The ultimate state of polymeric materials and laminated and fibrous composites under asymmetric high-cycle loading

The prediction of the high-cycle fatigue strength of polymeric and composite materials in asymmetric loading is considered. The problem is solved on the basis of a nonlinear model of ultimate state allowing us to describe all typical forms of the diagrams of ultimate stresses. The material constants of the model are determined from the results of fatigue tests in symmetric reversed cycling, in a single fatigue test with the minimum stress equal to zero, and in a short-term strength test. The fatigue strength characteristics of some polymers, glass-fiber laminates, glass-fiber-reinforced plastics, organic-fiber-reinforced plastics, and wood laminates in asymmetric tension-compression, bending, and torsion have been calculated and approved experimentally. __________ Translated from Mekhanika Kompozitnykh Materialov, Vol. 44, No. 1, pp. 87–102, January–February, 2008.     

Model checking for a general linear model with nonignorable missing covariates

In this paper, we investigate the model checking problem for a general linear model with nonignorable missing covariates. We show that, without any parametric model assumption for the response probability, the least squares method yields consistent estimators for the linear model even if only the complete data are applied. This makes it feasible to propose two testing procedures for the corresponding model checking problem: a score type lack-of-fit test and a test based on the empirical process. The asymptotic properties of the test statistics are investigated. Both tests are shown to have asymptotic power 1 for local alternatives converging to the null at the rate n-r, 0 ≤ r < 1/2 . Simulation results show that both tests perform satisfactorily.     

On tests of hypotheses about treatment effects and treatmentX places interactions,in two heteroscedastic experiments

Summary Suppose we have two independent experiments conducted with a set of ‘t’ treatments each, at different places. This paper deals with two interesting problems of testing of hypotheses associated with these experiments. The first problem deals with the test of the equality of the respective treatment effects in the two experiments. The second problem is concerned with the testing of the equality of treatment into places interactions. Though we assume normality, the variance σ ₁ ² in one experiment is assumed different from the variance σ ₂ ² in the other experiment. When no information is available aboutR=σ ₁ ² /(σ ₁ ² +σ ₂ ² ) except that 0≦R≦1, tests known as ‘bilateral tests’ are proposed in the literature, to test the hypotheses mentioned above. This paper studies some important small sample properties of these bilateral tests. More specifically we study the probability of the first and second kind of error of these bilateral tests as a function ofR. When the two experiments have the same number of observations on each treatment, the bilateral test is shown to control the first kind of error. Fort=1,2, the level of these tests is a strictly convex function ofR and hence these tests can be very conservative. Some power properties of these tests are also obtained. Two tests which are equivalent to the bilateral tests for large sample sizes, and which are superior to the bilateral tests for small sample sizes, are obtained.     

A posteriori error estimates for mixed finite element approximations of elliptic problems

We derive residual based a posteriori error estimates of the flux in L ²-norm for a general class of mixed methods for elliptic problems. The estimate is applicable to standard mixed methods such as the Raviart–Thomas–Nedelec and Brezzi–Douglas–Marini elements, as well as stabilized methods such as the Galerkin-Least squares method. The element residual in the estimate employs an elementwise computable postprocessed approximation of the displacement which gives optimal order.     

Markov model for analyzing the residual static strength of a fiber-reinforced composite

The possibility of using a unified mathematical model, based on the theory of Markov chains, to describe the distribution of the conditional fatigue limit at a fixed number of loading cycles and the residual strength of a specimen after cyclic loading is demonstrated. Numerical examples are presented. __________ Translated from Mekhanika Kompozitnykh Materialov, Vol. 44, No. 4, pp. 559–568, July–August, 2008.     

Markov model of connection between the distribution of static strength and fatigue life of a fibrous composite

Within the framework of a unified mathematical model based on the Markov chain theory, an attempt is made to describe the distribution of static strength, the fatigue curve, and the accumulation of fatigue damages. It is assumed that the fatigue failure of a test specimen occurs after the destruction of some its critical microvolume consisting of two — elastic (brittle fibers) and plastic (matrix) — parts. In the second part, plastic strains accumulate as soon as the cyclic load exceeds some level. Numerical examples are presented. __________ Translated from Mekhanika Kompozitnykh Materialov, Vol. 42, No. 5, pp. 615–630, September–October, 2006.     

