sample size and power calculations for open cohort longitudinal cluster randomized trials when calculating sample size or power for stepped wedge or other types of longitudinal cluster randomized trials it is critical that the planned sampling structure be accurately specified one common assumption is that participants will provide measurements in each trial period that is a closed cohort and another is that each participant provides only one measurement during the course of the trial however some studies have an open cohort sampling structure where participants may provide measurements in variable numbers of periods to date sample size calculations for longitudinal cluster randomized trials have not accommodated open cohorts feldman and mckinlay 1994 provided some guidance stating that the participant level autocorrelation could be varied to account for the degree of overlap in different periods of the study but did not indicate precisely how to do so we present sample size and power formulas that allow for open cohorts and discuss the impact of the degree of openness on sample size and power we consider designs where the number of participants in each cluster will be maintained throughout the trial but individual participants may provide differing numbers of measurements our results are a unification of closed cohort and repeated cross sectional sample results of hooper et al 2016 and indicate precisely how participant autocorrelation of feldman and mckinlay should be varied to account for an open cohort sampling structure we discuss different types of open cohort sampling schemes and how open cohort sampling structure impacts on power in the presence of decaying within cluster correlations and autoregressive participant level errors

some existence and nonexistence theorems for solutions of degenerate parabolic equations abstract the initial and boundary value problem for the degenerate parabolic equation v t v f v in the cylinder 0 r n bounded for a certain class of point functions satisfying v 0 e g v v m sign v is considered in the case that f v sign v c 1 v the equation has a global time solution the same is true for 1 provided the measure of is sufficiently small in the case that f v v is nondecreasing a condition is given on the initial state v x 0 which implies that the solution must blow up in finite time the existence of such initial states is discussed

accounting for a decaying correlation structure in cluster randomized trials with continuous recruitment a requirement for calculating sample sizes for cluster randomized trials crts conducted over multiple periods of time is the specification of a form for the correlation between outcomes of subjects within the same cluster encoded via the within cluster correlation structure previously proposed within cluster correlation structures have made strong assumptions for example the usual assumption is that correlations between the outcomes of all pairs of subjects are identical uniform correlation more recently structures that allow for a decay in correlation between pairs of outcomes measured in different periods have been suggested however these structures are overly simple in settings with continuous recruitment and measurement we propose a more realistic continuous time correlation decay structure whereby correlations between subjects outcomes decay as the time between these subjects measurement times increases we investigate the use of this structure on trial planning in the context of a primary care diabetes trial where there is evidence of decaying correlation between pairs of patients outcomes over time in particular for a range of different trial designs we derive the variance of the treatment effect estimator under continuous time correlation decay and compare this to the variance obtained under uniform correlation for stepped wedge and cluster randomized crossover designs incorrectly assuming uniform correlation will underestimate the required sample size under most trial configurations likely to occur in practice planning of crts requires consideration of the most appropriate within cluster correlation structure to obtain a suitable sample size

the blow up rate for a degenerate parabolic equation with a non local source abstract in this paper we establish the local existence of the solution and the finite time blow up result for the equation v v m xx a l l v q dx x l l 0 moreover it is proved that the solution has global blow up and there exist constants c 1 c 2 such that c 1 t 1 1 q 1 max x l l v x c 2 t 1 1 q 1 where t 1 is the blow up time

positive solutions of quasilinear parabolic systems with nonlinear boundary conditions abstract the aim of this paper is to investigate the existence uniqueness and asymptotic behavior of solutions for a coupled system of quasilinear parabolic equations under nonlinear boundary conditions including a system of quasilinear parabolic and ordinary differential equations also investigated is the existence of positive maximal and minimal solutions of the corresponding quasilinear elliptic system as well as the uniqueness of a positive steady state solution the elliptic operators in both systems are allowed to be degenerate in the sense that the density dependent diffusion coefficients d i u i may have the property d i 0 0 for some or all i our approach to the problem is by the method of upper and lower solutions and its associated monotone iterations it is shown that the time dependent solution converges to the maximal solution for one class of initial functions and it converges to the minimal solution for another class of initial functions and if the maximal and minimal solutions coincide then the steady state solution is unique and the time dependent solution converges to the unique solution applications of these results are given to three model problems including a porous medium type of problem a heat transfer problem and a two component competition model in ecology these applications illustrate some very interesting distinctive behavior of the time dependent solutions between density independent and density dependent diffusions

a new result for the porous medium equation derived from the ricci flow given r 2 with a good complete metric we show that the unique solution of the ricci flow approaches a soliton at time infinity solitons are solutions of the ricci flow which move only by diffeomorphism the ricci flow on r 2 is the limiting case of the porous medium equation when m is zero the results in the ricci flow may therefore be interpreted as sufficient conditions on the initial data which guarantee that the corresponding unique solution for the porous medium equation on the entire plane asymptotically behaves like a soliton solution

