asymptotics for euclidean minimal spanning trees on random points asymptotic results for the euclidean minimal spanning tree onn random vertices inrd can be obtained from consideration of a limiting infinite forest whose vertices form a poisson process in allrd in particular we prove a conjecture of robert bland the sum of thed th powers of the edge lengths of the minimal spanning tree of a random sample ofn points from the uniform distribution in the unit cube ofrd tends to a constant asn

model inconsistency illustrated by the cox proportional hazards model we consider problems involving the comparison of two or more treatments where we have the opportunity to adjust for relevant covariates either conditionally in a regression model or implicitly in repeated measures data for example in crossover trials it is seen that for data arising from non normal distributions there is the possibility that models adjusting for covariates and those not adjusting for covariates will be inconsistent that is at most one of the models can be valid alternatively even if conditional and unconditional models are valid parameters in each model may have different interpretations we note that this presents difficulties for the specification and interpretation of the analysis it is also clear that model validation is critical specific attention is paid to survival data analysed by the cox proportional hazards model

non parametric analysis of covariance for confirmatory randomized clinical trials to evaluate dose response relationships in confirmatory randomized clinical trials that are designed to compare multiple doses of a test treatment with a control group and with one another there are often statistical issues regarding compound hypotheses and multiple comparisons which need to be considered in most cases the analysis plan needs a clear specification for the proposed order for conducting statistical tests or for managing the overall significance level which statistical methods will be used and whether adjustment for covariates will be performed there are several benefits of specifying non parametric analysis of covariance ancova for performing the primary confirmatory analyses only minimal assumptions are needed beyond randomization in the study design whereas regression model based methods have assumptions about model fit for which departures may require modifications that are incompatible with a fully prespecified analysis plan non parametric methods provide traditionally expected results of ancova namely a typically small adjustment to the estimate for a treatment comparison so as to account for random imbalance of covariates between treatment groups and variance reduction for this estimate when covariates are strongly correlated with the response of interest the application of non parametric ancova is illustrated for two randomized clinical trials the first has a 3 x 4 factorial response surface design for the comparison of 12 treatments that is combinations of three doses of one drug and four doses of a second drug for change in blood pressure and the second example addresses the comparison of three doses of test treatment and placebo for time to disease progression this clinical trial has comparisons among treatments made for a dichotomous criterion wilcoxon rank scores and averages of cumulative survival rates in each example the non parametric covariance method provides variance reduction relative to its unadjusted counterpart

global attractivity for discrete models of nonautonomous logistic equations consider the following discrete model of a nonautonomous logistic equation n n 1 n n exp c n j 0mb j n n n j n 0 n 0 n 0 0andn j n j 0 1 j m where c n and b j n 0 j m n 0 are bounded and c n 0 b 0 n 0 b j n 0 1 j m n 0 in this paper using some kind of iterative method to the above equation we establish sufficient conditions that ensure the global attractivity for solutions the result is an extension of the former work k uesugi y muroya e ishiwata on the global attractivity for a logistic equation with piecewise constant arguments j math anal appl 294 2004 560 580 to the nonautonomous case

multivariate non parametric methods for mann whitney statistics to analyse cross over studies with two treatment sequences a non parametric strategy for the analysis of ordinal data from cross over studies with two treatment sequences and d or 2 periods is examined through mann whitney rank measures of association for each period these statistics estimate the probability of larger response for a randomly selected patient in one group relative to a randomly selected patient in the other group such estimates are as well formed for comparisons between groups for u pairs of periods with the same treatment methods for u statistics are used to produce a consistent estimate of the covariance matrix for the d u mann whitney estimates the effects of periods and treatments on the respective mann whitney estimates are evaluated through linear or log linear models for estimation of the parameters in these models a modified weighted least squares method is applied through a 2d 1 or d u dimensional basis which effectively addresses potentially near singularities in the estimated covariance matrix of the mann whitney estimates the proposed methods are applicable to response variables with an interval or an ordered categorical scale their scope additionally has capabilities for controlling strata in the design of a cross over study or concomitant variables for which covariance adjustment is of interest for reduction of variance applications of the methods are illustrated through three cross over studies with different specifications for the two sequences of two treatments during two to four periods

