exponential splittings of products of matrices and accurately computing singular values of long products accurately computing the singular values of long products of matrices is important for estimating lyapunov exponents i limn 1 n log i an a1 algorithms for computing singular values of products in fact compute the singular values of a perturbed product an en a1 e1 the question is how small are the relative errors of the singular values of the product with respect to these factorwise perturbations in general the relative errors in the singular values can be quite large however if the product has an exponential splitting then the error in the singular values is o n2maxi 2 ai ei f uniformly in n the exponential splitting property is not directly comparable with the notion of hyperbolicity in dynamical systems but is similar in philosophy

importance sampling for estimating exact probabilities in permutational inference abstract this article discusses importance sampling as an alternative to conventional monte carlo sampling for estimating exact significance levels in a broad class of two sample tests including all of the linear rank tests with or without censoring homogeneity tests based on the chi squared hypergeometric and likelihood ratio statistics the mantel haenszel trend test and zelen s test for a common odds ratio in several 2 2 contingency tables inference is based on randomly selecting 2 k contingency tables from a reference set of all such tables with fixed marginals through a network algorithm the tables are selected in proportion to their importance for reducing the variance of the estimated monte carlo p value spectacular gains up to four orders of magnitude are achieved relative to conventional monte carlo sampling the technique is illustrated on four real data sets

nonparametric tests under restricted treatment assignment rules abstract this article considers comparative clinical trials with r r 2 treatments and allocation designs that aim to balance the assignment of treatments while controlling selection bias a general form of these restricted designs was presented by smith 1984a his class of designs encompasses most of the other designs including efron s 1971 biased coin design bcd widespread use of these designs is impeded by the lack of knowledge concerning the proper statistical analyses of the resulting data in particular little is known about the small sample randomization distributions of test statistics thus current approaches rely on large sample approximations simulation techniques and distributional results derived under classical schemes such as the complete randomization design clearly none of these approaches is exact a recursion procedure is presented for computing the exact conditional randomization distribution of a wide class of test statistics under these restricted designs this proced

minimize the use of minimization with unequal allocation summary minimization as an alternative to randomization is gaining popularity for small clinical trials in response to critics questions about the proper analysis of such a trial proponents have argued that a rerandomization approach akin to a permutation test with conventional randomization can be used however they add that this computationally intensive approach is not necessary because its results are very similar to those of a t test or test of proportions unless the sample size is very small we show that minimization applied with unequal allocation causes problems that challenge this conventional wisdom

relative perturbation bound for invariant subspaces of graded indefinite hermitian matrices abstract we give a bound for the perturbations of invariant subspaces of graded indefinite hermitian matrix h d ad which is perturbed into h h d a a d such relative perturbations include an important case where h is given with an element wise relative error application of our bounds requires only the knowledge of the size of relative perturbation a and not the perturbation a itself this typically occurs when data are given with relative uncertainties when the matrix is being stored into computer memory and when analyzing some numerical algorithms subspace perturbations are measured in terms of perturbations of angles between subspaces and our bound is therefore a relative variant of the well known davis kahan sin theorem our bounds generalize some of the recent relative perturbation results

relatively robust representations of symmetric tridiagonals let ldlt be the triangular factorization of an unreduced symmetric tridiagonal matrix t i small relative changes in the nontrivial entries of l and d may be represented by diagonal scaling matrices 1 and 2 ldlt 2l 1d 1lt 2 the effect of 2 on the eigenvalues i is benign in this paper we study the inner perturbations induced by 1 suitable condition numbers govern the relative changes in the eigenvalues i we show that when j is an eigenvalue then the relative condition number of m j m j is the same for all n twisted factorizations one of which is ldlt that could be used to represent t i see section 2 we prove that as j the smallest eigenvalue has relative condition number relcond 1 o j each relcond is a rational function of we identify the poles and then use orthogonal polynomial theory to develop upper bounds on the sum of the relconds of all the eigenvalues these bounds require o n operations for an n n matrix we show that the sum of all the relconds is bounded by trace l d lt and conjecture that n ldlt the quantity trace l d lt ldlt is a natural measure of element growth in the context of this paper an algorithm for computing numerically orthogonal eigenvectors without recourse to the gram schmidt process is sketched it requires that there exist values of close to each cluster of close eigenvalues such that all the relconds belonging to the cluster are modest say 10 the sensitivity of the other eigenvalues is not important for this reason we develop o n bounds on the sum of the relconds associated with a cluster none of our bounds makes reference to the nature of the distribution of the eigenvalues within a cluster which can be very complicated

a theory for testing hypotheses under covariate adaptive randomization the covariate adaptive randomization method was proposed for clinical trials long ago but little theoretical work has been done for statistical inference associated with it practitioners often apply test procedures available for simple randomization which is controversial since procedures valid under simple randomization may not be valid under other randomization schemes in this paper we provide some theoretical results for testing hypotheses after covariate adaptive randomization we show that one way to obtain a valid test procedure is to use a correct model between outcomes and covariates including those used in randomization we also show that the simple two sample t test without using any covariate is conservative under covariate adaptive biased coin randomization in terms of its type i error and that a valid bootstrap t test can be constructed the powers of several tests are examined theoretically and empirically our study provides guidance for applications and sheds light on further research in this area

relative perturbation theory for hyperbolic eigenvalue problem we give relative perturbation bounds for eigenvalues and perturbation bounds for eigenspaces of a hyperbolic eigenvalue problem hx jx where h is a positive definite matrix and j is a diagonal matrix of signs we consider two types of perturbations when a graded matrix h d ad is perturbed in a graded sense to h h d a a d and the multiplicative perturbations of the form h h i e h i e our bounds are simple to compute compare well to the classical results and can be used when analyzing numerical algorithms

multiple representations to compute orthogonal eigenvectors of symmetric tridiagonal matrices in this paper we present an o nk procedure algorithm mr3 for computing k eigenvectors of an n n symmetric tridiagonal matrix t a salient feature of the algorithm is that a number of different ldlt products l unit lower triangular d diagonal are computed in exact arithmetic each ldlt is a factorization of a translate of t we call the various ldlt products representations of t and roughly speaking there is a representation for each cluster of close eigenvalues the unfolding of the algorithm for each matrix is well described by a representation tree we present the tree and use it to show that if each representation satisfies three prescribed conditions then the computed eigenvectors are orthogonal to working accuracy and have small residual norms with respect to the original matrix t

a relative perturbation bound for positive definite matrices abstract we give a sharp estimate for the eigenvectors of a positive definite hermitian matrix under a floating point perturbation the proof is elementary