on blow up results for solutions of inhomogeneous evolution equations and inequalities abstract the main purpose of this work is to obtain blow up results for solutions of the inequality u t u u u q x on the half space r 1 r n where n 1 0 q 1 and r n 0 is a nontrivial locally integrable function and in particular to establish that for n 2 the critical blow up exponent q n 1 n 2 belongs to the blow up case

the nonlinear heat equation with absorption effects of variable coefficients abstract we consider the nonnegative solutions to the nonlinear degenerate parabolic equation ut d x t um 1ux x b x t up with m 1 0

the role of non linear diffusion in non simultaneous blow up we study a parabolic system of two non linear reaction diffusion equations completely coupled through source terms and with power like diffusivity under adequate hypotheses on the initial data we prove that non simultaneous blow up is sometimes possible i e one of the components blows up while the other remains bounded the conditions for non simultaneous blow up rely strongly on the diffusivity parameters and significant differences appear between the fast diffusion and the porous medium case surprisingly flat homogeneous in space solutions are not always a good guide to determine whether non simultaneous blow up is possible

uniform convergence of hankel transforms abstract we investigate necessary and or sufficient conditions for the pointwise and uniform convergence of the weighted hankel transforms l f r r 0 r t f t j r t d t 1 2 r 0 where r are such that 0 3 2 we subdivide these transforms into two classes in such a way that the uniform convergence criteria is remarkably different on each class in more detail we have the transforms satisfying 0 such as the classical hankel transform that generalize the cosine transform and those satisfying 0 3 2 generalizing the sine transform

quasilinear reaction diffusion systems with nonlinear boundary conditions this paper deals with the existence and nonexistence of global positive solutions of quasilinear reaction diffusion systems with nonlinear boundary conditions necessary and sufficient conditions on the global existence of all positive solutions are obtained

blow up for a degenerate parabolic equation with a nonlocal source abstract in this paper we investigate the blowup properties of the positive solutions to the following nonlocal degenerate parabolic equation v x v m xx 0 l v p 1 dx kv q 1 with homogeneous dirichlet boundary conditions in the interval 0 l where 0 m 1 we first establish the local existence and uniqueness of its classical solutions then we show that the positive solution blows up in finite time if the initial datum is sufficient large finally we prove that the blow up set is the whole interval and we also obtain the estimates of the blow up rate

blow up for the porous media equation with source term and positive initial energy abstract we study the cauchy dirichlet problem for the porous media equation with nonlinear source term in a bounded subset of r n the problem describes the propagation of thermal perturbations in a medium with a nonlinear heat conduction coefficient and a heat source depending on the temperature the aim of the paper is to extend the unstable set to a part of the positive energy region a phenomenon which was known only for linear conduction

singular reaction diffusion equations of porous medium type abstract a class of reaction diffusions equations with nonlinear boundary conditions and porous medium type of diffusion is investigated by the method of upper and lower solutions the diffusion term in the equation includes the so called slow diffusion m 1 fast diffusion 0 m 1 and super diffusion m 0 and the reaction term may involve a singular function the aim of this paper is to show the existence of a global time dependent solution the existence of positive minimal and maximal steady state solutions including the uniqueness of the positive solution and the asymptotic behavior of the time dependent solution in relation to the positive steady state solutions of particular concern is the convergence of the time dependent solution to a unique positive steady state solution and the global attraction property of the reaction diffusion problem applications are given to several specific problems which are of considerable interest in the current literature

stability and instability for solutions of the convective porous medium equation with a nonlinear forcing at the boundary ii abstract we continue our study of the long time behavior of nonnegative solutions for the degenerate parabolic equation u t u m xx n u n x 0 x t 0 subject to nonlinear boundary conditions here we consider solutions of this equation subject to the boundary conditions here we consider solutions of this equation subject to the boundary conditions u m x 0 t au p 0 t u 1 t 0 t 0 as in part i j r anderson j differential equations to appear we give the bifurcation diagrams for the stationary solutions due to the existence of singular stationary states some of these diagrams are three dimensional we also examine the stability properties of these states when n p m 1

quasilinear parabolic and elliptic equations with nonlinear boundary conditions abstract this paper is concerned with a class of quasilinear parabolic and elliptic equations in a bounded domain with both dirichlet and nonlinear neumann boundary conditions the equation under consideration may be degenerate or singular depending on the property of the diffusion coefficient the consideration of the class of equations is motivated by some heat transfer problems where the heat capacity and thermal conductivity are both temperature dependent the aim of the paper is to show the existence and uniqueness of a global time dependent solution of the parabolic problem existence of maximal and minimal steady state solutions of the elliptic problem including conditions for the uniqueness of a solution and the asymptotic behavior of the time dependent solution in relation to the steady state solutions applications are given to some heat transfer problems and an extended logistic reaction diffusion equation