a sharper bonferroni procedure for multiple tests of significance summary a simple procedure for multiple tests of significance based on individual p values is derived this simple procedure is sharper than holm s 1979 sequentially rejective procedure both procedures contrast the ordered p values with the same set of critical values holm s procedure rejects an hypothesis only if its p value and each of the smaller p values are less than their corresponding critical values the new procedure rejects all hypotheses with smaller or equal p values to that of any one found less than its critical value

the persistence and global attractivity in general nonautonomous discrete single species kolmogorov model with delays in this paper general nonautonomous discrete time single species kolmogorov model with delays is studied some sufficient conditions on the boundedness persistence permanence and global attractivity of solutions are established as application the discrete time nonautonomous delayed allee effect model is discussed the sufficient conditions on the permanence and global attractivity are obtained

persistence and global stability of positive periodic solutions of three species food chains with omnivory in this paper we investigate the existence and global stability of the positive periodic solutions of delayed discrete food chains with omnivory with the help of continuation theorem in coincidence degree theory and lyapunov functions some sufficient conditions are obtained the results show that for such a system with omnivory more conditions are required for the persistence of the positive periodic solutions than that of a linear food chain on the other hand omnivory makes no differences on the stability of the solutions

indistinguishability of percolation clusters we show that when percolation produces infinitely many infinite clusters on a cayley graph one cannot distinguish the clusters from each other by any invariantly defined property this implies that uniqueness of the infinite cluster is equivalent to nondecay of connectivity a k a long range order we then derive applications concerning uniqueness in kazhdan groups and in wreath products and inequalities for p u

on closed test procedures for dose response analysis this paper concerns the testing of a dose response effect in medical studies we consider two situations the first is when the drug effect on a parameter of interest is likely to increase or decrease with increasing doses we propose a procedure based on the closure principle of peritz combined with an application of a closed testing procedure of marcus et al to a test proposed by turkey et al the second situation is the analysis of a drug effect where one might observe a reversal at higher doses for this situation we propose simultaneous examination of contrasts among the doses at each of several stages of the testing scheme we calculate critical values for this procedure incorporating the correlation structure among the contrasts both procedures strongly control the family wise error rate at the pre determined alpha level they also provide information about the shape of the dose response lacking in other commonly used procedures for testing against an ordered alternative hypothesis we illustrate the procedures on two datasets

covariate imbalance and random allocation in clinical trials a model is developed to estimate the effect of covariate imbalance on the size of a test of treatment efficacy in randomized clinical trials comparing two treatments when dispersion parameters are known it is concluded that tests of homogeneity on the covariates should not be performed that covariate imbalance is just as much a problem for large studies as for small ones in terms of effect on size and that the effect of correlation between covariates and measures of efficacy is more complex than has previously been suggested the best way to adjust for covariate imbalance is by an analysis of covariance

infinite clusters in dependent automorphism invariant percolation on trees we study dependent bond percolation on the homogeneous tree t n of order n 2 under the assumption of automorphism invariance excluding a trivial case we find that the number of infinite clusters a s is either 0 or furthermore each infinite cluster a s has either 1 2 or infinitely many topological ends and infinite clusters with infinitely many topological ends have a s a branching number greater than 1 we also show that if the marginal probability that a single edge is open is at least 2 n 1 then the existence of infinite clusters has to have positive probability several concrete examples are considered

existence of positive periodic solutions of nonlinear discrete model exhibiting the allee effect in this paper we shall consider the discrete nonlinear delay population model with allee effectx n 1 x n exp a n b n x p n w c n x q n w where a n b n and c n are positive sequences of period w and p and q are positive integers by using the coincidence degree theorem as well as some priori estimates we will establish a sufficient condition for the existence of positive periodic solution x n