The Exact Hausdorff Measure Function of the Level Sets of Multi-parameter Symmetric Stable Process

In this paper, we investigate the Hausdorff measure for level sets of N-parameter Rd-valued stable processes, and develop a means of seeking the exact Hausdorff measure function for level sets of N-parameter Rd-valued stable processes. We show that the exact Hausdorff measure function of level sets of N-parameter Rd-valued symmetric stable processes of index α is Ф（r） = r^N-d/α （log log l/r）d/α when Nα 〉 d. In addition, we obtain a sharp lower bound for the Hausdorff measure of level sets of general （N, d, α） strictly stable processes.     

Reliability and efficiency of algorithms for computing the significance of the Mann–Whitney test

Motivated by recent applications of the Mann–Whitney U test to large data sets we took a critical look at current methods for computing its significance. Surprisingly, we found that the two fastest and most popular tools for exact computation of the test significance, Dinneen and Blakesley’s and Harding’s, can exhibit large numerical errors even in moderately large datasets. In addition, another method proposed by Pagano and Tritchler also suffers from a similar numerical instability and can produce inaccurate results. This motivated our development of a new algorithm, mw-sFFT, for the exact computation of the Mann–Whitney test with no ties. Among the class of exact algorithms that are numerically stable, mw-sFFT has the best complexity: O(m ² n) versus O(m ² n ²) for others, where m and n are the two sample sizes. This asymptotic efficiency is also reflected in the practical runtime of the algorithm. In addition, we also present a rigorous analysis of the propagation of numerical errors in mw-sFFT to derive an error guarantee for the values computed by the algorithm. The reliability and efficiency of mw-sFFT make it a valuable tool in compuational applications and we plan to provide open-source libraries for it in C++ and Matlab.     

Correlation between the strength of fiber-reinforced plastics and the adhesive strength of fiber—Matrix joints

The relationship between the strength (σc) of unidirectional fiber-reinforced plastics in different stressed states and the interfacial strength of their components is investigated. The shear adhesive strength (τ0) of fiber—matrix joints determined by the pull-out technique is used as a measure of the interfacial strength. To obtain the correlation curves betweenσc andτ0, the experimental results are used, where both the plastic and adhesive strength change under the influence of a single factor. In this case, such factors are the fiber surface treatment, nature and composition of polymer matrices, and test temperature. It is shown that the strength of the glass, carbon, and boron plastics increases practically linearly with increased interfacial strength. Such a behavior is observed in any loading conditions (tension, shear, bending, and compression). Sometimes, a small (10–20%) increase in the adhesive strength induces a significant (50–70%) growth in the material strength. Therefore, the interface is the “weak link” in these composites. The shape of theσc—τ0 curves for composites based on the high-strength and high-modulus aramid fibers and different thermoreactive matrices depends on the nature of the fiber and the type of stress state. In many cases, the composite strength does not depend on the interfacial strength. Then, the fiber itself is the “weak link” in these composites. Submitted to the 11th International Conference on Mechanics of Composite Materials (Riga, June 11–15, 2000). Translated from Mekhanika Kompozitnykh Materialov, Vol. 36, No. 3, pp. 291–304, May–June, 2000.     

Raman spectroscopy investigation of stiffness change and residual strains due to matrix cracking

The matrix cracking models developed for cross-ply composite laminates have been poorly extended in the past to more complex geometries used in practice, and they are still under development. In this paper, a new detailed analysis of the effect of matrix cracking on the behaviour of cross-ply and [0/45]_s laminates under uniaxial tension is attempted. The model used in this work is applicable both to cross-ply laminates and unbalanced systems. It gives exact closed-form expressions for all thermomechanical properties of a general symmetric laminate with cracks in arbitrary layers. The theoretical approach is backed by experimental data obtained by microscopic strain-state variation measurements within a specimen, with using the technique of laser Raman spectroscopy. Glass-fibre-reinforced epoxy systems were investigated. Embedded aramid fibres-sensors within the 0° ply and near the 0°/θ ° interface were necessary due to the poor Raman signal of glass. Using experimental Raman data, the residual strain and the stiffness reduction are determined as functions of increase in crack density. The stiffness reduction is predicted with a high accuracy, whereas the measured residual strains are larger than predicted. The good results for the reduction in the elastic modulus show that the basic assumption of the model is accurate. The difference is explained by the viscoelastic-viscoplastic behaviour of the off-axis layer in shear, which in creases the “apparent” residual strain. Russian translation published in Mekhanika Kompozitnykh Materialov, Vol. 42, No. 6, pp. 771–786, November–December, 2006.     