fast enclosure of matrix eigenvalues and singular values via rounding mode controlled computation abstract modifications of bauer fike type and weyl type perturbation theorems are presented for matrix eigenvalue and singular value problems it is shown that the conditions of the presented theorems can be rigorously checked by floating point computation with rounding mode control it is stressed that verification programs can be easily constructed on usual numerical softwares like matlab computational cost of obtaining rigorous error bounds for computed eigenvalues is shown to be 6n 3 flops for a real symmetric n n matrix

wave propagation in a piezoelectric solid cylinder of arbitrary cross section in this article the wave propagation in an infinite piezoelectric solid cylinder of an arbitrary cross section is studied in the analysis the boundary conditions are satisfied by the use of nagaya s fourier expansion collocation method the frequency equations for symmetric and antisymmetric cases are obtained and analyzed numerically for longitudinal and flexural waves for the piezoceramic material pzt 4 the vibrations of circular and elliptical cylinders are investigated and the results are tabulated the results obtained for a circular cross section are compared with the available results the frequency versus the aspect ratio the ratio of the semimajor and semiminor axes of the elliptical cylinder is plotted and the dispersion curve is also presented

the transient responses of a special non homogeneous magneto electro elastic hollow cylinder for axisymmetric plane strain problem by virtue of the introduction of new dependent variable and the separation of variables technique the transient responses of a special non homogeneous magneto electro elastic hollow cylinder are transformed to two volterra integral equations of the second kind of about two functions with respect to time these integral equations can be solved successfully by means of the interpolation method then the complete solutions of displacements stresses electric potential electric displacements magnetic potential and magnetic inductions are obtained the present method is suitable for a magneto electro elastic hollow cylinder with an arbitrary thickness subjected to arbitrary axisymmetric mechanical and electromagnetic loads numerical results are finally presented

constructing exact significance tests with restricted randomization rules summary in a clinical trial comparing two treatments suppose that subjects are assigned sequentially to the treatment groups by a restricted randomization rule under the randomization model an efficient recursive algorithm is provided to generate the exact permutational distribution for linear rank statistics given the final imbalance between the number of patients in the two groups if only the significance level of the test is required the efficiency of the algorithm can be improved significantly in general by the time the size of the trial reaches 30 the large sample approximation to the permutation distribution wei smythe smith 1986 is satisfactory the new procedure is illustrated by censored survival data from a prostatic cancer clinical trial we show that if the study subjects are not a random sample from a well defined population then the treatment allocation rule used at the design stage should not be ignored in the analysis

for tridiagonals t replace t with ldl t abstract the same number of parameters determine a tridiagonal matrix t and its triangular factors l d and u the mapping t ldu is not well defined for all tridiagonals but in finite precision arithmetic l d and u determine the entries of t to more than working precision for the solution of linear equations ldu x b the advantages of factorization are clear recent work has shown that ldu is also preferable for the eigenproblem particularly in the symmetric case this essay describes two of the ideas needed to compute eigenvectors that are orthogonal without recourse to the gram schmidt procedure when some of the eigenvalues are tightly clustered in the symmetric case we must replace t or a translate of t by its triangular factors ldl t

dynamic asymmetry of piezoelectric shell structures abstract the non symmetry and asymmetric dynamic characteristics of piezoelectric shell structures are studied analytically and numerically the basic formulations of piezoelectric structures are firstly given in the tensor form the differential equations for displacements and electric potential are derived from the formulations which validate that the description in curvilinear coordinates with the electrical and mechanical coupling induces the non symmetry in generalized stiffness and then the dynamic asymmetry of piezoelectric structures also expanding the structural displacements and electric potential with various spatial characteristics can induce the non symmetry in generalized stiffness matrix of the converted ordinary differential equations then for the multi degree of freedom system with asymmetric stiffness matrix the conventional right modes or eigenvectors are proved to have not the auto orthogonality relations with the mass or stiffness weight however the left and right eigenvectors have the cross orthogonality relations with the mass or stiffness weight which can be used for uncoupling the asymmetric systems furthermore the non symmetric stiffness matrix is divided into the corresponding symmetric and anti symmetric stiffness matrices the algebraic equations for eigenvalues and singular values of the asymmetric symmetric and anti symmetric systems are given it is obtained that the eigenvalues are equal to the corresponding singular values for the symmetric system the singular values are equal to the products of the unit imaginary number and corresponding eigenvalues for the anti symmetric system and the differences of the eigenvalues and corresponding singular values for the asymmetric system depend on the anti symmetric stiffness also the upper limits of the absolute and relative differences of the singular values of the asymmetric system to the corresponding eigenvalues of the symmetric system and the upper and lower limits of the singular values of the asymmetric system are obtained respectively finally the spherically symmetric piezoelectric shell described in spherical coordinates is studied in detail to show the asymmetric dynamic characteristics the differential equations for the radial displacement and electric potential of the shell structure are obtained to illustrate the non symmetric generalized stiffness involving elastic and piezoelectric constants eliminating the electric potential expanding the displacement in space and using the galerkin method yield the ordinary differential equations which represent a multi degree of freedom dynamic system with the asymmetric generalized stiffness matrix numerical results are given to illustrate the eigenvalues and modes of the asymmetric system different from those of the corresponding symmetric system the relative differences of the eigenvalues of the asymmetric to symmetric system for different piezoelectric constants and geometric parameters and the non orthogonality of the left or right eigenvectors for the asymmetric system the analytical and numerical results on the non symmetric dynamics of piezoelectric shell structures are useful for accurate analysis and design

inference under covariate adaptive randomization this article studies inference for the average treatment effect in randomized controlled trials with covariate adaptive randomization here by covariate adaptive randomization we mean randomizati

k treatment comparisons with restricted randomization rules in clinical trials in the course of conducting a clinical trial to compare k 2 treatments it is often desirable to balance the trial with respect to the assignments of patients to treatments on the other hand some form of randomization of treatment assignments is essential for reducing experimental bias in this article the large sample approximation to the null distribution of k sample randomization tests generated from a broad class of restricted randomization rules is derived the implication of this result for conditional inference is also discussed

