policy. A model from this class is formulated as a stochastic fluid model (SFM), where all flows are described by stochastic rate processes with piecewise constant sample paths, subject to very mild regularity assumptions that merely preclude accumulation points of jumps with probability 1. Other than that, the MTS model in SFM setting is nonparametric in that it assumes no specific form for the underlying probability law, and as such is quite general. The paper proceeds to derive formulas for the (stochastic) IPA (Infinitesimal Perturbation Analysis) derivatives of the sample-path time averages of the inventory level and backorders level with respect to the base-stock level and a parameter of the production rate. These formulas are comprehensive in that they are exhibited for any initial condition of the system, and include right and left derivatives (when they do not coincide). The derivatives derived are then shown to be unbiased and their formulas are seen to be amenable to fast computation. The generality of the model and comprehensiveness of the IPA derivative formulas hold out the promise of gradient-based applications. More specifically, since the base-stock level and production rate are the key control parameters of MTS systems, the results provide the theoretical underpinnings for optimizing the design of MTS systems and for devising prospective on-line adaptive control algorithms that employ IPA derivatives. The paper concludes with a discussion of those issues.     

阻尼系统重特征对导数的计算

提出了一种计算阻尼系统重特征值及其特征向量导数的方法．该方法利用n维空间的特征向量计算特征对的导数,避免了状态空间中特征向量的使用,从而节省了计算量,提高了计算效率．最后以一个5自由度的非比例阻尼系统对所提方法进行了数值试验,数值结果表明方法是有效的．     

An Efficient Approximation Technique for Solving a Class of Fractional Optimal Control Problems

In this paper, we discuss a class of fractional optimal control problems, where the system dynamical constraint comprises a combination of classical and fractional derivatives. The necessary optimality conditions are derived and shown that the conditions are sufficient under certain assumptions. Additionally, we design a well-organized algorithm to obtain the numerical solution of the proposed problem by exercising Laguerre polynomials. The key motive associated with the present approach is to convert the concerned fractional optimal control problem to an equivalent standard quadratic programming problem with linear equality constraints. Given examples illustrate the computational technique of the method together with its efficiency and accuracy. Graphical representations are provided to analyze the performance of the state and control variables for distinct prescribed fractions.     

The re-iterated Galerkin method

The iterated Galerkin method introduced by Sloan is modifiedto allow the improvement in the Galerkin approximation whichiteration produces to be repeated. It is shown in principle,and verified in illustration, that the repeated improvementsmay be achieved under weaker conditions than those requiredby Sloan. The sequence of approximations obtained is appliedto variational principles and in the case when a maximum principleis available it is shown that each step in the process improvesthe estimate of the maximum value of the associated functional.Applications to integral equations are given.     

Generalized Normal Derivatives and Their Applications in DDMs with Nonmatching Grids and DG Methods

A class of normal-like derivatives for functions with low regularity defined on Lipschitz domains are introduced and studied.It is shown that the new normal-like derivatives,which are called the generalized normal derivatives,preserve the major prop- erties of the existing standard normal derivatives.The generalized normal derivatives are then applied to analyze the convergence of domain decomposition methods (DDMs) with nonmatching grids and discontinuous Galerkin (DG) methods for second-order el- liptic problems.The approximate solutions generated by these methods still possess the optimal energy-norm error estimates,even if the exact solutions to the underlying elliptic problems admit very low regularities.     

Favard’s interpolation problem in one or more variables

Given scattered data on the real line, Favard [4] constructed an interpolant which depends linearly and locally on the data and whose nth derivative is locally bounded by the nth divided differences of the data times a constant depending only on n. It is shown that the (n —1)th derivative of Favard’s interpolant can be likewise bounded by divided differences, and that one can bound at best two consecutive derivatives of any interpolant by the corresponding divided differences. In this sense, Favard’s univariate interpolant is the best possible. Favard’s result has been extended [8] to a special case in several variables, and here the extent to which this can be repeated in a more general setting is proven exactly.     

A simplified version of strong connectivity in geometries

If there are less than b distinct blocks in a BIB design with b blocks then we say the design has repeated blocks. The set of distinct blocks of a design is called the support of the design. BIB designs with repeated blocks, besides being optimal, have special applications in the design of experiments and controlled samplings. Construction of BIB(ν, b, r, k, λ) designs with repeated blocks becomes complicated whenever the three parameters b, r, and λ are relatively prime. BIB(8, 56, 21, 3, 6) designs are examples of such designs with the smallest number of varieties. BIB(10, 30, 9, 3, 2) designs are such designs with the smallest number of blocks. We make an interesting observation about BIB(8, 56, 21, 3, 6) designs and give a table of such designs with 30 different support sizes. We prove, by construction, that a BIB(10, 30, 9, 3, 2) design exists if and only if the support size belongs to {21, 23, 24, 25, 26, 27, 28, 29, 30}. Other results are also given.     

Iterative Reconstructions in Irregular Sampling With Derivatives

Finite energy band-limited functions are reconstructed iteratively from nonuniform sample values of the functions and its derivatives. It is shown that the maximum gap allowed between the sampling points increases linearly with the number of derivatives considered. Moreover, a more precise result is presented for the first derivative case and another reconstruction of the functions using the frame algorithm is deduced.     

Differentiated shift-invariant integral operators,univariate case

In a recent paper [1] the author, along with H. Gonska, introduced some wavelet type integral operators over the whole real line and studied their properties such as shift-invariance, global smoothness preservation, convergence to the unit, and preservation of probability distribution functions. These operators are very general and they are introduced through a convolution-like iteration of another general operator with a scaling type function. In this paper the author provides sufficient conditions, so that the derivatives of the above operators enjoy the same nice properties as their originals. A sufficient condition is also given so that the “global smoothness preservation” related inequality becomes sharp. At the end several applications are given, where the derivatives of the very general specialized operators are shown to fulfill all the above properties. In particular it is shown that they preserve continuous probability density functions.