the simes method for multiple hypothesis testing with positively dependent test statistics abstract the simes method for testing intersection of more than two hypotheses is known to control the probability of type i error only when the underlying test statistics are independent although this method is more powerful than the classical bonferroni method it is not known whether it is conservative when the test statistics are dependent this article proves that for multivariate distributions exhibiting a type of positive dependence that arise in many multiple hypothesis testing situations the simes method indeed controls the probability of type i error this extends some results established very recently in the special case of two hypotheses

applications of extensions of bivariate rank sum statistics to the crossover design to compare two treatments through four sequence groups this article describes applications of extensions of bivariate rank sum statistics to the crossover design with four sequence groups for two treatments a randomized clinical trial in ophthalmology provides motivating background for the discussion the bilateral design for this study has four sequence groups t t t p p t and p p respectively for t as test treatment or p as placebo in the corresponding order for the left and right eyes this article describes how to use the average of the separate wilcoxon rank sum statistics for the left and right eyes for the overall comparison between t and p with the correlation between the two eyes taken into account an extension of this criterion with better sensitivity to potential differences between t and p through reduction of the applicable variance has discussion in terms of a conceptual model with constraints for within side homogeneity of groups with the same treatment and between side homogeneity of the differences between t and p goodness of fit for this model can have assessment with test statistics for its corresponding constraints simulation studies for the conceptual model confirm better power for the extended test statistic with its full invocation than other criteria without this property the methods summarized here are illustrated for the motivating clinical trial in ophthalmology but they are applicable to other situations with the crossover design with four sequence groups for either two locations for two treatments at the same time for a patient or two successive periods for the assigned treatments for a recurrent disorder this article also notes that the methods based on its conceptual model can have unsatisfactory power for departures from that model where the difference between t and p via the t t and p p groups is not similar to that via the t p and p t groups as might occur when t has a systemic effect in a bilateral trial for this situation more robust test statistics have identification but there is recognition that the parallel groups design with only the t t and p p groups may be more useful than the bilateral design with four sequence groups

a stagewise rejective multiple test procedure based on a modified bonferroni test summary simes 1986 has proposed a modified bonferroni procedure for the test of an overall hypothesis which is the combination of n individual hypotheses in contrast to the classical bonferroni procedure it is not obvious how statements about individual hypotheses are to be made for this procedure in the present paper a multiple test procedure allowing statements on individual hypotheses is proposed it is based on the principle of closed test procedures marcus peritz gabriel 1976 and controls the multiple level a

non parametric covariance methods for incidence density analyses of time to event data from a randomized clinical trial and their complementary roles to proportional hazards regression the principal response criteria for many clinical trials involve time to event variables usual methods of analysis for this type of response criterion include product limit estimators of cumulative survival for the treatment groups stratified logrank tests to compare treatments and proportional hazards regression models with treatment and relevant covariates when adjustment for covariates is of some importance the relative roles of these methods may be of some concern particularly for confirmatory clinical trials which must provide convincing findings to regulatory agencies unadjusted methods may have lower power but there are issues regarding adjustment for covariates that may be controversial these issues include applicability of proportional hazards assumptions whether the correct model has been specified and whether there is parallelism between treatments for relationships with covariates one way to address these issues is to use non parametric analysis of covariance strategies with extensions to log incidence density estimation the principal basis for this method is no association between covariates and treatment groups as provided by randomized assignment of patients to groups the background theory and strategies for computation are described for this method aspects of its application are illustrated for a clinical trial with two treatment groups and 722 patients the objective of analysis for this clinical trial is evaluation of treatment effects with and without adjustment for 22 a priori covariates and a stratification for three geographical regions

a quantitative study of the bias in estimating the treatment effect caused by omitting a balanced covariate in survival models this paper discusses the quantitative aspects of bias in estimates of treatment effect in survival models when there is failure to adjust on balanced prognostic variables a simple numerical example of this bias is given along with approximate formulae for its calculation in the multiplicative exponential survival model the accuracy of the formulae is checked by simulation in addition approximate calculations and simulations of power loss and the effects of omitting more than one prognostic covariate are presented the weibull and cox models are also examined using simulation study of this bias is pertinent to much applied work and shows that the effect of omitting balanced covariates can be modest unless the variables are strongly prognostic or many in number this work emphasizes the need for thorough comparisons of adjusted and unadjusted analyses for sensible interpretation of treatment effects