accurate computation of the product induced singular value decomposition with applications we present a new algorithm for floating point computation of the singular value decomposition svd of the product b tau c where b and c are full row rank matrices the algorithm replaces the pair b c with an equivalent pair b c and then it uses the jacobi svd algorithm to compute the svd of the explicitly computed matrix b tau c in this way each nonzero singular value sigma is approximated with some sigma delta sigma where the relative error delta sigma sigma is up to a factor of the dimensions of order boldmath varepsilon unboldmath min delta in cal d kappa 2 delta b min delta in cal d kappa 2 delta c where cal d denotes the set of diagonal nonsingular matrices kappa 2 cdot denotes the spectral condition number and boldmath varepsilon unboldmath is the roundoff unit of floating point arithmetic the new algorithm is applied to the eigenvalue problem hm x lambda x with symmetric positive definite h and m it is shown that each eigenvalue lambda is computed with high relative accuracy and that the relative error delta lambda lambda of the computed approximation lambda delta lambda is up to a factor of the dimension of order boldmath varepsilon unboldmath min delta in cal d kappa 2 delta h delta min delta in cal d kappa 2 delta m delta the new algorithm can also be used for accurate svd computation of a single matrix g that admits an accurate factorization g b tau c

accurate singular values and differential qd algorithms in september 1991 j w demmel and w m kahan were awarded the second siam prize in numerical linear algebra for their paper accurate singular values of bidiagonal matrices 1 referred to as dk hereafter among several valuable results was the observation that the standard bidiagonal qr algorithm used in linpack 2 and in many other svd programs can be simplified when the shift is zero and of greater importance no subtractions occur the last feature permits very small singular values to be found with almost all the accuracy permitted by the data and at no extra cost

dynamic solution of a multilayered orthotropic piezoelectric hollow cylinder for axisymmetric plane strain problems the dynamic solution of a multilayered orthotropic piezoelectric infinite hollow cylinder in the state of axisymmetric plane strain is obtained by the method of superposition the solution is divided into two parts one is quasi static and the other is dynamic the quasi static part is derived by the state space method and the dynamic part is obtained by the separation of variables method coupled with the initial parameter method as well as the orthogonal expansion technique by using the obtained quasi static and dynamic parts and the electric boundary conditions as well as the electric continuity conditions a volterra integral equation of the second kind with respect to a function of time is derived which can be solved successfully by means of the interpolation method the displacements stresses and electric potentials are finally obtained the present method is suitable for a multilayered orthotropic piezoelectric infinite hollow cylinder consisting of arbitrary layers and subjected to arbitrary axisymmetric dynamic loads numerical results are finally presented and discussed

approaches to expanding the two arm biased coin randomization to unequal allocation while preserving the unconditional allocation ratio the paper discusses three methods for expanding the biased coin randomization bcr to unequal allocation while preserving the unconditional allocation ratio at every step the first method originally proposed in the contexts of bcr and minimization is based on mapping from an equal allocation multi arm bcr despite the improvement proposed in this paper to ensure tighter adherence to the targeted unequal allocation this method still distributes the probability mass at least as wide as the permuted block randomization pbr this works for smaller block sizes but for larger block sizes a tighter control of the imbalance in the treatment assignments is desired the second method which has two versions allows to tighten the distribution of the imbalance compared with that achieved with the pbr however the distribution of the imbalance remains considerably wider than that of the brick tunnel randomization the unequal allocation procedure with the tightest possible imbalance distribution among all allocation ratio preserving procedures with the same allocation ratio finally the third method the bcr with a preset proportion of maximal forcing mimics the properties of the equal allocation bcr with maximum forcing it approaches the brick tunnel randomization similar to how 1 1 bcr approaches 1 1 pbr with the permuted block size of 2 the equal allocation procedure with the lowest possible imbalance when the bias approaches 1 with minimum forcing the bcr with a preset proportion of maximal forcing approaches complete randomization similar to 1 1 bcr copyright 2017 john wiley sons ltd

optimal perturbation bounds for the hermitian eigenvalue problem there is now a large literature on structured perturbation bounds for eigenvalue problems of the formvbox hx mx where h and m are hermitian these results give relative error bounds on the ith eigenvalue i of the form i i i and bound the error in the ith eigenvector in terms of the relative gap minj i i j i j 1 2 in general this theory usually restricts h to be nonsingular and m to be positive definite we relax this restriction by allowing h to be singular for our results on eigenvalues weallow m to be positive semi definite and for a few results we allow it to be more general for these problems for eigenvalues that are not zero or infinity under perturbation it is possible to obtain local relative error bounds thus a wider class of problems may be characterized by this theory although it is impossible to give meaningful relative error bounds on eigenvalues that are not bounded away from zero we show that the error in the subspace associated with those eigenvalues can be characterized meaningfully

conditional monte carlo randomization tests for regression models we discuss the computation of randomization tests for clinical trials of two treatments when the primary outcome is based on a regression model we begin by revisiting the seminal paper of gail tan and piantadosi 1988 and then describe a method based on monte carlo generation of randomization sequences the tests based on this monte carlo procedure are design based in that they incorporate the particular randomization procedure used we discuss permuted block designs complete randomization and biased coin designs we also use a new technique by plamadeala and rosenberger 2012 for simple computation of conditional randomization tests like gail tan and piantadosi we focus on residuals from generalized linear models and martingale residuals from survival models such techniques do not apply to longitudinal data analysis and we introduce a method for computation of randomization tests based on the predicted rate of change from a generalized linear mixed model when outcomes are longitudinal we show by simulation that these randomization tests preserve the size and power well under model misspecification

estimating exact p values by the method of control variates or monte carlo rescue abstract despite algorithmic advances in exact nonparametric inference problems often occur that are too large for exact p value computations but too sparse for reliable asymptotic results in these situations monte carlo methods are a good compromise they bound the true p value within a confidence interval but a factor discouraging the use of monte carlo p values is their sensitivity to the random number sequence one can overcome this drawback by computing a 99 c i confidence interval of width less than 0001 then the estimated p values become insensitive to the random number sequence up to three decimals for all practical purposes these estimates are invariant to random number sequences the usual monte carlo method requires millions of samples to yield such an invariant estimate the monte carlo scheme presented here decreases the sample size by two to three orders of magnitude we illustrate this method with tests for r c tables two sample survival data and stratified 2 2 tables

exact and asymptotically robust permutation tests given independent samples from p and q two sample permutation tests allow one to construct exact level tests when the null hypothesis is p q on the other hand when comparing or testing particular parameters of p and q such as their means or medians permutation tests need not be level or even approximately level in large samples under very weak assumptions for comparing estimators we provide a general test procedure whereby the asymptotic validity of the permutation test holds while retaining the exact rejection probability in finite samples when the underlying distributions are identical a quite general theory is possible based on a coupling construction as well as a key contiguity argument for the binomial and hypergeometric distributions the ideas are broadly applicable and special attention is given to a nonparametric k sample behrens fisher problem whereby a permutation test is constructed which is exact level under the hypothesis of identical distributions but has asymptotic rejection probability under the more general null hypothesis of equality of means a monte carlo simulation study is performed 2010 msc subject classifications primary 62e20 secondary 62g10

