fractals in surface growth with power law noise the authors present a microscopic description of interface growth with power law noise distribution in the form p eta eta 1 approximately 1 mu which exhibits non universal roughening for the mu d 1 case in d 1 dimensions the existence of a fractal pattern in the bulk of the aggregate is explained leading trivially to the proof of the identity alpha z 2 for the roughening and the dynamical scaling exponents alpha and z respectively investigations on the distribution of step sizes of the discretized interface and the saturated growth speed further support the arguments

recursive combination tests we present a method that extends the flexibility of adaptive designs to the number of interim analyses and to the choice of decision boundaries at each stage of the trial the design of the next stage can be determined using all of the information gathered so far additionally one can specify the next stage as the final stage or can plan a further interim analysis the method is based on a recursive application of the two stage combination tests for p values the crucial point is the appropriate definition of a p value function that combines p values from two separate stages to a single p value formally we start with a two stage combination test however the p value of the second stage can be replaced by the p value of a further combination test this applies if the experimenter decides in the first interim analysis to perform another interim analysis thereby extending the design to at least three stages obviously this can be also done in a recursive way in the following interim analyses the test

on the universality of crossing probabilities in two dimensional percolation six percolation models in two dimensions are studied percolation by sites and by bonds on square hexagonal and triangular lattices rectangles of widtha and heightb are superimposed on the lattices and four functions representing the probabilities of certain crossings from one interval to another on the sides are measured numerically as functions of the ratioa b in the limits set by the sample size and by the conventions and on the range of the ratioa b measured the four functions coincide for the six models we conclude that the values of the four functions can be used as coordinates of the renormalization group fixed point

stochastic renormalization group in percolation i fluctuations and crossover abstract a generalization of the renormalization group which describes order parameter fluctuations in finite systems is developed in the specific context of percolation this stochastic renormalization group srg expresses statistical self similarity through a non stationary branching process the srg provides a theoretical basis for analytical or numerical approximations both at and away from criticality whenever the correlation length is much larger than the lattice spacing regardless of the system size for example the srg predicts order parameter distributions and finite size scaling functions for the complete crossover between phases for percolation the simplest srg describes structural quantities conditional on spanning such as the total cluster mass or the minimum chemical distance between two boundaries in these cases the central limit theorem for independent random variables holds at the stable off critical fixed points while a fractal central limit theorem describing long range correlations holds at the unstable critical fixed point this first part of a series of articles explains these basic concepts and a general theory of crossover subsequent parts will focus on limit theorems and comparisons of small cell srg approximations with simulation results

modeling λ invariants by p adic random matrices how often is the p adic invariant of an imaginary quadratic field equal to m we model this by the statistics of random p adic matrices and test these predictions numerically 2011 wiley periodicals inc

confidence intervals for a binomial parameter following a multistage test with application to mil std 105d and medical trials this paper describes a method for constructing confidence intervals for a binomial parameter upon termination of a sequential or multistage test tables are presented for use with the mil std 105d multiple sampling plans for acceptance sampling also given are tables for use with some three stage schemes that have been proposed in connection with biomedical trials the results are compared with confidence intervals calculated as if the sampling plan had been one with a fixed sample size

universality at the three dimensional percolation threshold the fraction of samples spanning a lattice at its percolation threshold is found via simulations of bond and site bond percolation to have a universal value of about 0 42 in three dimensions the ratio of shift of the finite to infinite size threshold to width of the threshold distribution is also compatible with universality lattices with up to 10013 sites and or very good statistics were needed to obtain clear results in these monte carlo tests preliminary studies with smaller lattices were inconclusive

test of universal finite size scaling in two dimensional site percolation traditional two scale factor universality concerning the number of clusters within a correlation volume near the percolation threshold is reconfirmed by a comparison of site percolation on square and triangular lattices we also test universality at the threshold e g of the ratio where is the correlation length and l the lattice size at the threshold the result is sensitive to the system s shape and aspect ratio boundary conditions the algorithm the detailed units measuring and l and possibly other factors as yet unexplored

amplitude ratios for critical systems in the c 2 universality class we study the finite size corrections of the critical dense polymer cdp and the dimer models on n rectangular lattice we find that the finite size corrections in the cdp and dimer models depend in a crucial way on the parity of n and a change of the parity of n is equivalent to the change of boundary conditions we present a set of universal amplitude ratios for amplitudes in finite size correction terms of critical systems in the universality class with central charge c 2 the results are in perfect agreement with a perturbated conformal field theory under the assumption that all analytical corrections coming from the operators which belongs to the tower of the identity our results inspire many interesting problems for further studies