the persistence of a general nonautonomous single species kolmogorov system with delays abstract in this paper we establish some sufficient conditions for the persistence of system x t x t f t x t without the condition that f t is monotonously decreasing with respect to this partly answers the open problems proposed by teng

on permanence and global stability in a general gilpin ayala competition predator prey discrete system abstract in this paper we propose a discrete multispecies general gilpin ayala competition predator prey model by using new difference inequality and developing the analysis technique of chen and zhou y chen z zhou stable periodic solution of a discrete periodic lotka volterra competition system j math anal appl 277 2003 358 366 sufficient conditions are established for the permanence and the global stability an example shows the feasibility of our results

monotonicity of uniqueness for percolation on cayley graphs all infinite clusters are born simultaneously consider site or bond percolation with retention parameter p on an infinite cayley graph in response to questions raised by grimmett and newman 1990 and benjamini and schramm 1996 we show that the property of having almost surely a unique infinite open cluster is increasing in p moreover in the standard coupling of the percolation models for all parameters a s for all p2 p1 pc each infinite p2 cluster contains an infinite p1 cluster this yields an extension of alexander s 1995 simultaneous uniqueness theorem as a corollary we obtain that the probability v p that a given vertex v belongs to an infinite cluster is depends continuously on p throughout the supercritical phase p pc all our results extend to quasi transitive infinite graphs with a unimodular automorphism group

the supercritical phase of percolation is well behaved we prove a general result concerning the critical probabilities of subsets of a lattice l it is a consequence of this result that the critical probability of a percolation process on l equals the limit of the critical probability of a slice of l as the thickness of the slice tends to infinity this verification of one of the standard hypotheses of the subject settles many questions concerning supercritical percolation

a review of some statistical methods for covariance analysis of categorical data three general methods for covariance analysis of categorical data are reviewed and applied to an example from a clinical trial in rheumatoid arthritis the three methods considered are randomization model nonparametric procedures maximum likelihood logistic regression and weighted least squares analysis of correlated marginal functions a fourth heuristic approach the unweighted linear model analysis is an approximate procedure but it is easy to implement the assumptions and statistical issues for each method are discussed so as to emphasize philosophical differences between their rationales attention is given to computational differences but it is shown that the methods lead to similar results for analogous problems it is argued that the essential differences between the methods lie in their underlying assumptions and the generality of the conclusions which may be drawn

multiple testing in clinical trials the basic ideas of multiple testing are outlined and the problem of how to control the probability of erroneous decisions is discussed the main emphasis is on the concept of the multiple level of significance controlling the experiment or family error in the strong sense which can be achieved by applying the principle of closed tests various practical situations encountered in multiple testing in clinical trials are considered more than one end point more than two treatments such as comparisons with a single control comparisons using ordered alternatives all pairwise comparisons and contrast methods and more than one trial tests based on global statistics the union intersection principle and other criteria are discussed the application of the multiple test concept in sequential sampling is investigated finally some comments are made on multiple power multiple confidence intervals and directed decisions

qualitative analysis of a discrete logistic equation with several delays in this paper we study the qualitative behavior of solutions of the discrete delay logistic equation with several delaysx n 1 x email protected i 1 mr i1 x n k ik n 0 1 2 where r email protected 0 for i 1 m k i i 1 2 m are positive integers and email protected 0 we obtain sufficient conditions for the global attractivity of all positive solutions about the positive equilibrium furthermore we show that such model is uniformly persistent and that all its eventually positive solutions are bounded the oscillation about the positive equilibrium is also discussed