weyl type relative perturbation bounds for eigensystems of hermitian matrices we present a weyl type relative bound for eigenvalues of hermitian perturbations a e of not necessarily definite hermitian matrices a this bound given in function of the quantity a 1 2ea 1 2 2 that was already known in the definite case is shown to be valid as well in the indefinite case we also extend to the indefinite case relative eigenvector bounds which depend on the same quantity as a consequence new relative perturbation bounds for singular values and vectors are also obtained using matrix differential calculus techniques we obtain for eigenvalues a sharper first order bound involving the logarithm matrix function which is smaller than not only for small e as expected but for any perturbation

preserving the allocation ratio at every allocation with biased coin randomization and minimization in studies with unequal allocation the demand for unequal allocation in clinical trials is growing most commonly the unequal allocation is achieved through permuted block randomization however other allocation procedures might be required to better approximate the allocation ratio in small samples reduce the selection bias in open label studies or balance on baseline covariates when these allocation procedures are generalized to unequal allocation special care is to be taken to preserve the allocation ratio at every allocation step this paper offers a way to expand the biased coin randomization to unequal allocation that preserves the allocation ratio at every allocation the suggested expansion works with biased coin randomization that balances only on treatment group totals and with covariate adaptive procedures that use a random biased coin element at every allocation balancing properties of the allocation ratio preserving biased coin randomization and minimization are described through simulations it is demonstrated that these procedures are asymptotically protected against the shift in the rerandomization distribution identified for some examples of minimization with 1 2 allocation the asymptotic shift in the rerandomization distribution of the difference in treatment means for an arbitrary unequal allocation procedure is explicitly derived in the paper

on computing accurate singular values and eigenvalues of matrices with acyclic graphs abstract it is known that small relative perturbations in the entries of a bidiagonal matrix only cause small relative perturbations in its singular values independentof the values of the matrix entries in this paper we show that a matrix has this property if and only if its associated bipartite graph is acyclic we also show how to compute the singular values of such a matrix to high relative accuracy the same algorithm can compute eigenvalues of symmetric matrices with acyclic graphs with tiny componentwise relative backward error this class includes tridiagonal matrices arrow matrices and exponentially many others

an overview of relative sin t theorems for invariant subspaces of complex matrices abstract relative perturbation bounds for invariant subspaces of complex matrices are reviewed with emphasis on bounding the sines of the largest principal angle between two subspaces i e sin theorems the goal is to provide intuition as well as an idea for why the bounds hold and why they look the way they do relative bounds have the advantage of being better at exploiting structure in a perturbation than absolute bounds therefore the reaction of subspaces to relative perturbations can be different than to absolute perturbations in particular there are certain classes of relative perturbations to which subspaces of indefinite hermitian matrices can be more sensitive than subspaces of definite matrices

spectral sensitivity of products of bidiagonals abstract simple formulae are derived for certain relative condition numbers tridiagonal matrices may be represented as products of bidiagonals in various ways depending on properties such as symmetry and positive definiteness the condition numbers give the amplification factor for relative changes in a nonzero eigenvalue caused by relative changes in an entry of a bidiagonal factor the formulae show that in many but not all cases these condition numbers are of modest size several examples illustrate the results and raise new questions

significance tests with restricted randomization design in an experiment comparing the effectiveness of two treatments a and b subjects arrive at an experimental site sequentially and must be assigned immediately to one of the two treatment groups the complete randomization scheme tossing a fair coin each time independently to decide which treatment will be used for the next subject has been widely employed because this scheme is free of different kinds of experimental bias efron 1971 and it also provides a basis for statistical inference lehmann 1975 chapter 1 but in a small experiment complete randomization may result in a severe imbalance between the numbers of subjects assigned to the different treatment groups a simple way to attain a perfectly balanced experiment is to use the systematic design abab however this plan can easily bias the results of the experiment in a number of ways for example if the experimenter is predisposed toward treatment a and if he knows or guesses that treatment a will be the next assignment then he may tend to choose a subject with a high expected response because of this bias at the end of the experiment treatment a might appear to be significantly better than treatment b even if there is in fact no difference between the two obviously the systematic design maximizes this kind of bias several compromises between a perfectly balanced assignment scheme and a completely randomized one have been proposed for example the permuted block design the biased coin design efron 1971 the urn design wei 1977 1978a the adaptive biased coin design wei 1978b and the random allocation design stigler 1969 wei 1978c simon 1979 has given an excellent review in particular the urn and adaptive biased coin designs force a small experiment to be balanced but they tend to the complete randomization scheme as the size of the experiment increases recently a specific urn design has been shown to be d optimal by atkinson 1982 sometimes it may be inappropriate to postulate a population from which the subjects in the experiment have been obtained by random sampling instead a randomization model lehmann 1975 chapter 1 is utilized when the data are analysed with this approach the significance tests used for testing the null hypothesis ho that there is no difference between a and b must be only those tests generated by the experimental randomization design actually employed for any given sequence of responses from the subjects one can tabulate all possible patterns of treatment assignments to subjects using the restricted randomization rule and calculate the corresponding treatment

stochastic response analysis of piezoelectric axisymmetric hollow cylinders the stochastic response of a piezoelectric thick axisymmetric hollow cylinder in plane strain under boundary stochastic excitations is analyzed and calculated the stochastic stress and electric potential boundary conditions of the piezoelectric hollow cylinder are converted into homogeneous boundary conditions by transformations that yields the electrical and mechanical coupling partial differential equations of motion with damping and stochastic excitations the equation for electric potential is integrated radially to obtain the electric potential as a function of displacement and the displacement is expanded as a series in terms of the legendre polynomials the partial differential equation for displacement is further converted into ordinary differential equations by using the galerkin method which represent a stochastic multi degree of freedom system with asymmetric stiffness matrix due to the asymmetric electrical and mechanical coupling and the transformed boundary conditions the frequency response function matrix and correlation function matrix of the system response are derived from these equations based on the theory of random vibration the expressions of mean square displacement and electric potential of the piezoelectric hollow cylinder are finally obtained and illustrated by numerical results for non white stochastic excitations the frequency response characteristics and electrical and mechanical coupling properties are explored

