phase transitions in dynamical random graphs we study a large time limit of a markov process whose states are finite graphs the number of the vertices is described by a supercritical branching process and the dynamics of edges is determined by the rates of appending and deleting we find a phase transition in our model similar to the one in the random graph model g n p we derive a formula for the line of critical parameters which separates two different phases one is where the size of the largest component is proportional to the size of the entire graph and another one where the size of the largest component is at most logarithmic with respect to the size of the entire graph in the supercritical phase we find the asymptotics for the size of the largest component

principal eigenvalues and the dirichlet problem for fully nonlinear elliptic operators abstract we study uniformly elliptic fully nonlinear equations of the type f d 2 u d u u x f x we show that convex positively 1 homogeneous operators possess two principal eigenvalues and eigenfunctions and study these objects we obtain existence and uniqueness results for nonproper operators whose principal eigenvalues in some cases only one of them are positive finally we obtain an existence result for nonproper isaac s equations

fujita type exponent for fully nonlinear parabolic equations and existence results abstract in this paper we prove that a class of parabolic equations involving a second order fully nonlinear uniformly elliptic operator has a fujita type exponent these exponents are related with an eigenvalue problem in all r n and play the same role in blow up theorems as the classical p 1 2 n introduced by fujita for the laplacian we also obtain some associated existence results

scaling properties of scale free evolving networks continuous approach the scaling behavior of scale free evolving networks arising in areas such as communications scientific citations collaborations etc is studied we derive universal scaling relations describing properties of such networks and indicate the limits of their validity we show that the main properties of scale free evolving networks may be described in the framework of a simple continuous approach the simplest models of networks growing according to a mechanism of preferential attachment of links to nodes are used we consider different forms of this preference and demonstrate that the range of preferential attachments producing scale free networks is wide we also obtain scaling relations for networks with nonlinear accelerating growth and describe the temporal evolution of the arising distributions size effects char22 the cutoffs of these distributions char22 introduce restrictions for the observation of power law dependences mainly we discuss the so called degree distribution i e the distribution of the number of connections of nodes a scaling form of the distribution of links between pairs of individual nodes for a growing network of citations is also studied we describe the effects of differences between nodes the aging of nodes changes the exponents of the distributions the appearance of a single node with high fitness changes the degree distribution of a network dramatically if its fitness exceeds some threshold value this node captures a finite part of all links of the network we show that permanent random damage to a growing scale free network char22 a permanent deletion of some links char22 radically changes the values of the scaling exponents results of other kinds of permanent damage are described

a multiple step selection procedure with sequential protection of preferred treatments dunnett 1955 journal of the american statistical association 50 1096 1121 proposed a one step many one test to compare several experimental treatment means with a control however after the control treatment is rejected one would want to make comparisons among the experimental treatments we assume that there is an order of decreasing preference among the experimental treatments based on other considerations we propose in this paper a multiple step selection procedure an extension of dunnett s one step many one test that will give protection to the preferred treatments and guarantee a high selection probability of the correct treatment when that selection is appropriate we also provide methods to calculate the sample size required to carry out this test for normal binomial and exponential data with random censoring

solutions to an inhomogeneous equation involving infinity laplacian abstract we obtain existence and uniqueness results of viscosity solutions to the dirichlet boundary value problem for a nonlinear highly degenerate elliptic equation of the form 1 d u h u f where 0 h 2 and u denotes the so called infinity laplacian operator given by u i j 1 n u x i u x j 2 u x i x j we also give an asymptotic behavior of the viscosity solutions of two kinds of inhomogeneous infinity laplace equations near an isolated singular point

a tour of the theory of absolutely minimizing functions a detailed analysis of the class of absolutely minimizing functions in euclidean spaces and the relationship to the infinity laplace equation

on a controlled eigenvalue problem abstract we consider the problem of minimizing the asymptotic exit rate of a diffusion from a bounded domain and derive the associated hamilton jacobi bellman equation this turns out to be a nonlinear eigenvalue problem a verification theorem for optimal control is also provided