On the k-sample Behrens-Fisher problem for high-dimensional data

For several decades, much attention has been paid to the two-sample Behrens-Fisher (BF) problem which tests the equality of the means or mean vectors of two normal populations with unequal variance/covariance structures. Little work, however, has been done for the k-sample BF problem for high dimensional data which tests the equality of the mean vectors of several high-dimensional normal populations with unequal covariance structures. In this paper we study this challenging problem via extending the famous Scheffe’s transformation method, which reduces the k-sample BF problem to a one-sample problem. The induced one-sample problem can be easily tested by the classical Hotelling’s T ² test when the size of the resulting sample is very large relative to its dimensionality. For high dimensional data, however, the dimensionality of the resulting sample is often very large, and even much larger than its sample size, which makes the classical Hotelling’s T ² test not powerful or not even well defined. To overcome this difficulty, we propose and study an L ²-norm based test. The asymptotic powers of the proposed L ²-norm based test and Hotelling’s T ² test are derived and theoretically compared. Methods for implementing the L ²-norm based test are described. Simulation studies are conducted to compare the L ²-norm based test and Hotelling’s T ² test when the latter can be well defined, and to compare the proposed implementation methods for the L ²-norm based test otherwise. The methodologies are motivated and illustrated by a real data example. The work was supported by the National University of Singapore Academic Research Grant (Grant No. R-155-000-085-112)     

A Translation and Rotation Invariant Gauss–Newton Like Scheme for Image Registration

A Gauss–Newton like method is considered to obtain a d–dimensional displacement vector field , which minimizes a suitable distance measure D between two images. The key to find a minimizer is to substitute the Hessian of D with the Sobolev-H²(Ω)^d norm for . Since the kernel of the associated semi-norm consists only of the affine linear functions we can show in this way, that the solution of each Newton step is a linear combination of an affine linear transformation and an affine-free nonlinear deformation. Our approach is based on the solution of a sequence of quadratic subproblems with linear constraints. We show that the resulting Karush–Kuhn–Tucker system, with a 3×3 block structure, can be solved uniquely and the Gauss–Newton like scheme can be separated into two separated iterations. Finally, we report on synthetic as well as on real-life data test runs. AMS subject classification (2000) 65F20, 68U10     

Initial value problem and relaxation limits of the hamer model for radiating gases in several space variables

We study the initial value problem for a hyperbolic-elliptic coupled system with L ^∞ initial data. We prove global-in-time existence and uniqueness for that model by means of contraction and comparison properties. Moreover, after suitable scalings, we analyze both the hyperbolic–hyperbolic and the hyperbolic–parabolic relaxation limits for the model itself.     

Polynomial quasisolutions of linear second-order differential-difference equations

We consider the scalar linear second-order differential-difference equation with delay {fx159-01}. This equation is investigated by the method of polynomial quasisolutions based on the representation of an unknown function in the form of a polynomial {ie159-01}. Upon the substitution of this polynomial in the original equation, the residual Δ(t) = O(t ^N−1) appears. An exact analytic representation of this residual is obtained. We show the close connection between a linear differential-difference equation with variable coefficients and a model equation with constant coefficients, the structure of whose solution is determined by the roots of the characteristic quasipolynomial. __________ Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 60, No. 1, pp. 140–152, January, 2008.     

Rate of Approximation of Regular Convex Bodies by?Convex Algebraic Level Surfaces

We consider the problem of the approximation of regular convex bodies in ℝ ^d by level surfaces of convex algebraic polynomials. Hammer (in Mathematika 10, 67–71, 1963) verified that any convex body in ℝ ^d can be approximated by a level surface of a convex algebraic polynomial. In Jaen J. Approx. 1, 97–109, 2009 and subsequently in J. Approx. Theory 162, 628–637, 2010 a quantitative version of Hammer’s approximation theorem was given by showing that the order of approximation of convex bodies by convex algebraic level surfaces of degree n is \frac1n\frac{1}{n}. Moreover, it was also shown that whenever the convex body is not regular (that is, there exists a point on its boundary at which the convex body possesses two distinct supporting hyperplanes), then \frac1n\frac{1}{n} is essentially the sharp rate of approximation. This leads to the natural question whether this rate of approximation can be improved further when the convex body is regular. In this paper we shall give an affirmative answer to this question. It turns out that for regular convex bodies a o(1/n) rate of convergence holds. In addition, if the body satisfies the condition of C ²-smoothness the rate of approximation is O(\frac1n²)O(\frac{1}{n^{2}}).     