dynamic treatment allocation adjusting for prognostic factors for more than two treatments methods of sequential allocation of one of k treatments to patients while controlling for important prognostic factors are developed and compared we focus on methods that are based on optimality theory begg and iglewicz 1980 biometrics 36 81 90 atkinson 1982 biometrika 69 61 67 the permuted block procedure zelen 1974 journal of chronic diseases 27 365 375 and the compromise method faraggi and reiser 1991 communication in statistics simulation and computation 20 243 254 these methods are extended to the k treatments case and are evaluated in terms of efficiency and balance it is shown that each method achieved the best results in the criterion it was designed to optimize i e within stratum balance for the permuted block allocation and efficiency for the allocations that are based on optimality theory but did not do well with other criteria the compromise method on the other hand has good overall properties in terms of both balance and efficiency

coefficient of maass forms and the siegel zero with an appendix by dorian goldfeld jeffrey hoffstein and daniel lieman

optimum biased coin designs for sequential clinical trials with prognostic factors patients in a clinical trial arrive sequentially and are assigned to one of t treatments this assignment should maintain a balance between the numbers receiving each treatment yet should be sufficiently random to avoid any suspicion of conscious or unconscious cheating to balance these requirements efron 1971 introduced biased coin designs for the comparison of two treatments in which allocation of the treatment is determined probabilistically but with a bias towards the underrepresented treatment one disadvantage of efron s scheme is that it does not include balance over covariates or prognostic factors which may affect the response of the patient to the treatment biased coin schemes which do force balance over both treatments and prognostic factors are given by pocock simon 1975 and efron 1980 the properties of the designs have been elucidated by numerical studies and they are now being increasingly used in clinical trials reviews of the literature and discussions of the practical implications of the designs are given by pocock 1979 and simon 1979 however the designs suffer from the disadvantage that they rely on arbitrary functions to achieve the desired balance the procedures thus lack a firm theoretical framework an alternative approach in the presence of prognostic factors begg iglewicz 1980 uses optimum design theory to suggest a deterministic design criterion which is then modified for computational convenience in this paper i use optimum design theory to provide a procedure of the biased coin type for an arbitrary number of treatments in the presence or absence of prognostic factors this has the theoretical advantage of obviating dependence on a series of arbitrary functions the necessary optimum design theory is presented in 2 and in 3 applied to biased coin experiments the simplest case that of two treatments in the absence of prognostic factors is studied in 4 the extension to three or more treatments is in 5 followed in 6 by the allowance for prognostic factors although in the biased coin designs allocation of treatments is made probabilistically the results of the paper can be used for the sequential construction of da optimum designs in which randomization is not introduced by the experimenter all the designs mentioned so far are sequential but not adaptive in the useful distinction made by pericchi 1981 they are data dependent but not outcomedependent each allocation depends on the previous allocations and prognostic factors

sequential monitoring with conditional randomization tests sequential monitoring in clinical trials is often employed to allow for early stopping and other interim decisions while maintaining the type i error rate however sequential monitoring is typically described only in the context of a population model we describe a computational method to implement sequential monitoring in a randomization based context in particular we discuss a new technique for the computation of approximate conditional tests following restricted randomization procedures and then apply this technique to approximate the joint distribution of sequentially computed conditional randomization tests we also describe the computation of a randomization based analog of the information fraction we apply these techniques to a restricted randomization procedure efron s biometrika 58 1971 403 417 biased coin design these techniques require derivation of certain conditional probabilities and conditional covariances of the randomization procedure we employ combinatoric techniques to derive these for the biased coin design 1 introduction sequential monitoring refers to analyzing data periodically during the course of a clinical trial with the purpose of detecting early evidence in support of or against a hypothesis a desirable feature of such a monitoring plan would be flexible inspections of the data that can occur at arbitrary time points at the same time sequentially tested hypotheses must maintain the overall probability of type i error at the prespecified level since repeated testing is known to inflate it the lan and demets 1983 error spending approach for sequential monitoring allows this the approach makes use of a type i error spending function which depends on the amount of statistical information available at the time of the interim inspection in

conditional inference for restricted randomization designs on verifie la conjecture de wei smythe et smith 1986 sous les memes regles de randomisation restreinte et une hypothese additionnelle douce sur les scores

magneto thermo electro elastic transient response in a piezoelectric hollow cylinder subjected to complex loadings abstract the article presents an analytical solution for magneto thermo electro elastic problems of a piezoelectric hollow cylinder placed in an axial magnetic field subjected to arbitrary thermal shock mechanical load and transient electric excitation using an interpolation method solves the volterra integral equation of the second kind caused by interaction among magnetic thermal electric and mechanical fields the electric displacement is determined thus the exact expressions for the transient responses of displacement stresses electric displacement electric potential and perturbation of the magnetic field vector in the piezoelectric hollow cylinder are obtained by means of hankel transforms laplace transforms and inverse laplace transforms from sample numerical calculations it is seen that the present method is suitable for a piezoelectric hollow cylinder subjected to arbitrary thermal shock mechanical load and transient electric excitation and the result carried out may be used as a reference to solve other transient coupled problems of magneto thermo electro elasticity

wave propagation in a piezoelectric rod of 6mm symmetry abstract the wave propagation in a piezoelectric rod of 6 mm symmetry is investigated by applying a 3 d piezoelectric elastic model a self adjoint method is introduced to solve this problem this method avoids calculating the generalized eigenvalue equation it completely draws the dispersion curves in the forms of quasi p wave quasi sv wave and quasi sh wave under the self adjoint boundary condition and it can evaluate the dispersion curves of all kinds of boundary conditions as an example the dispersion curves of plt 5h are completely drawn we also found the quasi sv wave has standing wave phenomenon in the plt 5h rod in addition the relation of dispersion curves among different boundary conditions is discussed and an experiment method is introduce to decide the dispersion curves for another boundary conditions

relative perturbation theory iv sin 2θ theorems abstract the double angle theorems of davis and kahan bound the change in an invariant subspace when a hermitian matrix a is subject to an additive perturbation a a a a this paper supplies analogous results when a is subject to a congruential or multiplicative perturbation a a d ad the relative gaps that appear in the bounds involve the spectrum of only one matrix either a or a in contrast to the gaps that appear in the single angle bounds the double angle theorems do not directly bound the difference between the old invariant subspace s and the new one s but instead bound the difference between s and its reflection j s where the mirror is s and j reverses s the orthogonal complement of s the double angle bounds are proportional to the departure from the identity and from orthogonality of the matrix d def d 1 jdj note that d is invariant under the transformation d d for 0 whereas the single angle theorems give bounds proportional to d s departure from the identity and from orthogonality the corresponding results for the singular value problem when a nonsquare matrix b is perturbed to b d 1 bd 2 are also presented