dynamical random graphs with memory we study the large time dynamics of a markov process whose states are finite but unbounded graphs the number of vertices is described by a supercritical branching process and the edges follow a certain mean field dynamics determined by the rates of appending and deleting the older an edge is the lesser is the probability that it is still in the graph the lifetime of any edge is distributed exponentially we call its mean value common for all edges a parameter of memory since it shows for how long the system keeps a particular connection between the vertices in the graph we show that our model provides a bridge between two well known models when the parameter of memory goes to infinity this is a generalized model of random growth and when this parameter is zero i e no memory our model behaves as a random graph thus by introducing a general class of dynamical graphs we have a unified overview on rather different models and the relations between them we find all the critical values of the parameters at which our model exhibits phase transitions and describe the properties of the phase diagram finally we compare and discuss the efficiency of the corresponding networks

an adaptive confirmatory trial with interim treatment selection practical experiences and unbalanced randomization this article has been motivated by an ongoing international adaptive confirmatory trial at the interim analysis of this two stage trial none one or two active treatment regimens are selected for further study in the second stage a combination test approach is used in this practical setting with an extension of the theory to unbalanced randomization we show that a combination test with suitable weights can still preserve the overall type i error rate provided that the test statistic is chosen appropriately and the unpooled z test for proportions is not used the accuracy of stagewise p values is also discussed in a more general framework monte carlo simulations confirm the validity of the approach retained and evaluate the necessary sample size additional issues addressed during the design of the trial are also examined such as multiplicity due to testing hypotheses on key secondary endpoints a non inferiority comparison to an active treatment and covariate adjusted analyses for various types of outcome

bifurcation and optimal stochastic control abstract two questions concerning bifuraction theory and optimal stochastic control are considered first in a few examples we give the interpretation of a bifurcation in terms of optimal stochastic control next we introduce the analogue of the lowest eigenvalue for the nonlinear operator associated with the hamilton jacobi bellman equations of optimal stochastic control author

two stage selection and testing designs for comparative clinical trials summary a two stage design which selects the best of several experimental treatments and compares it to a standard control is proposed the design allows early termination with acceptance of the global null hypothesis optimal sample size and cut off parameters are obtained by minimizing expected total sample size for fixed significance level and power with many other factors moreover the usual error rate computations associated with comparative testing do not account for the preliminary selection process consequently the overall procedure may be neither effective nor efficient for identifying an experimental treatment which is an improvement over c in this paper we propose a new approach to the problem of identifying the best of k experimental treatments and determining whether it is superior to a control we deal with the binomial setting where patient response may be characterized as either success or failure with ok the success probability for ek k 0 corresponds to c for ease of notation assume 01 ok we propose a two stage procedure which allows early termination with acceptance of h0 o 01 ok with design parameters chosen to minimize expected total sample size the design and accompanying generalized definitions of size a and power 1 3 are given in 2 the algorithm used for optimization is described in 3 numerical results are presented in 4 followed by a discussion of the relative merits of the proposed design

user s guide to viscosity solutions of second order partial differential equations the notion of viscosity solutions of scalar fully nonlinear partial differential equations of second order provides a framework in which startling comparison and uniqueness theorems existence theorems and theorems about continuous dependence may now be proved by very efficient and striking arguments the range of important applications of these results is enormous this article is a self contained exposition of the basic theory of viscosity solutions

a strict inequality for the random triangle model the random triangle model on a graph g is a random graph model where the usual i i d measure is perturbed by a factor qt where q 1 is a constant and t is the number of triangles in the random subgraph here we consider the case where g is the usual two dimensional triangular lattice for which there exists a percolation threshold pc q such that the probability of getting an infinite connected component of retained edges is 0 for p pc q it has previously been shown that pc q is a decreasing function of q here we strengthen this by showing that pc q is strictly decreasing this confirms a conjecture by haggstrom and jonasson

