efficient sequential designs with binary data abstract a class of sequential designs for estimating the percentiles of a quantal response curve is proposed its updating rule is based on an efficient summary of all of the data available via a parametric model the logit mle version of the proposed designs can be viewed as a natural analog of the robbins monro procedure in the case of binary data it is shown to be asymptotically consistent optimal and nonparametric via its connection with the latter procedure for certain choices of initial designs the proposed method performs very well in a simulation study for sample sizes up to 35 a nonparametric sequential design via the spearman karber estimator for estimating the median is also proposed

design and analysis of phase i clinical trials the phase i clinical trial is a study intended to estimate the so called maximum tolerable dose mtd of a new drug although there exists more or less a standard type of design for such trials its development has been largely ad hoc as usually implemented the trial design has no intrinsic property that provides a generally satisfactory basis for estimation of the mtd in this paper the standard design and several simple alternatives are compared with regard to the conservativeness of the design and with regard to point and interval estimation of an mtd 33rd percentile with small sample sizes using a markov chain representation we found several designs to be nearly as conservative as the standard design in terms of the proportion of patients entered at higher dose levels in monte carlo simulations two two stage designs are found to provide reduced bias in maximum likelihood estimation of the mtd in less than ideal dose response settings of the three methods considered for determining confidence intervals the delta method a method based on fieller s theorem and a likelihood ratio method none was able to provide both usefully narrow intervals and coverage probabilities close to nominal

global bifurcation and nodal solutions for fourth order problems with sign changing weight in this paper we shall establish unilateral global bifurcation result for a class of fourth order eigenvalue problems with sign changing weight under some natural hypotheses on perturbation function we show that m k n 0 is a bifurcation point of the above problems and there are two distinct unbounded continua c k n and c k n consisting of the bifurcation branch c k n from m k n 0 where m k n is the kth positive or negative eigenvalue of the linear problem corresponding to the above problems email protected as the applications of the above result we study the existence of nodal solutions for a class of fourth order eigenvalue problems with sign changing weight moreover we also establish the sturm type comparison theorem for fourth order problems with sign changing weight

algebraic families of analytic functions i the aim of this paper is to consider analytic functions arising in connection with algebraic differential equations from the following point of view taylor coefficients of such functions usually turn out to be polynomials or rational functions in the parameters of the problem following pioneering work of bautin 1 we relate analytic properties of the functions considered with algebraic properties of various ideals generated by their taylor coefficients bautin s approach has been successfully applied to the study of the poincare mapping of the planar polynomial systems in 4 9 27 and to the study of the period function of polynomial hamiltonian systems in 5 6 12 13 however the study of the poincare mapping of the polynomial differential equations closely related to the second part of hilbert 16th problem is probably the most difficult among a wide variety of problems to which bautin s approach can be applied in particular counting zeroes of solutions of algebraic differential equations is an important and mostly open problem to our best knowledge the first application of bautin s method to this problem was given in 26 in the present paper we systematically apply this method in the following two situations to algebraic functions and to solutions of the first order polynomial differential equations higher order linear equations are also considered in section 5 article no de963250

periodic solutions for a class of fourth order nonlinear differential equations in this paper by using inequality techniques and coincidence degree theory some sufficient conditions are established for ensuring the existence of t periodic solutions for a class of fourth order nonlinear differential equations two illustrative examples are provided to demonstrate that the results of this paper hold under weaker conditions than the existing ones and are more effective

on the nonlinear elastic beam equation we consider the existence of solutions and the continuous dependence on a parameter results for fourth order dirichlet problem connected with the elastic beam equation we introduce a dual variational method

multiple positive solutions to superlinear periodic boundary value problems with repulsive singular forces abstract the existence of multiple positive solutions to superlinear periodic boundary value problems with repulsive singular forces is discussed in this paper our nonlinearity may be singular in its dependent variable and our analysis relies on a nonlinear alternative of leray schauder type and on a fixed point theorem in cones

coherence principles in dose finding studies of the methods this paper shows examples in which coherence is violated and discusses how the coherence principles may be applied to calibrate a two stage design and to deal with situations with delayed toxicity