lattice shapes and scaling functions for bond random percolation on honeycomb lattices we use a histogram monte carlo simulation method to calculate the scaling functions of the existence probability ep and the percolation probability p of the bond random percolation model on honeycomb lattices with aspect ratios of 0 5 and 2 0 we find that such different aspect ratios give quite different scaling functions near the critical region however they give a consistent critical point critical exponents and the thermodynamic order parameter from renormalization group calculations some interesting theoretical problems related to this work are discussed

adaptive and nonadaptive group sequential tests summary methods have been proposed for redesigning a clinical trial at an interim stage in order to increase power in order to preserve the type i error rate methods for unplanned design change have to be defined in terms of nonsufficient statistics and this calls into question their efficiency and the credibility of conclusions reached we evaluate schemes for adaptive redesign extending the theoretical arguments for use of sufficient statistics of tsiatis mehta 2003 and assessing the possible benefits of preplanned adaptive designs by numerical computation of optimal tests these optimal adaptive designs are concrete examples of optimal sequentially planned sequential tests proposed by schmitz 1993 we conclude that the flexibility of unplanned adaptive designs comes at a price and we recommend that the appropriate power for a study should be determined as thoroughly as possible at the outset then standard error spending tests possibly with unevenly spaced analyses provide efficient designs but it is still possible to fall back on flexible methods for redesign should study objective change unexpectedly once the trial is under way

the efficient determination of the percolation threshold by a frontier generating walk in a gradient the frontier in gradient percolation is generated directly by a type of self avoiding random walk the existence of the gradient permits one to generate an infinite walk on a computer of finite memory from this walk the percolation threshold pc for a two dimensional lattice can be determined with apparently maximum efficiency for a naive monte carlo calculation or n 12 for a square lattice the value pc 0 592745 or 0 000002 is found for a simulation of n 2 6 1011 total steps occupied and blocked perimeter sites the power of the method is verified on the kagome site percolation case

critical exponents of two dimensional potts and bond percolation models critical exponents of the two dimensional q state potts model are calculated by means of finite size scaling and transfer matrix techniques for continuous q results for the temperature exponent agree accurately with the conjecture of den nijs 1979 the magnetic exponent is found to behave in accordance with the conjecture of nienhuis et al 1980

adaptive group sequential designs for clinical trials combining the advantages of adaptive and of classical group sequential approaches a general method is presented integrating the concept of adaptive interim analyses into classical group sequential testing this allows the researcher to represent every group sequential plan as an adaptive trial design and to make design changes during the course of the trial after every interim analysis in the same way as with adaptive designs the concept of adaptive trial designing is thereby generalized to a large variety of possible sequential plans

discrete sequential boundaries for clinical trials summary pocock 1977 o brien fleming 1979 and slud wei 1982 have proposed different methods to construct discrete sequential boundaries for clinical trials these methods require that the total number of decision times be specified in advance in the present paper we propose a more flexible way to construct discrete sequential boundaries the method is based on the choice of a function a t which characterizes the rate at which the error level ac is spent the boundary at a decision time is determined by a t and by past and current decision times but does not depend on the future decision times or the total number of decision times

boundary conditions and scaling functions of percolation models we use a histogram monte carlo simulation method to calculate the scaling functions of the existence probability ep and the percolation probability p of the site percolation model on square lattices with free and periodic boundary conditions we find that different boundary conditions give quite different scaling functions near the critical region however they give the consistent critical point critical exponents and the thermodynamic order parameter from renormalization group calculations similar results are found for other percolation models the implications of our calculated results for some theoretical problems of current interest are discussed

group sequential tests for delayed responses with discussion summary group sequential methods are used routinely to monitor clinical trials and to provide early stopping when there is evidence of a treatment effect a lack of an effect or concerns about patient safety in many studies the response of clinical interest is measured some time after the start of treatment and there are subjects at each interim analysis who have been treated but are yet to respond we formulate a new form of group sequential test which gives a proper treatment of these pipeline subjects these tests can be applied even when the continued accrual of data after the decision to stop the trial is unexpected we illustrate our methods through a series of examples we define error spending versions of these new designs which handle unpredictable group sizes and provide an information monitoring framework that can accommodate nuisance parameters such as an unknown response variance by studying optimal versions of our new designs we show how the benefits of lower expected sample size that are normally achieved by a group sequential test are reduced when there is a delay in response the loss of efficiency for larger delays can be ameliorated by incorporating data on a correlated short term end point fitting a joint model for the two end points but still making inferences on the original longer term end point we derive p values and confidence intervals on termination of our new tests