uniform convergence and integrability of fourier integrals abstract firstly we study the uniform convergence of cosine and sine fourier transforms secondly we obtain pitt boas type results on l p integrability of fourier transforms with the power weights the solutions of both problems are written as criteria in terms of general monotone functions

asymptotic behavior of the nonlinear diffusion equation nt n 1nx x the asymptotic behavior of the equation nt ln n xx is studied on the finite interval 0 x 1 with the boundary conditions n 0 t n 1 t n0 and initial data n x 0 n0 we prove that asymptotically ln n x t n0 a exp 2t n0 21 2 sin x and also provide rigorous upper and lower bounds on the asymptotic amplitude a in terms of integrals of nonlinear functions of the initial data the rigorous bounds are compared to values of a obtained from computer experiments the lower bound l 23 2 exp li 1 q where li is the logarithmic integral is euler s constant and q 2 f n x 0 n0 1 sin x dx is found to be the best known estimate of a

sample size calculation for stepped wedge and other longitudinal cluster randomised trials the sample size required for a cluster randomised trial is inflated compared with an individually randomised trial because outcomes of participants from the same cluster are correlated sample size calculations for longitudinal cluster randomised trials including stepped wedge trials need to take account of at least two levels of clustering the clusters themselves and times within clusters we derive formulae for sample size for repeated cross section and closed cohort cluster randomised trials with normally distributed outcome measures under a multilevel model allowing for variation between clusters and between times within clusters our formulae agree with those previously described for special cases such as crossover and analysis of covariance designs although simulation suggests that the formulae could underestimate required sample size when the number of clusters is small whether using a formula or simulation a sample size calculation requires estimates of nuisance parameters which in our model include the intracluster correlation cluster autocorrelation and individual autocorrelation a cluster autocorrelation less than 1 reflects a situation where individuals sampled from the same cluster at different times have less correlated outcomes than individuals sampled from the same cluster at the same time nuisance parameters could be estimated from time series obtained in similarly clustered settings with the same outcome measure using analysis of variance to estimate variance components copyright 2016 john wiley sons ltd

global existence for nonlinear diffusion equations solutions of nonlinear possibly degenerate reaction diffusion models are known to exist for all time if the asymptotic growth of the reaction term is not greater than that of the diffusion term via concavity methods it is also known that negating such a condition results in solutions which blow up in finite time on the other hand two recent studies regarding the case of linear diffusion have reported on the ability for convective terms in a model to create global existence of solutions where no such result is true in their absence in this paper we develop a theory of global existence for a general class of reaction diffusion convection models which only requires a similar balance of diffusion and reaction our result extends previous work to models including convection but it does not reveal the exact role of convection in yielding global solutions further analysis of a slightly simplified model establishes the global existence of all solutions if the reaction term grows asymptotically at a rate less than that of either diffusion or convection

on the uniform convergence of sine integrals abstract the classical theorem of chaundry and jolliffe states that the sine series k 1 a k sin k x with coefficients a 1 a 2 a k 0 converges uniformly in x if and only if k a k 0 as k recently the monotonicity condition has been relaxed by a number of authors an analysis of the proofs of these results reveals that condition is sufficient for the uniform convergence even in the case of complex coefficients under appropriately modified conditions but our main achievement is the extension of these results for the sine integral 0 f x sin t x d x where f r c is a measurable function with the property x f x l loc 1 r

design and analysis considerations for cohort stepped wedge cluster randomized trials with a decay correlation structure a stepped wedge cluster randomized trial is a type of longitudinal cluster design that sequentially switches clusters to intervention over time until all clusters are treated while the traditional posttest only parallel design requires adjustment for a single intraclass correlation coefficient the stepped wedge design allows multiple outcome measurements from the same cluster and so additional correlation parameters are necessary to characterize the within cluster correlation structure although a number of studies have differentiated between the concepts of within period and between period correlations only a few studies have allowed the between period correlation to decay over time in this article we consider the proportional decay correlation structure for a cohort stepped wedge design and provide a matrix adjusted quasi least squares approach to accurately estimate the correlation parameters along with the marginal intervention effect we further develop the sample size and power procedures accounting for the correlation decay and investigate the accuracy of the power procedure with continuous outcomes in a simulation study we show that the empirical power agrees well with the prediction even with as few as nine clusters when data are analyzed with matrix adjusted quasi least squares concurrently with a suitable bias corrected sandwich variance two trial examples are provided to illustrate the new sample size procedure