long paths and cycles in dynamical graphs we study the large time dynamics of a markov process whose states are finite directed graphs the number of the vertices is described by a supercritical branching process and the edges follow a certain mean field dynamics determined by the rates of appending and deleting we find sufficient conditions under which asymptotically a s the order of the largest component is proportional to the order of the graph a lower bound for the length of the longest directed path in the graph is provided as well we derive an explicit formula for the limit as time goes to infinity of the expected number of cycles of a given finite length finally we study the phase diagram

a weighted eigenvalue problem of the degenerate operator associated with infinity laplacian abstract in this paper we study a weighted eigenvalue problem of the degenerate operator associated with infinity laplacian h u a x u h 1 u 0 in u 0 on where h 1 h u d u h 3 u is the h homogeneous infinity laplacian and a x is a positive continuous bounded function in we prove the existence of the principal eigenvalue and a corresponding positive eigenfunction the approach to the weighted eigenvalue problem is based on the maximum principle and when a parameter is less than the principal eigenvalue some existence and uniqueness results related to this problem are established

singular solutions of fully nonlinear elliptic equations and applications we study the properties of solutions of fully nonlinear positively homogeneous elliptic equations near boundary points of lipschitz domains at which the solution may be singular we show that these equations have two positive solutions in each cone of mathbb r n and the solutions are unique in an appropriate sense we introduce a new method for analyzing the behavior of solutions near certain lipschitz boundary points which permits us to classify isolated boundary singularities of solutions which are bounded from either above or below we also obtain a sharp phragmen lindelof result as well as a principle of positive singularities in certain lipschitz domains

a composite design for transition from a preliminary to a full scale study in drug development a sequence of studies are conducted to evaluate effectiveness or efficacy and safety such as a phase ii study to assess pharmacological activity or safety that is then followed by a definitive phase iii study to assess clinical effectiveness rather than conducting separate successive studies we describe a design in which the patients enrolled in a preliminary e g phase ii study are continued into a subsequent full scale e g phase iii study this design also applies to a study that uses an internal pilot with a preliminary assessment of efficacy or safety the combined preliminary to full scale design potentially reduces the total numbers of patients required and the total duration of the program the design allows a futility or safety stopping boundary for the preliminary study result that is specified in terms of a lower critical z value z l and the pursuant type ii error probability incurred under a specified alternative hypothesis of a beneficial effect or no toxicity at that stage this boundary also leads to a deflation of the type i error probability for the final test at the completion of the full scale study such that a critical value for the final test z f can be determined that provides the desired level of the type i error probability exactly thus it is possible to determine sample sizes at the two stages and critical values z l and z f that provide specified type i and ii error probabilities for the combined study we describe the design using large sample normally distributed z tests at the two phases including a test for means or proportions or survival times or combinations thereof such as a test for means at phase ii followed by a test for proportions at phase iii we compare the properties of this design versus the conduct of two successive studies and explore the factors that influence the operating characteristics of the design we also discuss the practical considerations in the implementation of the design

uniformly minimum variance conditionally unbiased estimation in multi arm multi stage clinical trials multi arm multi stage clinical trials compare several experimental treatments with a control treatment with poorly performing treatments dropped at interim analyses this leads to inferential challenges including the construction of unbiased treatment effect estimators a number of estimators unbiased conditional on treatment selection have been proposed but are specific to certain selection rules may ignore the comparison to the control and are not all minimum variance we obtain estimators for treatment effects compared to the control that are uniformly minimum variance unbiased conditional on selection with any specified rule or stopping for futility

an eigenvalue approach to the risk sensitive control problem in near monotone case abstract risk sensitive control problem under near monotonicity condition is considered we prove existence of a solution to the corresponding hamilton jacobi bellman equation by an eigenvalue approach existence of an optimal control has also been proved