persistence and global stability in discrete models of lotka volterra type abstract in this paper we establish new sufficient conditions for global asymptotic stability of the positive equilibrium in the following discrete models of lotka volterra type n i p 1 n i p exp c i a i n i p j 1 n a i j n j p k i j p 0 1 i n n i p n i p 0 p 0 and n i 0 0 1 i n where each n i p for p 0 each c i a i and a i j are finite and a i 0 a i a i i 0 1 i n and k i j 0 1 i j n applying the former results y muroya persistence and global stability for discrete models of nonautonomous lotka volterra type j math anal appl 273 2002 492 511 on sufficient conditions for the persistence of nonautonomous discrete lotka volterra systems we first obtain conditions for the persistence of the above autonomous system and extending a similar technique to use a nonnegative lyapunov like function offered by y saito t hara and w ma y saito t hara w ma necessary and sufficient conditions for permanence and global stability of a lotka volterra system with two delays j math anal appl 236 1999 534 556 for n 2 to the above system for n 2 we establish new conditions for global asymptotic stability of the positive equilibrium in some special cases that k i j k j j 1 i j n and j 1 n a j i a j k 0 i k these conditions become a i j 1 n a j i 2 1 i n and improve the well known stability conditions a i j 1 n a j i 1 i n obtained by k gopalsamy k gopalsamy global asymptotic stability in volterra s population systems j math biol 19 1984 157 168

phase transitions on nonamenable graphs we survey known results about phase transitions in various models of statistical physics when the underlying space is a nonamenable graph most attention is devoted to transitive graphs and trees

the components of the wired spanning forest are recurrent we show that a s all of the connected components of the wired spanning forest are recurrent proving a conjecture of benjamini lyons peres and schramm our analysis relies on a simple martingale involving the effective conductance between the endpoints of an edge in a uniform spanning tree we believe that this martingale is of independent interest and will find further applications in the study of uniform spanning trees and forests

density and uniqueness in percolation two results on site percolation on thed dimensional lattice d 1 arbitrary are presented in the first theorem we show that for stationary underlying probability measures each infinite cluster has a well defined density with probability one the second theorem states that if in addition the probability measure satisfies the finite energy condition of newman and schulman then there can be at most one infinite cluster with probability one the simple arguments extend to a broad class of finite dimensional models including bond percolation and regular lattices

minimal spanning forests minimal spanning forests on infinite graphs are weak limits of minimal spanning trees from finite subgraphs these limits can be taken with free or wired boundary conditions and are denoted fmsf free minimal spanning forest and wmsf wired minimal spanning forest respectively the wmsf is also the union of the trees that arise from invasion percolation started at all vertices we show that on any cayley graph where critical percolation has no infinite clusters all the component trees in the wmsf have one end a s in z d this was proved by alexander ann probab 23 1995 87 104 but a different method is needed for the nonamenable case we also prove that the wmsf components are thin in a different sense namely on any graph each component tree in the wmsf has p c 1 a s where p c denotes the critical probability for having an infinite cluster in bernoulli percolation on the other hand the fmsf is shown to be thick on any connected graph the union of the fmsf and independent bernoulli percolation with arbitrarily small parameter is a s connected in conjunction with a recent result of gaboriau this implies that in any cayley graph the expected degree of the fmsf is at least the expected degree of the fsf the weak limit of uniform spanning trees we also show that the number of infinite clusters for bernoulli p u percolation is at most the number of components of the fmsf where p u denotes the critical probability for having a unique infinite cluster finally an example is given to show that the minimal spanning tree measure does not have negative associations

oscillatory and asymptotic behavior of positive periodic solutions of nonlinear discrete model exhibiting the allee effect in this paper we shall consider the discrete nonlinear delay population model with allee effectx n 1 x n exp a n b n x p email protected c n x q email protected where a n b n and c n are positive sequences of period w and p and q are positive integers we will establish some sufficient conditions for the oscillation of all positive solutions about its positive periodic solution x n and prove that every nonoscillatory solution converges to x n monotonically as n

choosing a spanning tree for the integer lattice uniformly consider the nearest neighbor graph for the integer lattice zd in d dimensions for a large finite piece of it consider choosing a spanning tree for that piece uniformly among all possible subgraphs that are spanning trees as the piece gets larger this approaches a limiting measure on the set of spanning graphs for zd this is shown to be a tree if and only if d 4 in this case the tree has only one topological end that is there are no doubly infinite paths when d 2 5 the spanning forest has infinitely many components almost surely with each component having one or two topological ends