free vibrations of piezoelectric cylindrical shells filled with compressible fluid abstract three displacement functions are introduced to represent each mechanical displacement according to the three dimensional theory after expanding these functions and the electric potential with orthogonal series the free vibration equation of piezoelectric cylindrical shells satisfying ss3 edge conditions can be obtained the equation was solved by utilizing bessel functions with complex arguments the effects of compressible fluid on shells are considered and some new phenomena which are exclusive for piezoelectric cylindrical shells are reported results of empty infinite piezoelectric cylindrical shells are compared to those presented in relative references some lowest frequencies that were missed by paul and venkatesan paul h s and venkatesan m 1987 vibrations of hollow circular cylinder of piezoelectric ceramics journal of the acoustic society of america82 852 856 were discovered

analysis of axially polarized piezoelectric cylinders with arbitrary boundary conditions on flat surfaces a method to analyze axially polarized piezoelectric ceramic cylinders with arbitrary length to radius ratio and arbitrary boundary conditions on the flat surfaces is presented the method is based on the use of functions that individually exactly satisfy the axisymmetric equations of motion and the gauss electrostatic conditions the axial and radial components of displacement and the potential are expressed as weighted sums of these functions the functions form complete sets in the radial direction and arbitrary boundary conditions can therefore be satisfied on the flat surfaces on the curved surface certain uniform boundary conditions can be exactly satisfied the weights are easily determined by using the orthogonal property of the functions the input electrical admittance is a function of only the average boundary conditions and is very easily determined several special cases and numerical results are presented for electrically and mechanically excited cylinders to illustrate the method the se

wave propagation in axially polarized piezoelectric hollow cylinders of sector cross section in this paper the problem of electroelastic waves propagating in piezoelectric hollow cylinders of sector cross section is studied for the case when the boundary surfaces of sector cut are covered by non extensible membranes the three dimensional linear equations of motion for the piezoelectric cylinder are analytically integrated and different boundary conditions on the cylindrical surfaces yield frequency equations which relate the frequencies of elastic waves to their wavenumbers numerical results for waveguides with various boundary conditions are presented to illustrate the approach analysis of the dispersion spectra is carried out and cutoff frequencies are obtained and characterized mode asymptotic behavior and amplitude distributions of wave characteristics are analyzed the main effects of their transformation by variation of the angular measure and the ratio of inner and outer radii are discussed the results obtained are in good agreement with the results for the special case of a hollow semicircular cylinder

relative perturbation techniques for singular value problems a technique is presented for deriving bounds on the relative change in the singular values of a real matrix or the eigenvalues of a real symmetric matrix due to a perturbation as well as bounds on the angles between the unperturbed and perturbed singular vectors or eigenvectors the class of perturbations considered consists of all delta b for which b delta b d l bd r for some nonsingular matrices d l and d r this class includes componentwise relative perturbations of a bidiagonal or biacyclic matrix and perturbations that annihilate the off diagonal block in a block triangular matrix many existing relative perturbation and deflation bounds are derived from results for this general class of perturbations also some new relative perturbation and deflation results for the singular values and vectors of biacyclic triangular and shifted triangular matrices are presented

exact optimum coin bias in efron s randomization procedure efron s biased coin design is a restricted randomization procedure that has very favorable balancing properties yet it is fully randomized in that subjects are always randomized to one of two treatments with a probability less than 1 the parameter of interest is the bias p of the coin which can range from 0 5 to 1 in this note we propose a compound optimization strategy that selects p based on a subjected weighting of the relative importance of the two fundamental criteria of interest for restricted randomization mechanisms namely balance between the treatment assignments and allocation randomness we use exact and asymptotic distributional properties of efron s coin to find the optimal p under compound criteria involving imbalance variability expected imbalance selection bias and accidental bias for both small moderate trials and large samples

recursive transformed component statistical analysis for incipient fault detection abstract this paper presents a new data driven process monitoring method called recursive transformed component statistical analysis rtcsa for the purpose of incipient fault detection without space partition rtcsa processes data in sliding windows to obtain orthogonal transformed components tcs recursively using rank one modification the statistical information of tcs can reveal some important process features implying that faults can be detected by monitoring the statistics of tcs with second order statistics the detection index reduces to relative changes of ordered eigenvalues of the sample covariance matrix fault detectability is analyzed in a statistical sense leading to the analysis of the eigenvalues of stochastic matrices including the closed form expressions for the probability distribution function of the arbitrary l th largest eigenvalue of a class of real uncorrelated wishart matrices it indicates that a scaled ordered eigenvalue is sensitive to small changes the structure of the detection index ensures that rtcsa is sensitive to incipient faults compared with existing multivariate statistical process monitoring approaches such as principal component analysis pca and its variants the superior detectability of rtcsa is illustrated by a numerical example and the tennessee eastman process

interference between units in randomized experiments in a randomized experiment comparing two treatments there is interference between units if applying the treatment to one unit may affect other units interference implies that treatment effects are not comparisons of two potential responses that a unit may exhibit one under treatment and the other under control but instead are inherently more complex interference is common in social settings where people communicate compete or spread disease in studies that treat one part of an organism using a symmetrical part as control in studies that apply different treatments to the same organism at different times and in many other situations available statistical tools are limited for instance fisher s sharp null hypothesis of no treatment effect implicitly entails no interference and so his randomization test may be used to test no effect but conventional ways of inverting the test to obtain confidence intervals say for an additive effect are not applicable with interference another commonly used

designing experiments with respect to standardized optimality criteria we introduce a new class of standardized optimality criteria which depend on standardized covariances of the least squares estimators and provide an alternative to the commonly used criteria in design theory besides a nice statistical interpretation the new criteria satisfy an extremely useful invariance property which allows an easy calculation of optimal designs on many linearly transformed design spaces

dynamic responses of piezoelectric hollow cylinders in an axial magnetic field abstract this paper presents an analytical solution for the interaction of electric potentials electric displacements elastic deformations and mechanical loads and describes electromagnetoelastic responses and perturbation of the magnetic field vector in a piezoelectric hollow cylinder subjected to sudden mechanical load and electric potential an interpolation method is applied to solve the volterra integral equation of the second kind caused by interactions between different physical fields by means of finite integral transforms laplace transforms and their inverse transforms the exact expressions for the dynamic responses of stresses electric displacements electric potentials and perturbation response of an axial magnetic field vector in the piezoelectric hollow cylinders are obtained the present method is suitable for piezoelectric hollow cylinders in an axial magnetic field subjected to arbitrary mechanical loads and electrical potential shocks finally numerical results are carried out and discussed