inhomogeneous infinity laplace equation abstract we present the theory of the viscosity solutions of the inhomogeneous infinity laplace equation x i u x j u x i x j 2 u f in domains in r n we show existence and uniqueness of a viscosity solution of the dirichlet problem under the intrinsic condition f does not change its sign we also discover a characteristic property which we call the comparison with standard functions property of the viscosity sub and super solutions of the equation with constant right hand side applying these results and properties we prove the stability of the inhomogeneous infinity laplace equation with nonvanishing right hand side which states the uniform convergence of the viscosity solutions of the perturbed equations to that of the original inhomogeneous equation when both the right hand side and boundary data are perturbed in the end we prove the stability of the well known homogeneous infinity laplace equation x i u x j u x i x j 2 u 0 which states the viscosity solutions of the perturbed equations converge uniformly to the unique viscosity solution of the homogeneous equation when its right hand side and boundary data are perturbed simultaneously

the infinity laplacian aronsson s equation and their generalizations the infinity laplace equation u 0 arose originally as a sort of euler lagrange equation governing the absolute minimizer for the l variational problem of minimizing the functional ess sup u du the more general functional ess sup u f x u du leads similarly to the so called aronsson equation a f u 0 in this paper we show that these pde operators and various interesting generalizations also appear in several other contexts seemingly quite unrelated to l variational problems including two person game theory with random order of play rapid switching of states in control problems etc the resulting equations can be parabolic and inhomogeneous equation types precluded in conventional l variational problems

dynamic random networks in dynamic populations we consider a random network evolving in continuous time in which new nodes are born and old may die and where undirected edges between nodes are created randomly and may also disappear the node population is markovian and so is the creation and deletion of edges given the node population each node is equipped with a random social index and the intensity at which a node creates new edges is proportional to the social index and the neighbour is either chosen uniformly or proportional to its social index in a modification of the model we derive properties of the network as time and the node population tends to infinity in particular the degree distribution is shown to be a mixed poisson distribution which may exhibit a heavy tail e g power law if the social index distribution has a heavy tail the limiting results are verified by means of simulations and the model is fitted to a network of sexual contacts

solvability of uniformly elliptic fully nonlinear pde we get existence uniqueness and non uniqueness of viscosity solutions of uniformly elliptic fully nonlinear equations of the hamilton jacobi bellman isaacs type with unbounded ingredients and quadratic growth in the gradient without hypotheses of convexity or properness some of our results are new even for equations in divergence form

the eigenvalue problem the euler lagrange equation of the nonlinear rayleigh quotient left int omega nabla u p dx right bigg left int omega u p dx right is div left nabla u p 2 nabla u right lambda p p u p 2 u where lambda p p is the minimum value of the quotient the limit as p to infty of these equations is found to be max left lambda infty frac nabla u x u x delta infty u x right 0 where the constant lambda infty lim p to infty lambda p is the reciprocal of the maximum of the distance to the boundary of the domain

on viscosity solutions of fully nonlinear equations with measurable ingredients we study fully nonlinear uniformly elliptic equations with measurable ingredients caffarelli s recent work on w2 p estimates for viscosity solutions has led to significant progress in this area here we present a unified treatment of this theory based on an appropriate notion of viscosity solution for instance it is shown that strong solutions are viscosity solutions that viscosity solutions are twice differentiable a e and that the pointwise derivatives satisfy the equation a e an important consequence of our approach is the possibility of passage to various kinds of limits in fully nonlinear equations this extends results of this type due to evans and krylov our work is to some extent expository the main purpose being to provide an easily accessible set of tools and techniques to study equations with measurable ingredients 1996 john wiley sons inc

mean field theory for scale free random networks random networks with complex topology are common in nature describing systems as diverse as the world wide web or social and business networks recently it has been demonstrated that most large networks for which topological information is available display scale free features here we study the scaling properties of the recently introduced scale free model that can account for the observed power law distribution of the connectivities we develop a mean eld method to predict the growth dynamics of the individual vertices and use this to calculate analytically the connectivity distribution and the scaling exponents the mean eld method can be used to address the properties of two variants of the scale free model that do not display power law scaling c 1999 elsevier science b v all rights reserved

