how long are the arms in dla the author shows that the maximal length of the arms of dla can grow at most at a rate of n2 3 in dimension 2 and at a rate of n2 d in dimension d 2 here n denotes the volume or mass of the aggregate

limit laws for sums of functions of subtrees of random binary search trees we consider sums of functions of subtrees of a random binary search tree and obtain general laws of large numbers and central limit theorems these sums correspond to random recurrences of the quicksort type x n stackrel cal l x i n x n 1 i n y n n ge 1 where in is uniformly distributed on 0 1 n 1 yn is a given random variable x k stackrel cal l x k for all k and given in xin and x n 1 in are independent conditions are derived such that x n mu n sigma sqrt n stackrel cal l rightarrow cal n 0 1 the normal distribution for some finite constants mu and sigma

permanence and global attractivity of a discrete multispecies lotka volterra competition predator prey systems in this paper we propose a discrete multispecies lotka volterra competition predator prey systems for general non autonomous 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 a globally stable positive periodic solution of the system are obtained

multi particle diffusion limited aggregation we consider a stochastic aggregation model on mathbb z d start with particles distributed according to the product bernoulli measure with parameter mu in addition start with an aggregate at the origin non aggregated particles move as continuous time simple random walks obeying the exclusion rule whereas aggregated particles do not move the aggregate grows by attaching particles to its surface whenever a particle attempts to jump onto it this evolution is called multi particle diffusion limited aggregation our main result states that if on d 1 the initial density of particles is large enough then with positive probability the aggregate has linearly growing arms that is there exists a constant c 0 so that at time t the aggregate contains a point of distance at least ct from the origin for all t the key conceptual element of our analysis is the introduction and study of a new growth process consider a first passage percolation process called type 1 starting from the origin whenever type 1 is about to occupy a new vertex with positive probability instead of doing it it gives rise to another first passage percolation process called type 2 which starts to spread from that vertex each vertex gets occupied only by the process that arrives to it first this process may have three phases extinction type 1 gets eventually surrounded by type 2 coexistence infinite clusters of both types emerge and strong survival type 1 produces an infinite cluster which entraps all type 2 clusters understanding the various phases of this process is of mathematical interest on its own right we establish the existence of a strong survival phase and use this to show our main result

asymptotically fast factorization of integers the paper describes a probabilistic algorithm for finding a factor of any large composite integer n the required input is the integer n together with an auxiliary sequence of random numbers it is proved that the expected number of operations which will be required is o exp 83qn n in in n l 2 for some constant f 0 asymptotically this algorithm is much faster than any previously analyzed algorithm for factoring integers earlier algorithms have all required o na operations where a 1 5

some remarks on computing the square parts of integers abstract let n be a positive integer and suppose n p i a i is its prime factorization let n p i a i 1 so that n n is the largest squarefree factor of n we show that is deterministic polynomial time turing reducible to the euler function we also show that is reducible to the carmichael function we survey other recent work on computing the square part of an integer and give upper and lower bounds on the complexity of solving the problem

asymptotical growth of a class of random trees we study three rules for the development of a sequence of finite subtrees t n of an infinite m ary tree t independent realizations omega n of a stationary ergodic process omega on m letters are used to trace out paths in t in the first rule t n is formed by adding a node to t n 1 at the first location where the path defined by omega n leaves t n 1 the second and third rules are similar but more complicated for each rule the height l n of the added node is shown to grow in probability as ln n divided by h the entropy per symbol of the generic process a typical retrieval time has the same behavior on the other hand lim inf nl n ln n sigma 1 lim sup n l n ln n sigma 2 a s where the constants sigma 1 sigma 2 are in general different depend on the rule in use and sigma 1 1 h sigma 2 it is proven along the way that the height of t n grows as sigma 2 ln n with probability one

probabilistic analysis of bucket recursive trees we introduce the bucket recursive tree a generalization of recursive trees the tree grows from a succession of integer labels that join the multitype nodes of the tree according to a stochastic rule we study the multivariate structure of the tree and obtain a multivariate central limit theorem for the joint distribution of the number of nodes of different types for trees with bucket size b 26 for trees with b 26 a phase change in the distribution is detected and the central limit theorem does not hold recent results on the extended polya urn models smythe 1995 provide the basic tool to reach these results two kinds of distances are also studied the tree height and the depth of the nth label strong laws for the height are obtained via a technique based on introducing ghost nodes then removing them the ghost nodes create a companion tree that relates our tree to biggins 1976 first and last birth problem in a multitype process a weak law for the depth is obtained by formulating and asymptotically manipulating the depth probability generating function

reductions among number theoretic problems abstract many number theoretic problems such as integer factorization and the discrete logarithm problem have defied all attempts to classify their complexities thirteen such problems are considered none of which is known either to have a deterministic polynomial time solution or to be complete for any natural complexity class failing this the next best goal is to determine which among these are the easiest and which are the hardest problems toward this end this paper gives an overview of reductions among the problems two reductions are new a deterministic polynomial time reduction from squarefreeness to euler s function n and a probabilistic polynomial time reduction from order modulo a prime power to discrete logarithm modulo a prime power