universality of critical existence probability for percolation on three dimensional lattices using a histogram monte carlo simulation method we calculate the existence probability for bond percolation on simple cubic sc and body centred cubic bcc lattices and site percolation on sc lattices with free boundary conditions the spanning rule considered by reynolds stanley and klein is used to define percolating clusters we find that for such systems has very good finite size scaling behaviour and the value of at the critical point is universal and is about

self similarity in the classification of finite size scaling functions for toroidal boundary conditions the conventional periodic boundary conditions in two dimensions are extended to general boundary conditions prescribed by primitive vector pairs that may not coincide with the coordinate axes this extension is shown to be unambiguously specified by the twisting scheme equivalent relations between different twist settings are constructed explicitly the classification of finite size scaling functions is discussed based on the equivalent relations a self similar pattern for distinct classes of finite size scaling functions is shown to appear on the plane that parametrizes the toroidal geometry

percolation fractals and the finite size scaling of existence probability abstract based on the connection between phase transitions of thermodynamic systems and percolation transitions we obtain the geometrical meaning of critical exponents using finite size scaling of the percolation probability p and the existence probability e p for percolating clusters and percolating subgraphs respectively where e p is the ratio of the total probability weight for percolating subgraphs to the total probability weight for all subgraphs we use a histogram monte carlo simulation method to calculate e p for a q state bond correlated percolation model qbcpm which is corresponding to the q state potts model on the square lattice with various linear dimensions we find that for a given q e p has very good scaling behavior but e p for different q has different scaling functions

issues in designing flexible trials we outline the general framework of adaptive combination tests and discuss their relationship to flexible group sequential designs an important field of applications is sample size reassessment we discuss reassessment rules based on conditional power arguments using either the observed or the prefixed effect size these rules tend to lead to large expected sample sizes for small actual effects however the application of a maximal bound for the second stage sample size leads to more favourable properties additionally we consider an optimized reassessment rule in terms of expected sample sizes since the adaptive design does not use the classical test statistics for some types of sample size reassessments the adaptive test may reject the null hypothesis while the classical one sample test does not we characterize sample size reassessment rules where such inconsistencies are avoided finally the extension of flexibility to the number of stages is explored in the first interim analysis a second interim analysis is only planned if the chance to achieve a decision there is high this leads to savings in the average number of interim analysis performed without paying a noticeable price in terms of expected sample size

scaling functions of the q state potts model on planar lattices abstract based on the subgraph expansion of the q state potts model qpm in an external field it has been shown that the qpm is corresponding to a q state bond correlated percolation model qbcpm the histogram monte carlo simulation method proposed by hu is used to calculate the existence probability e p g p q and the percolation probability p g p q of the qbcpm on the honeycomb the kagome and the plane triangular lattices with various linear dimensions from e p g p q and p g p q we obtain scaling functions of the qpm and qbcpm we find that as q or the coordination number of the lattices increases the widths of the scaling functions also increase the implication of this study is discussed

exact confidence intervals following a group sequential trial a comparison of methods summary with interest in designing group sequential clinical trials increasing methodology for analysing the data arising in such trials is needed we discuss several approaches to constructing confidence intervals for the parameter of interest in a group sequential trial after the trial has ended we compare these met1fods to one already in the literature and point out interesting differences

exact partition functions of the ising model on m n planar lattices with periodic aperiodic boundary conditions the grassmann path integral approach is used to calculate exact partition functions of the ising model on m n square sq plane triangular pt and honeycomb hc lattices with periodic periodic pp periodic antiperiodic pa antiperiodic periodic ap and antiperiodic antiperiodic aa boundary conditions the partition functions are used to calculate and plot the specific heat c kb as a function of temperature kbt j we find that for the n n sq lattice c kb for pa and ap boundary conditions are different from those for aa boundary conditions but for the n n pt and hc lattices c kb for ap pa and aa boundary conditions have the same values our exact partition functions might also be useful for understanding the effects of lattice structures and boundary conditions on critical finite size corrections of the ising model