updating the singular value decomposition the spectral decomposition of a symmetric matrix a with small off diagonal and distinct diagonal elements can be approximated using a direct scheme of r davies and modi linear algebra appl 77 1986 61 in this paper a generalization of this method for computing the singular value decomposition of close to diagonal a rm n is presented when a has repeated or close singular values it is possible to apply the direct method to split the problem in two with one part containing the well separated singular values and one requiring the computation of the close singular values

randomization based interval estimation in randomized clinical trials randomization based interval estimation takes into account the particular randomization procedure in the analysis and preserves the confidence level even in the presence of heterogeneity it is distinguished from population based confidence intervals with respect to three aspects definition computation and interpretation the article contributes to the discussion of how to construct a confidence interval for a treatment difference from randomization tests when analyzing data from randomized clinical trials the discussion covers i the definition of a confidence interval for a treatment difference in randomization based inference ii computational algorithms for efficiently approximating the endpoints of an interval and iii evaluation of statistical properties ie coverage probability and interval length of randomization based and population based confidence intervals under a selected set of randomization procedures when assuming heterogeneity in patient outcomes the method is illustrated with a case study

compound optimal allocation for individual and collective ethics in binary clinical trials in recent years several authors have investigated response adaptive allocation rules for comparative clinical trials in order to favour at each stage of the trial the treatment that appears to be best in this paper we define admissible allocations namely treatment assignments that cannot be simultaneously improved upon with respect to both a specific design criterion reflecting the inferential properties of the experiment and the proportion of patients assigned to the best treatment or treatments we survey existing designs from this viewpoint we also suggest combining information and ethical considerations by taking a suitable weighted mean of two corresponding standardized criteria with weights that depend on the actual treatment effects this compound criterion leads to a locally optimal allocation that can be targeted by some response adaptive randomization rule the paper mainly deals with the case of two treatments but the suggested methodology is shown to extend to more than two

asymptotic properties of covariate adaptive randomization balancing treatment allocation for influential covariates is critical in clinical trials this has become increasingly important as more and more biomarkers are found to be associated with different diseases in translational research genomics proteomics and metabolomics stratified permuted block randomization and minimization methods pocock and simon biometrics31 1975 103 115 etc are the two most popular approaches in practice however stratified permuted block randomization fails to achieve good overall balance when the number of strata is large whereas traditional minimization methods also suffer from the potential drawback of large within stratum imbalances moreover the theoretical bases of minimization methods remain largely elusive in this paper we propose a new covariate adaptive design that is able to control various types of imbalances we show that the joint process of within stratum imbalances is a positive recurrent markov chain under certain conditions therefore this new procedure yields more balanced allocation the advantages of the proposed procedure are also demonstrated by extensive simulation studies our work provides a theoretical tool for future research in this area

implicit standard jacobi gives high relative accuracy we prove that the jacobi algorithm applied implicitly on a decomposition a xdx t of the symmetric matrix a where d is diagonal and x is well conditioned computes all eigenvalues of a to high relative accuracy the relative error in every eigenvalue is bounded by o epsilon kappa x where epsilon is the machine precision and kappa x equiv x 2 cdot x 1 2 is the spectral condition number of x the eigenvectors are also computed accurately in the appropriate sense we believe that this is the first algorithm to compute accurate eigenvalues of symmetric indefinite matrices that respects and preserves the symmetry of the problem and uses only orthogonal transformations

perturbation theory for the ldu factorization and accurate computations for diagonally dominant matrices we present a structured perturbation theory for the ldu factorization of row diagonally dominant matrices and we use this theory to prove that a recent algorithm of ye math comp 77 264 2195 2230 2008 computes the l d and u factors of these matrices with relative errors less than 14n 3 u where u is the unit roundoff and n n is the size of the matrix the relative errors for d are componentwise and for l and u are normwise with respect the max norm a m max ij a ij these error bounds guarantee that for any diagonally dominant matrix a we can compute accurately its singular value decomposition and the solution of the linear system ax b for most vectors b independently of the magnitude of the traditional condition number of a and in o n 3 flops

multi objective optimal designs in comparative clinical trials with covariates the reinforced doubly adaptive biased coin design the present paper deals with the problem of allocating patients to two competing treatments in the presence of covariates or prognostic factors in order to achieve a good trade off among ethical concerns inferential precision and randomness in the treatment allocations in particular we suggest a multipurpose design methodology that combines efficiency and ethical gain when the linear homoscedastic model with both treatment covariate interactions and interactions among covariates is adopted the ensuing compound optimal allocations of the treatments depend on the covariates and their distribution on the population of interest as well as on the unknown parameters of the model therefore we introduce the reinforced doubly adaptive biased coin design namely a general class of covariate adjusted response adaptive procedures that includes both continuous and discontinuous randomization functions aimed to target any desired allocation proportion the properties of this proposal are described both theoretically and through simulations

brick tunnel randomization for unequal allocation to two or more treatment groups studies with unequal allocation to two or more treatment groups often require a large block size for permuted block allocation this could present a problem in small studies multi center studies or adaptive design dose finding studies in this paper an allocation procedure which generalizes the maximal procedure by berger ivanova and knoll to the case of k 2 treatment groups and any allocation ratio is offered brick tunnel bt randomization requires the allocation path drawn in the k dimensional space to stay close to the allocation ray that corresponds to the targeted allocation ratio specifically it requires the allocation path to be confined to the set of the k dimensional unitary cubes that are pierced by the allocation ray the brick tunnel the important property of the bt randomization is that the transition probabilities at each node within the tunnel are defined in such a way that the unconditional allocation ratio is the same for every allocation step this property is not necessarily met by other allocation procedures that implement unequal allocation

properties of biased coin designs in sequential clinical trials martingale methods and the martingale invariance principle are used to derive central limit theorems and related results for biased coin designs of the kind previously studied by efron wei and many others the results are applied to the study of selection bias the method is developed for the simplest two treatment case and then extended first to the case of several treatments and secondly to the case of two treatments with prognostic factors