multiple test procedures for dose finding the problem of identifying the lowest dose level for which the mean response differs from that at the zero dose level is considered a general framework for stepwise testing procedures that use contrasts among the dose level means is proposed using this framework several new procedures are derived these and some existing procedures including that of williams 1971 biometrics 27 103 117 1972 biometrics 28 519 531 are compared analytically and by an extensive simulation study for the normal theory balanced one way layout case it is pointed out that the procedures based on the so called step and basin contrasts proposed by ruberg 1989 journal of american statistical association 84 816 822 have excessively high type i familywise error rates fwes and hence they should not be used some findings of the simulation study are as follows for monotone dose mean configurations williams procedure and two step down test procedures based on helmert and linear contrasts offer the best performance for nonmonotone dose mean configurations the performance of williams procedure does degrade somewhat but the other two procedures are still the best for more complex designs a simple step down test procedure that uses any alpha level tests not necessarily t tests to compare each dose level with the zero dose level controls the fwe and is the only alternative available but its power is rather low especially under nonmonotone configurations step up procedures are generally dominated by step down procedures when the same contrasts are used although the differences are not great

uniform spanning forests we study uniform spanning forest measures on infinite graphs which are weak limits of uniform spanning tree measures from finite subgraphs these limits can be taken with free fsf or wired wsf boundary conditions pemantle proved that the free and wired spanning forests coincide in z d and that they give a single tree iff d 4 in the present work we extend pemantle s alternative to general graphs and exhibit further connections of uniform spanning forests to random walks potential theory invariant percolation and amenability the uniform spanning forest model is related to random cluster models in statistical physics but because of the preceding connections its analysis can be carried further among our results are the following the fsf and wsf in a graph g coincide iff all harmonic dirichlet functions on g are constant the tail fields of the wsf and the fsf are trivial on any graph on any cayley graph that is not a finite extension of z all component trees of the wsf have one end this is new in z d for d 5 on any tree as well as on any graph with spectral radius less than 1 a s all components of the wsf are recurrent the basic topology of the free and the wired uniform spanning forest measures on lattices in hyperbolic space h d is analyzed a cayley graph is amenable iff for all e 0 the union of the wsf and bernoulli percolation with parameter e is connected harmonic measure from infinity is shown to exist on any recurrent proper planar graph with finite codegrees we also present numerous open problems and conjectures

percolation in the hyperbolic plane the purpose of this paper is to study percolation in the hyperbolic plane and in transitive planar graphs that are quasi isometric to the hyperbolic plane

issues for covariance analysis of dichotomous and ordered categorical data from randomized clinical trials and non parametric strategies for addressing them analysis of covariance is an effective method for addressing two considerations for randomized clinical trials one is reduction of variance for estimates of treatment effects and thereby the production of narrower confidence intervals and more powerful statistical tests the other is the clarification of the magnitude of treatment effects through adjustment of corresponding estimates for any random imbalances between the treatment groups with respect to the covariables the statistical basis of covariance analysis can be either non parametric with reliance only on the randomization in the study design or parametric through a statistical model for a postulated sampling process for non parametric methods there are no formal assumptions for how a response variable is related to the covariables but strong correlation between response and covariables is necessary for variance reduction computations for these methods are straightforward through the application of weighted least squares to fit linear models to the differences between treatment groups for the means of the response variable and the covariables jointly with a specification that has null values for the differences that correspond to the covariables moreover such analysis is similarly applicable to dichotomous indicators ranks or integers for ordered categories and continuous measurements since non parametric covariance analysis can have many forms the ones which are planned for a clinical trial need careful specification in its protocol a limitation of non parametric analysis is that it does not directly address the magnitude of treatment effects within subgroups based on the covariables or the homogeneity of such effects for this purpose a statistical model is needed when the response criterion is dichotomous or has ordered categories such a model may have a non linear nature which determines how covariance adjustment modifies results for treatment effects insight concerning such modifications can be gained through their evaluation relative to non parametric counterparts such evaluation usually indicates that alternative ways to compare treatments for a response criterion with adjustment for a set of covariables mutually support the same conclusion about the strength of treatment effects this robustness is noteworthy since the alternative methods for covariance analysis have substantially different rationales and assumptions since findings can differ in important ways across alternative choices for covariables as opposed to methods for covariance adjustment the critical consideration for studies with covariance analyses planned as the primary method for comparing treatments is the specification of the covariables in the protocol or in an amendment or formal plan prior to any unmasking of the study