the method of lower and upper solutions for third order periodic boundary value problems abstract in this paper we develop the monotone method in the presence of lower and upper solutions for the problem l n u t t u t u i 0 u i 2 i r i 0 n 1 with ln an nth order linear operator and a caratheodory function we obtain sufficient conditions to guarantee the validity of the monotone method for this problem for this we study the sign of the green function of the operator t 1n with tnu lnu mu m 0 furthermore existence results are obtained we show that the results obtained are optimal given a lower solution and an upper solution we obtain the existence of solution between and when either or this general study is applied to the third order problem u t pu t qu t t u t u i 0 u i 2 i r i 0 1 2 with p q r for this problem we obtain the best value on the constant m to guarantee that the green function of the operator t 13 is positive

dose finding using the biased coin up and down design and isotonic regression we are interested in finding a dose that has a prespecified toxicity rate in the target population in this article we investigate five estimators of the target dose to be used with the up and down biased coin design bcd introduced by durham and flournoy 1994 statistical decision theory and related topics these estimators are derived using maximum likelihood weighted least squares sample averages and isotonic regression a linearly interpolated isotonic regression estimate is shown to be simple to derive and to perform as well as or better than the other target dose estimators in terms of mean square error and average number of subjects needed for convergence in most scenarios studied

periodic solutions of some fourth order nonlinear differential equations abstract in this paper using mawhin s continuation theorem we obtain existence results concerning t periodic solutions of fourth order nonlinear differential equations of the type u p u g t u e t where p r g r 2 r is continuous t periodic in t and e r r is continuous t periodic with mean value zero applications are given for the extended fisher kolmogorov and swift hohenberg equations with forcing term

existence and multiplicity of solutions to 2mth order ordinary differential equations abstract in this paper the existence and multiplicity of solutions are obtained for the 2mth order ordinary differential equation two point boundary value problems 1 m u 2 m t i 1 m 1 m i a i u 2 m i t f t u t for all t 0 1 subject to dirichlet neumann mixed and periodic boundary value conditions respectively where f is continuous a i r for all i 1 2 m since these four boundary value problems have some common properties and they can be transformed into the integral equation of form u i 1 m a i t i u t m f u we firstly deal with this nonlinear integral equation by using the strongly monotone operator principle and the critical point theory we establish some conditions on f which are able to guarantee that the integral equation has a unique solution at least one nonzero solution and infinitely many solutions furthermore we apply the abstract results on the integral equation to the above four 2mth order two point boundary problems and successfully resolve the existence and multiplicity of their solutions

existence multiplicity and infinite solvability of positive solutions to a nonlinear fourth order periodic boundary value problem abstract the existence of n and infinitely many positive solutions is proved for the nonlinear fourth order periodic boundary value problem u 4 t u t u t f t u t 0 t 1 u i 0 u i 1 i 0 1 2 3 where n is an arbitrary natural number and 2 2 0 1 2 2 2 2 4 2 1 0 this kind of fourth order boundary value problems usually describes the equilibrium state of an elastic beam with periodic boundary condition the main results show that the problem may have n or infinitely many positive solutions provided the growth rates of nonlinear term f t l are appropriate on some bounded subsets of its domain

bernstein inequalities and applications to analytic geometry and differential equations we introduce new methods of complex analysis inequalities of bernstein type to study projections of analytic sets these techniques are applied to problems of bifurcations of periodic orbits of differential equations such as the local hilbert s 16th problem

bayesian model averaging continual reassessment method in phase i clinical trials the continual reassessment method crm is a popular dose finding design for phase i clinical trials this method requires that practitioners prespecify the toxicity probability at each dose such prespecification can be arbitrary and different specifications of toxicity probabilities may lead to very different design properties to overcome the arbitrariness and further enhance the robustness of the design we propose using multiple parallel crm models each with a different set of prespecified toxicity probabilities in the bayesian paradigm we assign a discrete probability mass to each crm model as the prior model probability the posterior probabilities of toxicity can be estimated by the bayesian model averaging bma approach dose escalation or deescalation is determined by comparing the target toxicity rate and the bma estimates of the dose toxicity probabilities we examine the properties of the bma crm approach through extensive simulation studies and also compare this new method and its vari

on linear and nonlinear fourth order eigenvalue problems with indefinite weight abstract we determine the principal eigenvalues of the linear indefinite weight problem u x r g x u x x 0 1 u 0 u 1 u 0 u 1 0 moreover we investigate the existence of positive solutions for the corresponding nonlinear indefinite weight problem where g 0 1 r is a continuous function which attains both positive and negative values f c r r and r is a parameter