brick tunnel randomization and the momentum of the probability mass the allocation space of an unequal allocation permuted block randomization can be quite wide the development of unequal allocation procedures with a narrower allocation space however is complicated by the need to preserve the unconditional allocation ratio at every step the allocation ratio preserving arp property when the allocation paths are depicted on the k dimensional unitary grid where allocation to the l th treatment is represented by a step along the l th axis l 1 to k the arp property can be expressed in terms of the center of the probability mass after i allocations specifically for an arp allocation procedure that randomizes subjects to k treatment groups in w1 wk ratio w1 wk 1 the coordinates of the center of the mass are w1 i wk i in this paper the momentum with respect to the center of the probability mass expected imbalance in treatment assignments is used to compare arp procedures in how closely they approximate the target allocation ratio it is shown that the two arm and three arm brick tunnel randomizations btr are the arp allocation procedures with the tightest allocation space among all allocation procedures with the same allocation ratio the two arm btr is the minimum momentum two arm arp allocation procedure resident probabilities of two arm and three arm btr are analytically derived from the coordinates of the center of the probability mass the existence of the respective transition probabilities is proven probability of deterministic assignments with btr is found generally acceptable copyright 2015 john wiley sons ltd

does simple random sampling provide adequate balance une etude empirique montre que sous un echantillonnage aleatoire simple les intervalles de confiance du total peuvent avoir des probabilites de recouvrement conditionnelles qui sont imposantes de facon persistante lorsque la taille de l echantillon croit

efficient generation of constrained block allocation sequences selection bias is a potential problem especially in trials where blinding is not possible randomization with a constrained block of large size provides better protection from selection bias than using a sequence of blocks of small size with the same maximum imbalance we propose an algorithm that efficiently generates constrained block allocation sequences and we describe two clinical trials in which such a constrained randomization was used

optimal allocation to maximize the power of two sample tests for binary response summary we study allocations that maximize the power of tests of equality of two treatments having binary outcomes when a normal approximation applies the asymptotic power is maximized by minimizing the variance leading to a neyman allocation that assigns observations in proportion to the standard deviations this allocation which in general requires knowledge of the parameters of the problem is recommended in a large body of literature under contiguous alternatives the normal approximation indeed applies and in this case the neyman allocation reduces to a balanced design however when studying the power under a noncontiguous alternative a large deviations approximation is needed and the neyman allocation is no longer asymptotically optimal in the latter case the optimal allocation depends on the parameters but is rather close to a balanced design thus a balanced design is a viable option for both contiguous and noncontiguous alternatives finite sample studies show that a balanced design is indeed generally quite close to being optimal for power maximization this is good news as implementation of a balanced design does not require knowledge of the parameters

the lidskii mirsky wielandt theorem additive and multiplicative versions we use a simple matrix splitting technique to give an elementary new proof of the lidskii mirsky wielandt theorem and to obtain a multiplicative analog of the lidskii mirsky wielandt theorem which we argue is the fundamental bound in the study of relative perturbation theory for eigenvalues of hermitian matrices and singular values of general matrices we apply our bound to obtain numerous bounds on the matching distance between the eigenvalues and singular values of matrices our results strengthen and generalize those in the literature

qr factorization with complete pivoting and accurate computation of the svd a new algorithm of demmel et al for computing the singular value decomposition svd to high relative accuracy begins by computing a rank revealing decomposition rrd demmel et al analyse the use of gaussian elimination with complete pivoting gecp for computing the rrd we investigate the use of qr factorization with complete pivoting that is column pivoting together with row sorting or row pivoting as an alternative to gecp since this leads to a faster svd algorithm we derive a new componentwise backward error result for householder qr factorization and combine it with the theory of demmel et al to show that high relative accuracy in the computed svd can be expected for matrices that are diagonal scalings of a well conditioned matrix an a posteriori error bound is derived that gives useful estimates of the relative accuracy of the computed singular values numerical experiments confirm the theoretical predictions

dynamic randomization and a randomization model for clinical trials data randomization models are useful in supporting the validity of linear model analyses applied to data from a clinical trial that employed randomization via permuted blocks here a randomization model for clinical trials data with arbitrary randomization methodology is developed with treatment effect estimators and standard error estimators valid from a randomization perspective a central limit theorem for the treatment effect estimator is also derived as with permuted blocks randomization a typical linear model analysis provides results similar to the randomization model results when roughly unit effects display no pattern over time a key requirement for the randomization inference is that the unconditional probability that any patient receives active treatment is constant across patients when this probability condition is violated the treatment effect estimator is biased from a randomization perspective most randomization methods for balanced 1 to 1 treatment allocation satisfy this condition however many dynamic randomization methods for planned unbalanced treatment allocation like 2 to 1 do not satisfy this constant probability condition and these methods should be avoided copyright 2012 john wiley sons ltd

a block qr algorithm and the singular value decomposition abstract in this note we consider an iterative algorithm for moving a triangular matrix toward diagonality the algorithm is related to algorithms for refining rank revealing triangular decompositions and in a variant form to the qr algorithm it is shown to converge if there is a sufficient gap in the singular values of the matrix and the analysis provides a new approximation theorem for singular values and singular subspaces

elastic waves in piezoceramic cylinders of sector cross section abstract the propagation of elastic waves in piezoceramic cylindrical waveguides of circular cross sections with sector cut is investigated on the basis of the linear theory of electroelasticity dispersion functions are obtained from boundary conditions in an analytical form of functional determinants for each value of the generalized wave number a selected set of numerical results including real imaginary and complex branches of full dispersion spectrums with various symmetry of wave movements is presented to describe the essential characteristics of the waves leading effects of spectrums transformation by change of waveguide s angular measure are enlightened and wave asymptotic behavior is analyzed the variation of the cross section is considered as a mechanism to control the dispersion characteristics of waveguides

the covariate adaptive biased coin design for balancing clinical trials in the presence of prognostic factors the present paper deals with sequential designs intended to balance the allocations of two competing treatments in the presence of prognostic factors after giving a theoretical framework on the optimality of balanced designs that can arise when covariates are taken into account we propose a new family of covariate adaptive randomized designs that represents higher order approximation to balance treatments both globally and also across covariates we derive the theoretical properties of the suggested designs in terms of loss of precision and predictability the performance of this proposal is illustrated through a simulation study and compared with those of other procedures suggested in the literature copyright 2011 oxford university press

the transient responses of piezoelectric hollow cylinders for axisymmetric plane strain problems by virtue of the separation of variables technique the axisymmetric plane strain electroelastic dynamic problem of hollow cylinder is transferred to an integral equation about a function with respect to time which can be solved successfully by means of the interpolation method then the solution of the displacements stresses electric displacements and electric potentials are finally obtained the present method is suitable for the hollow cylinder with arbitrary thickness subjected to arbitrary mechanical and electrical loads numerical results are also presented