a note on the height of binary search trees let h n be the height of a binary search tree with n nodes constructed by standard insertions from a random permutation of 1 n it is shown that h n log n c 4 31107 in probability as n where c is the unique solution of c log 2 e c 1 c 2 also for all p 0 lim n e h p n log p n c p finally it is proved that s n log n c 0 3733 in probability where c is defined by c log 2 e c 1 c 1 and s n is the saturation level of the same tree that is the number of full levels in the tree

one dimensional long range diffusion limited aggregation i we examine diffusion limited aggregation generated by a random walk on zz with long jumps we derive upper and lower bounds on the growth rate of the aggregate as a function of the number of moments a single step of the walk has under various regularity conditions on the tail of the step distribution we prove that the diameter grows as n o 1 n o 1 with an explicitly given the growth rate of the aggregate is shown to have three phase transitions when the walk steps have finite third moment finite variance and conjecturally finite half moment

extinction in two species nonautonomous nonlinear competitive system a two species nonautonomous nonlinear competitive system is studied in this paper it is shown that if the coefficients are continuous bounded above and below by positive constants and satisfy certain inequalities then one of the components will be driven to extinction while the other one will stabilize at the certain positive solution of a nonlinear single species model our result not only generalizing but also improving the result of li and chen 2006

second phase changes in random m ary search trees and generalized quicksort convergence rates we study the convergence rate to normal limit law for the space requirement of random m ary search trees while it is known that the random variable is asymptotically normally distributed for 3 m 26 and that the limit law does not exist for m 26 we show that the convergence rate is o n 1 2 for 3 m 19 and is o n 3 3 2 where 4 3 3 2 is a parameter depending on m for 20 m 26 our approach is based on a renemen t to the method of moments and applicable to other recursive random variables we briey mention the applications to quicksort proper and the generalized quicksort of hennequin where more phase changes are given these results provide natural concrete examples for which the berry esseen bounds are not necessarily proportional to the reciprocal of the standard deviation local limit theorems are also derived

permanence of a delayed non autonomous gilpin ayala competition model abstract in this paper we consider a delayed non autonomous n species gilpin ayala competitive system which is more general and more realistic then classical lotka volterra competition model by means of ahmad and lazer s definitions of lower and upper averages of a function we give the average conditions for the permanence of the system it is shown that our result is the generalization of those of zhao et al j d zhao j f jiang a c lazer the permanence and global attractivity in a nonautonomous lotka volterra system nonlinear analysis real world applications 5 4 2004 265 276 our results also supplement the results of fan and wang m fan k wang global periodic solutions of a generalized n species gilpin ayala competition model computer and mathematics with applications 40 2000 1141 1151

the functional law of the iterated logarithm for stationary strongly mixing sequences let x i i z be a strictly stationary and strongly mixing sequence of real valued mean zero random variables let a n n 0 be the sequence of strong mixing coefficients we define the strong mixing function by t t and we denote by q the quantile function of x 0 assume that 0 1 1 t q 2 t dt where f 1 denotes the inverse of the monotonic function f the main result of this paper is that the functional law of the iterated logarithm lil holds whenever x i i z satisfies moreover it follows from doukhan massart and rio that for any positive there exists a stationary sequence x i i z with strong mixing coefficients n of the order of n such that the bounded lil does not hold if condition is violated the proof of the functional lil is mainly based on new maximal exponential inequalities for strongly mixing processes which are of independent interest

sorting algorithms for broadcast communications mathematical analysis we study three algorithms for sorting under the broadcast communication model two algorithms are based on the maximum finding and the loser selection strategy the third one is the naive strategy based on loser selection only while a precise description of the algorithms can be found in the paper 15 we concentrate here on the mathematical aspects of the analysis from these elementary considerations it follows that the average time complexities of the non naive algorithms are n we give precise expressions for the constants involved by writing them as contour integrals involving zeta functions the numerical evaluation via residues leads to slowly converging series and the acceleration of them is a non trivial task that is done in a slightly more general fashion in order to fit all the applications

random walks and percolation on trees there is a way to define an average number of branches per vertex for an arbitrary infinite locally finite tree it equals the exponential of the hausdorff dimension of the boundary in an appropriate metric its importance for probabilistic processes on a tree is shown in several ways including random walk and percolation where it provides points of phase transition

behavior near the extinction time in self similar fragmentations ii finite dislocation measures we study a markovian model for the random fragmentation of an object at each time the state consists of a collection of blocks each block waits an exponential amount of time with parameter given by its size to some power alpha independently of the other blocks every block then splits randomly into sub blocks whose relative sizes are distributed according to the so called dislocation measure we focus here on the case where alpha 0 in this case small blocks split intensively and so the whole state is reduced to dust in a finite time almost surely we call this the extinction time in this paper we investigate how the fragmentation process behaves as it approaches its extinction time in particular we prove a scaling limit for the block sizes which as a direct consequence gives us an expression for an invariant measure for the fragmentation process in an earlier paper we considered the same problem for another family of fragmentation processes the so called stable fragmentations the results here are similar but we emphasize that the methods used to prove them are different our approach in the present paper is based on markov renewal theory and involves a somewhat unusual spine decomposition for the fragmentation which may be of independent interest