existence and multiplicity of solutions for fourth order boundary value problems with parameters abstract in this paper we study the existence and multiplicity of the solutions for the fourth order boundary value problem bvp u 4 t u t u t f t u t 0 t 1 u 0 u 1 u 0 u 1 0 where f 0 1 r r is continuous r and r are parameters by means of the idea of the decomposition of operators shown by chen w y chen a decomposition problem for operators xuebao of dongbei renmin university 1 1957 95 98 see also m krasnosel skii topological methods in the theory of nonlinear integral equations gostehizdat moscow 1956 and the critical point theory we obtain that if the pair is on the curve 2 4 satisfying 2 2 then the above bvp has at least one two three and infinitely many solutions for being in different interval respectively

sequential implementation of stepwise procedures for identifying the maximum tolerated dose this article considers the problem of finding the maximum tolerated dose mtd of a drug in human trials the mtd is defined as the maximum test dose with toxicity probability less than or equal to a target toxicity rate we adopt the multiple test framework with step down tests used in an escalation stage and step up tests used in a deescalation stage to allow sequential dose assignments for ethical purposes by formulating the estimation problem as a testing problem the proposed procedures formally control the error probability of selecting an unsafe dose in addition we can control the probability of correctly selecting the mtd under a parameter subspace where no toxicity probability lies in an interval bracketed by the target toxicity rate and an unacceptably high toxicity rate the so called indifference zone this frequentist property which is currently lacking in the conduct of dose finding trials in humans is appealing from a regulatory standpoint we give the general expressions of the se

the hopf cyclicity of the centers of a class of quintic polynomial vector fields abstract we consider families of planar polynomial vector fields having a singularity with purely imaginary eigenvalues for which a basis of its bautin ideal b is known we provide an algorithm for computing an upper bound of the hopf cyclicity less than or equal to the bautin depth of b we also present a method for studying the cyclicity problem for the hamiltonian and the time reversible centers without the necessity of solving previously the dulac complex center problem associated to the larger complexified family as application we analyze the hopf cyclicity of the quintic polynomial family written in complex notation as z i z z z a z 3 b z 2 z c z z 2 d z 3

symmetric positive solutions of boundary value problems with integral boundary conditions under conditions weaker than those used by ma we establish various results on the existence and nonexistence of symmetric positive solutions to fourth order boundary value problems with integral boundary conditions the arguments are based upon a specially constructed cone and the fixed point theory in a cone our results improve and extend that obtained in h ma symmetric positive solutions for nonlocal boundary value problems of fourth order nonlinear anal 68 2008 645 651 we illustrate our results by one example which cannot be handled using the existing results

on the existence of maximum likelihood estimators for the binomial response models summary necessary and sufficient conditions are given for the existence of maximum likelihood estimators of the linear regression parameter in binomial response this includes logit and probit models the question of existence of maximum likelihood estimators mle for logit models arose in an analysis of the relationship of psychiatric caseness to scores on a psychiatric screening questionnaire tennant 1977 administered the ghq general health questionnaire goldberg 1972 to 120 patients attending a general practitioner s surgery and also gave each one a standardized psychiatric interview from the interview patients were classified as psychiatric case non case in a secondary analysis of tennant s data duncan jones and henderson 1978 fitted a logit regression of caseness on ghq score and obtained a good fit with the model

designs for single or multiple agent phase i trials phase i trials of cytotoxic agents in oncology are usually dose finding studies that involve a single cytotoxic agent many statistical methods have been proposed for these trials all of which are based on the assumption of a monotonic dose toxicity curve for single agent trials this is a valid assumption in many trials however investigators are interested in finding the maximally tolerated dose based on escalating multiple cytotoxic agents when there are multiple agents monotonicity of the dose toxicity curve is not clearly defined in this article we present a design for phase i trials in which the toxicity probabilities follow a partial order meaning that there are pairs of treatments for which the ordering of the toxicity probabilities is not known at the start of the trial we compare the new design to existing methods for simple orders and investigate the properties of the design for two partial orders