a two stage design for choosing among several experimental treatments and a control in clinical trials in clinical trials where several experimental treatments are of interest the goal may be viewed as identification of the best of these and comparison of that treatment to a standard control therapy however it is undesirable to commit patients to a large scale comparative trial of a new regimen without evidence that its therapeutic success rate is acceptably high we propose a two stage design in which patients are first randomized among the experimental treatments and the single treatment having the highest observed success rate is identified if this highest rate falls below a fixed cutoff then the trial is terminated otherwise the best new treatment is compared to the control at a second stage locally optimal values of the cutoff and the stage 1 and stage 2 sample sizes are derived by minimizing expected total sample size the design has both high power and high probability of terminating early when no experimental treatment is superior to the control numerical results for implementing the design are presented and comparison to dunnett s 1984 in design of experiments ranking and selection t j santner and a c tamhane eds 47 66 new york marcel dekker optimal one stage procedure is made

tug of war and the infinity laplacian we consider a class of zero sum two player stochastic games called tug of war and use them to prove that every bounded real valued lipschitz function f on a subset y of a length space x admits a unique absolutely minimal am extension to x i e a unique lipschitz extension u x for which lip u u lip u u for all open u x y

uniqueness of lipschitz extensions minimizing the sup norm of the gradient in this paper we examine the problem of minimizing the sup norm of the gradient of a function with prescribed boundary values geometrically this can be interpreted as finding a minimal lipschitz extension due to the weak convexity of the functional associated to this problem solutions are generally nonunique by adopting g aronsson s notion of absolutely minimizing we are able to prove uniqueness by characterizing minimizers as the unique solutions of an associated partial differential equation in fact we actually prove a weak maximum principle for this partial differential equation which in some sense is the euler equation for the minimization problem this is significantly difficult because the partial differential equation is both fully nonlinear and has very degenerate ellipticity to overcome this difficulty we use the weak solutions of m g crandall and p l lions also known as viscosity solutions in conjunction with some arguments using integration by parts

on the regularity theory of fully nonlinear parabolic equations ii recently m crandall and p l lions 3 developed a very successful method for proving the existence of solutions of nonlinear second order partial differential equations their method called the theory of viscosity solutions also applies to fully nonlinear equations in which even the second order derivatives can enter in nonlinear fashion solutions produced by the viscosity method are guaranteed to be continuous but not necessarily smooth here we announce smoothness results for viscosity solutions our methods extend those of 1 we obtain krylov safonov i e c estimates 8 c 1 schauder c and w estimates for viscosity solutions of uniformly parabolic equations in general form the results can be viewed as a priori estimates on the classical c solutions our method produces in particular regularity results for a broad new array of nonlinear heat equations including the bellman equation 6

existence uniqueness and qualitative properties of positive solutions of quasilinear elliptic equations we study the following quasilinear elliptic equation p pu x a x up 1 b x g u 0in rn where p 1 a b l rn b g 0 b 0 and lloc rn infrn we provide a sharp criterion in term of generalized principal eigenvalues for existence nonexistence of positive solution of p in suitable classes of functions uniqueness result for p in those classes is also derived under additional conditions on we further show that i either for every 0 nonexistence phenomenon occurs ii or there exists a threshold value 0 in the sense that for every 0 existence and uniqueness phenomenon occurs and for every nonexistence phenomenon occurs in the latter case we study the limits as 0 and of the sequence of positive solutions of p our results are new even in the case p 2