A permutation-coded evolutionary strategy for multi-objective GSM network planning

The base station placement problem, with n potential candidate sites is NP-Hard with 2 ⁿ solutions (Mathar and Niessen, Wirel. Netw. 6, 421–428, 2000). When dimensioned on m unknown variable settings (e.g., number of power settings + number of tilt settings, etc.) the computational complexity becomes (m+1) ⁿ (Raisanen, PhD. thesis, 2006). We introduce a novel approach to reduce the computational complexity by dimensioning sites only once to guarantee traffic hold requirements are satisfied. This approach works by determining the maximum set of service test points candidate sites can handle without exceeding a hard traffic constraint, T _MAX . Following this, the ability of two evolutionary strategies (binary and permutation-coded) to search for the minimum set cover are compared. This reverses the commonly followed approach of achieving service coverage first and then dimensioning to meet traffic hold. To test this approach, three realistic GSM network simulation environments are engineered, and a series of tests performed. Results indicate this approach can quickly meet network operator objectives.     

Uniformly robust tests in errors-in-variables models

Consider an ordinary errors-in-variables model. The true level α _n (θ*) of a test at nominal level α and sample size n is said to be pointwise robust if α _n (θ*) → α as n → ∞ for each parameter θ*. Let Ω* be a set of values of θ*. Define α _n = sup _{θ* ∈Ω*}α _n (θ*). The test is said to be uniformly robust over Ω* if α _n → α as n → ∞. Corresponding definitions apply to the coverage probabilities of confidence sets. It is known that all existing large-sample tests for the parameters of the errors-in-variables model are pointwise robust. However, they might not be uniformly robust over certain null parameter spaces. In this paper, we construct uniformly robust tests for testing the vector coefficient parameter and vector slope parameter in the functional errors-in-variables model. These tests are established through constructing the confidence sets for the same parameters in the model with similar desirable property. Power comparisons based on simulation studies between the proposed tests and some existing tests in finite samples are also presented.     

A simple GAP-canceling algorithm for the generalized maximum flow problem

We give a simple primal algorithm for the generalized maximum flow problem that repeatedly finds and cancels generalized augmenting paths (GAPs). We use ideas of Wallacher (A generalization of the minimum-mean cycle selection rule in cycle canceling algorithms, 1991) to find GAPs that have a good trade-off between the gain of the GAP and the residual capacity of its arcs; our algorithm may be viewed as a special case of Wayne’s algorithm for the generalized minimum-cost circulation problem (Wayne in Math Oper Res 27:445–459, 2002). Most previous algorithms for the generalized maximum flow problem are dual-based; the few previous primal algorithms (including Wayne in Math Oper Res 27:445–459, 2002) require subroutines to test the feasibility of linear programs with two variables per inequality (TVPIs). We give an O(mn) time algorithm for finding negative-cost GAPs which can be used in place of the TVPI tester. This yields an algorithm with O(m log(mB/ε)) iterations of O(mn) time to compute an ε-optimal flow, or O(m ² log (mB)) iterations to compute an optimal flow, for an overall running time of O(m ³ nlog(mB)). The fastest known running time for this problem is , and is due to Radzik (Theor Comput Sci 312:75–97, 2004), building on earlier work of Goldfarb et al. (Math Oper Res 22:793–802, 1997). David P. Williamson is supported in part by an IBM Faculty Partnership Award and NSF grant CCF-0514628.     

A minimum residual algorithm for solving linear systems

A minimum residual algorithm for solving a large linear system (I+S)x=b, with b∈ℂ ⁿ and S∈ℂ ^n×n being readily invertible, is proposed. For this purpose Krylov subspaces are generated by applying S and S ^-1 cyclically. The iteration can be executed with every linear system once the coefficient matrix has been split into the sum of two readily invertible matrices. In case S is a translation and a rotation of a Hermitian matrix, a five term recurrence is devised. In memory of Germund Dahlquist (1925–2005).AMS subject classification (2000) 65F10     

Chaotic time dependence in a disordered spin system

Stochastic Ising and voter models on ℤ ^d are natural examples of Markov processes with compact state spaces. When the initial state is chosen uniformly at random, can it happen that the distribution at time t has multiple (subsequence) limits as t→∞? Yes for the d = 1 Voter Model with Random Rates (VMRR) – which is the same as a d = 1 rate-disordered stochastic Ising model at zero temperature – if the disorder distribution is heavy-tailed. No (at least in a weak sense) for the VMRR when the tail is light or d≥ 2. These results are based on an analysis of the “localization” properties of Random Walks with Random Rates. Received: 10 August 1998