an approximate method for one dimensional structures with strong nonlinear and nonhomogenous boundary conditions abstract there is a lack of analytic methods for vibration analysis of structures with strong nonlinear and non homogenous boundary conditions this paper proposes such a method a distinct feature of the method is to introduce the generalized coordinates for nonlinear and nonhomogeneous boundary conditions the total solution of the structure consists of a sum of a spatial particular solution for the boundary conditions and a field solution satisfying linear and homogeneous boundary conditions the field solution can be obtained in terms of the modal expansion the modal coefficients together with the generalized coordinates for the boundary conditions satisfy a set of nonlinear ordinary differential equations which are solved with the harmonic balance method two examples are presented to study the convergence and accuracy of the solution the first example is the longitudinal vibration of a bar subject to a base excitation on the left and restrained by a damper on the right the wave method and the differential quadrature method dqm are compared with the proposed method and are found in good agreement the second example is a beam with nonlinear torsional boundary absorbers the method shows a good accuracy to deal with the complex boundary conditions as compared with the differential quadrature element method dqem the proposed method is also compared with the multi scale method msm and shows an advantage to deal with strong nonlinearities the proposed method is highly promising for vibration analysis of complex structures with nonlinear and nonhomogeneous boundary conditions

on the positive solutions of lidstone boundary value problems the existence nonexistence and multiplicity of positive solutions are considered for lidstone boundary value problem 1 email protected 2 n email protected t w t 0 0 by making use of krasnosel skii fixed point theorem some new results are obtained particularly it is proved that the lidstone boundary value problem has n positive solutions under suitable conditions where n is an arbitrary natural number

a random walk rule for phase i clinical trials we describe a family of random walk rules for the sequential allocation of dose levels to patients in a dose response study or phase i clinical trial patients are sequentially assigned the next higher same or next lower dose level according to some probability distribution which may be determined by ethical considerations as well as the patient s response it is shown that one can choose these probabilities in order to center dose level assignments unimodally around any target quantile of interest estimation of the quantile is discussed the maximum likelihood estimator and its variance are derived under a two parameter logistic distribution and the maximum likelihood estimator is compared with other nonparametric estimators random walk rules have clear advantages they are simple to implement and finite and asymptotic distribution theory is completely worked out for a specific random walk rule we compute finite and asymptotic properties and give examples of its use in planning studies having the finite distribution theory available and tractable obviates the need for elaborate simulation studies to analyze the properties of the design the small sample properties of our rule as determined by exact theory compare favorably to those of the continual reassessment method determined by simulation

interval estimation of the ld50 based on an up and down experiment it is well known that an up and down method can be more efficient than fixed sample methods in estimating the ld50 of a quantal response curve a problem that has not been addressed by many is that of obtaining a confidence interval for the ld50 from the up and down method dixon and mood 1948 journal of the american statistical association 43 109 126 proposed a confidence interval using a maximum likelihood approach but not much is known about its properties in this paper a new confidence interval for the ld50 based on turning points is obtained which uses the concept of phi mixing simulation results indicate that the coverage probabilities of both methods tend to be less than the nominal level unless the sample size is large even so when the tolerance distribution is normal the proposed confidence interval is found to be superior to dixon s interval in terms of the coverage the width and stability the advantages of the method do not appear to hold in the presence of nonnormal tolerance distribution

positive solutions of fourth order periodic boundary value problems abstract in this paper the existence results of positive solutions are obtained for fourth order periodic boundary value problem u 4 u u f t u 0 t 1 u i 0 u i 1 i 0 1 2 3 where f 0 1 r r is continuous r and satisfy 0 2 2 4 2 1 0 the discussion is based on a new maximum principle for operator l 4 u u 4 u u in periodic boundary condition and fixed point index theory in cones

the method of lower and upper solutions for second third fourth and higher order boundary value problems abstract in this paper we develop the monotone method in the presence of lower and upper solutions for the problem u n t t u t u i a u i b i r i 0 n 1 where is a caratheodory function we obtain necessary and sufficient conditions in to guarantee the existence of solutions between a lower solution and an upper solution for n 2 if n 3 either or and n 4 if furthermore we obtain sufficient conditions in for n 2 k 6 when

critical period bifurcations of a cubic system we study bifurcations of the period function of a linear centre perturbed by third degree homogeneous polynomials the approach is based on making use of algorithms of computational algebra