repeated confidence intervals for the median survival time summary we describe methods for the construction of a confidence interval for median survival time based on right censored data these methods are extended to the construction of repeated confidence intervals for the median based on accumulating data here the overall probability that all intervals contain the true median is guaranteed at a fixed level the use of repeated confidence intervals for median survival time in post marketing surveillance is discussed a confidence interval for median survival time provides a useful summary of the survival experience of a group of patients if confidence intervals are calculated repeatedly as data accumulates the probability that at least one interval fails to contain the median may be much higher than the error rate for a single interval and if these confidence intervals are used in a decision making process the probability of an incorrect decision increases accordingly jennison turnbull 1984 have proposed methods for calculating repeated confidence intervals appropriate to such situations similar ideas have also been discussed by lai 1984 in 2 we propose a new form of single sample nonparametric confidence interval this interval has asymptotically correct coverage probability and monte carlo simulations suggest it is superior to its competitors for small sample sizes repeated confidence intervals for the median are presented in 3 and their stnall sample size performance is assessed by monte carlo simulation an example of their use is given in 4 all the methods considered can easily be modified to give confidence intervals for other quantiles or for the survival probability at a fixed time

universal scaling functions and quantities in percolation models abstract we briefly review recent work on universal finite size scaling functions ufssfs and quantities in percolation models the topics under discussion include a ufssfs for the existence probability also called crossing probability e p the percolation probability p and the probability w n of the appearance of n percolating clusters b universal slope for average number of percolating clusters c ufssfs for a q state bond correlated percolation model corresponding to the q state potts model we also briefly mention some very recent related developments and discuss implications of our results

p values for tests using a repeated significance test design summary repeated significance testing can be used as a multistage testing procedure which has advantages over single sample tests and purely sequential tests repeated significance tests having a constant nominal level of significance have designs which are more easily understood by researchers and still have good power relative to designs using varying nominal levels of significance in this paper we present a method for defining an overall p value for repeated significance tests with constant significance levels which can aid researchers in interpreting the observed results of such tests a table of p values for various designs is given

minireview new results for old percolation abstract percolation theory deals with the formation of clusters in disordered media at the percolation threshold for the first time an infinite cluster appears however research of the last years showed that at this phase transition some old ideas were wrong there can also be two or three clusters spanning from top to bottom even in large two dimensional lattices and the probability of spanning is not given by simple real space renormalization ideas

group sequential analysis incorporating covariate information abstract in this article we survey existing results concerning the joint distribution of the sequence of estimates of the parameter vector when a model is fitted to accumulating data and provide a unified theory that explains the independent increments structure commonly seen in group sequential test statistics our theory covers normal linear models including the case of correlated observations and asymptotic results extend to generalized linear models and the proportional hazards regression model for survival data the asymptotic results are derived using standard methods for the nonsequential case and they hold as long as these nonsequential techniques are applicable at each individual analysis in all cases the joint distribution of the sequence of parameter estimates has the same form exactly or asymptotically as that of the sequence of means of an increasing number of independent identically distributed normal variables thus our results provide the formal basis for extending the scope of s

on cardy s formula for the critical crossing probability in 2d percolation cardy s formula for the probability r of crossing a rectangular critical percolation system in the vertical direction with free boundaries on the two sides is written explicitly in terms of the aspect ratio r heightlwidth the first three terms are given by n 24 33i r 4 2 e r 3 e 7n 13 e 3zr 3 i which with n 1 nv i r m sufficient to determine x to high accuracy for all r

designed extension of studies based on conditional power we propose a flexible method of extending a study based on conditional power the possibility for extension when the p value at the planned end is small but not statistically significant is built in to the design of the study the significance of the treatment difference at the planned end is used to determine the number of additional observations needed and the critical value necessary for use after accruing those additional observations it may therefore be thought of as a two stage procedure even though the observed treatment difference at stage 1 is used to make decisions the type i error rate is protected

internal pilot studies i type i error rate of the naive t test when sample size is recalculated using unblinded interim data use of the usual t test at the end of a study may lead to an elevated type i error rate this paper describes a numerical quadrature investigation to calculate the true probability of rejection as a function of the time of the recalculation the magnitude of the detectable treatment effect and the ratio of the guessed to the true variance we consider both restricted designs those that require final sample size at least as large as the originally calculated size and unrestricted designs those that permit smaller final sample sizes than originally calculated our results indicate that the bias in the type i error rate is often negligible especially in restricted designs some sets of parameters however induce non trivial bias in the unrestricted design