global stability of discrete population models with time delays and fluctuating environment the global stability of a discrete population model of volterra type is studied the model incorporates time delays and allows for a fluctuating environment by linearization of the model at positive solutions and construction of liapunov functionals sufficient conditions are obtained to ensure a positive solution of the model is stable and attracts all positive solutions

permanence and global attractivity of the discrete gilpin ayala type population model in this paper we propose a discrete gilpin ayala competition model and a discrete gilpin ayala type multispecies competition predator model for general nonautonomous case sufficient conditions which ensure the permanence and the global stability of the system are obtained for periodic case sufficient conditions which ensure the existence of an unique globally stable positive periodic solution of the system are obtained an example together with its numeric simulations shows the feasibility of the main results

stability of infinite clusters in supercritical percolation a recent theorem by haggstrom and peres concerning independent percolation is extended to all the quasi transitive graphs this theorem states that if 0 p1 p2 1 and percolation occurs at level p1 then every infinite cluster at level p2 contains some infinite cluster at level p1 consequences are the continuity of the percolation probability above the percolation threshold and the monotonicity of the uniqueness of the infinite cluster i e if at level p1 there is a unique infinite cluster then the same holds at level p2 these results are further generalized to graphs with a uniform percolation property the threshold for uniqueness of the infinite cluster is characterized in terms of connectivities between large balls

a necessary and sufficient condition for permanence of a lotka volterra discrete system with delays abstract in this paper we consider the permanence of the following lotka volterra discrete competition system with delays k1 k2 l1 and l2 x n 1 x n exp r1 1 x n k1 1y n k2 y n 1 y n exp r2 1 2x n l1 y n l2 we show the system is permanent for all nonnegative integers k1 k2 l1 and l2 if and only if 1

biased estimates of treatment effect in randomized experiments with nonlinear regressions and omitted covariates summary certain important nonlinear regression models lead to biased estimates of treatment effect even in randomized experiments if needed covariates are omitted the asymptotic bias is determined both for estimates based on the method of moments and for maximum likelihood estimates the asymptotic bias from omitting covariates is shown to be zero if the regression of the response variable on treatment and covariates is linear or exponential and in regular cases this is a necessary condition for zero bias many commonly used models do have such exponential regressions thus randomization ensures unbiased treatment estimates in a large number of important nonlinear models for moderately censored exponential survival data analysis with the exponential survival model yields less biased estimates of treatment effect than analysis with the proportional hazards model of cox if needed covariates are omitted simulations confirm that calculations of asymptotic bias are in excellent agreement with the bias observed in experiments of modest size

ends in free minimal spanning forests we show that for a transitive unimodular graph the number of ends is the same for every tree of the free minimal spanning forest this answers a question of lyons peres and schramm

persistence and global stability in discrete models of pure delay nonautonomous lotka volterra type abstract consider the persistence and the global asymptotic stability of the following discrete model of pure delay nonautonomous lotka volterra type n i p 1 n i p exp c i p j 1 n l 0 m a ij l p n j p k l p 0 1 2 1 i n n i p n i p 0 p 0 and n i 0 0 1 i n where each ci p and aijl p are bounded for p 0 and inf p 0 l 0 m a ii l p 0 a ij l p 0 i j n 1 i n and k l 0 1 l m in this paper for the above discrete system of pure delay type by improving the former work j math anal appl 273 2002 492 511 which extended the averaged condition offered by s ahmad and a c lazer nonlinear anal 40 2000 37 49 we offer conditions of persistence and considering a lyapunov like discrete function to the above discrete system we establish sufficient conditions of global asymptotic stability