are randomly grown graphs really random we analyze a minimal model of a growing network at each time step a new vertex is added then with probability ensuremath delta two vertices are chosen uniformly at random and joined by an undirected edge this process is repeated for t time steps in the limit of large t the resulting graph displays surprisingly rich characteristics in particular a giant component emerges in an infinite order phase transition at ensuremath delta 1 8 at the transition the average component size jumps discontinuously but remains finite in contrast a static random graph with the same degree distribution exhibits a second order phase transition at ensuremath delta 1 4 and the average component size diverges there these dramatic differences between grown and static random graphs stem from a positive correlation between the degrees of connected vertices in the grown graph older vertices tend to have higher degree and to link with other high degree vertices merely by virtue of their age we conclude that grown graphs however randomly they are constructed are fundamentally different from their static random graph counterparts

principal eigenvalue of a very badly degenerate operator and applications abstract in this paper we define and investigate the properties of the principal eigenvalue of the singular infinity laplace operator u d 2 u d u d u d u d u this operator arises from the optimal lipschitz extension problem and it plays the same fundamental role in the calculus of variations of l functionals as the usual laplacian does in the calculus of variations of l 2 functionals our approach to the eigenvalue problem is based on the maximum principle and follows the outline of the celebrated work of berestycki nirenberg and varadhan h berestycki l nirenberg s r s varadhan the principal eigenvalue and maximum principle for second order elliptic operators in general domains comm pure appl math 47 1 1994 47 92 in the case of uniformly elliptic linear operators as an application we obtain existence and uniqueness results for certain related non homogeneous problems and decay estimates for the solutions of the evolution problem associated to the infinity laplacian

eigenvalue and dirichlet problem for fully nonlinear operators in non smooth domains we study the maximum principle the existence of eigenvalue and the existence of solution for the dirichlet problem relative to operators which are fully nonlinear elliptic but presenting some singularity or degeneracy which are similar to those of the p laplacian we consider the equations in bounded domains which only satisfy the exterior cone condition

landesman lazer type results for second order hamilton jacobi bellman equations we study the boundary value problem f d2u du u x u f x u in u 0on where the second order differential operator f is of hamilton jacobi bellman type f is sub linear in u at infinity and rn is a regular bounded domain we extend the well known landesman lazer conditions to study various bifurcation phenomena taking place near the two principal eigenvalues associated to the differential operator we provide conditions under which the solution branches extend globally along the eigenvalue gap we also present examples illustrating the results and hypotheses

eigenvalues for radially symmetric fully nonlinear singular or degenerate operators abstract in this paper we present a one dimensional and radial theory for the existence of eigenvalues and eigenfunctions for fully nonlinear elliptic 1 homogeneous operators 1 a general theory for the first eigenvalue and eigenfunction exists in the frame of viscosity solutions but in this particular case a simpler theory can be established that extends via degree theory to obtain the complete set of eigenvalues and eigenfunctions characterized by the number of zeros

low complexity methods for estimation after parameter selection statistical inference of multiple parameters often involves a preliminary parameter selection stage the selection stage has an impact on subsequent estimation for example by introducing a selection bias the post selection maximum likelihood psml estimator is shown to reduce the selection bias and the post selection mean squared error psmse compared with conventional estimators such as the maximum likelihood ml estimator however the computational complexity of the psml is usually high due to the multi dimensional exhaustive search for a global maximum of the post selection log likelihood psll function moreover the psll involves the probability of selection that in general does not have an analytical form in this paper we develop new low complexity post selection estimation methods for a two stage estimation after parameter selection architecture the methods are based on implementing the iterative maximization by parts mbp approach which is based on the decomposition of the psll function into easily optimized and complicated parts the proposed second best psml method applies the mbp psml algorithm with a pairwise probability of selection between the two highest ranked parameters w r t the selection rule the proposed sa psml method is based on using stochastic approximation sa and monte carlo integrations to obtain a non parametric estimation of the gradient of the probability of selection and then applying the mbp psml algorithm on this approximation for low complexity performance analysis we develop the empirical post selection cram acute text e r rao type lower bound simulations demonstrate that the proposed post selection estimation methods are tractable and reduce both the bias and the psmse compared with the ml estimator while only requiring moderate computational complexity