on the internal structure of random recursive circuits we study the joint probability distribution of the number of nodes of fan out k in random recursive circuits for suitable norming we obtain a limiting multivariate normal distribution for the numbers of node of fan out at most k where we compute explicitly the limiting covariance matrix by solving a recurrence satisfied among its entries

binary search tree recursions with harmonic toll functions we study recursions that are traditional in the context of binary search trees and quicksort with the non standard toll function hn

permanence of a nonlinear prey competition model with delays abstract a nonlinear periodic predator prey model with m preys and n m predators and delays is studied which can be seen as the modification of the traditional lotka volterra prey competition model two sets of sufficient conditions which guarantee the permanence of the system are obtained one set is delay independent while the other set is delay dependent our results supplement those obtained by chen et al f d chen x d xie j l shi existence uniqueness and stability of positive periodic solution for a nonlinear prey competition model with delays journal of computational and applied mathematics 194 2 2006 368 387

universal limit laws for depths in random trees random binary search trees b ary search trees median of 2k 1 trees quadtrees simplex trees tries and digital search trees are special cases of random split trees for these trees we offer a universal law of large numbers and a limit law for the depth of the last inserted point as well as a law of large numbers for the height

the qualitative analysis of n species nonlinear prey competition systems in this paper we consider an n species nonlinear prey competition system using the differential inequality theorem and the v function method we obtain some sufficient conditions for the permanence and global asymptotic stability of the system

upper bounds for the growth rate of dla abstract we show that the maximal length of the arms in dla can grow at most at a rate n 2 d 1 where n denotes the volume or mass of the aggregate and d is the dimension a logarithmic correction factor is needed if d 3 we also give a lower bound for the number of sites in a sphere which are eventually occupied by the aggregate

existence uniqueness and stability of positive periodic solution for a nonlinear prey competition model with delays a nonlinear periodic predator prey model with m preys and n m predators and delays is proposed in this paper which can be seen as the modification of the traditional lotka volterra prey competition model sufficient conditions which guarantee the existence of a unique globally attractive positive periodic solution of the system are obtained

a diffusion limit for a class of randomly growing binary trees binary trees are grown by adding one node at a time an available node at height i being added with probability proportional to c i c 1 we establish both a strong law of large numbers and a central limit theorem for the vector x t x i t where x i t is the proportion of nodes of height i that are available at time t we show in fact that there is a deterministic process x i t such that sum x i t x i t text converges to 0 a text s text and such that if c2 tfrac 1 2 z i n t 2 n 2 x n 1 tc n x n 1 tc n and z n t z i n t then z n t converges weakly to a gaussian diffusion z t the results are applied to establish asymptotic normality in the unbiased coin tossing case for an entropy estimation procedure due to j ziv and to obtain results on the growth of the maximum height of the tree

diffusion limited aggregation on the hyperbolic plane we consider an analogous version of the diffusion limited aggregation model defined on the hyperbolic plane we prove that almost surely the aggregate viewed at time infinity will have a positive density

average conditions for kolmogorov systems in this paper we consider nonautonomous competitive kolmogorov systems which is a generalization of classical lotka volterra competition model applying ahmad and lazer s definitions of lower and upper averages of a function we give the average conditions for the persistence permanence and global attractivity of the system

on a nonlinear nonautonomous predator prey model with diffusion and distributed delay in this paper a nonlinear nonautonomous predator prey model with diffusion and continuous distributed delay is studied where all the parameters are time dependent the system which is composed of two patches has two species the prey can diffuse between two patches but the predator is confined to one patch we first discuss the uniform persistence and global asymptotic stability of the model after that by constructing a suitable lyapunov functional some sufficient conditions for the existence of a unique almost periodic solution of the system are obtained an example shows the feasibility of our main results

on a delayed nonautonomous ratio dependent predator prey model with holling type functional response and diffusion abstract a two species periodic ratio dependent predator prey model with holling type functional response and diffusion is investigated three different sets of sufficient conditions one set is diffusive independent and the other two sets are diffusive dependent which ensure the permanence of the system and a set of sufficient condition which ensure the extinction of the predator species are obtained examples together with their numeric simulations show the feasibility of the main results

global attractivity in an almost periodic multi species nonlinear ecological model abstract a nonlinear almost periodic predator prey model with n preys and m predators is studied in this paper which can be seen as the modification of the traditional multi species lotka volterra predator prey model for general nonautonomous case by using the differential inequality theory we obtain the sufficient conditions which guarantee the uniform persistence and nonpersistence of the system after that by constructing a suitable lyapunov function some sufficient conditions are obtained which ensure the global attractivity of the system for almost periodic case by constructing a suitable lyapunov function sufficient conditions which guarantee the existence of an unique globally attractive positive almost periodic solution of the system are obtained examples together with their numeric simulations show the feasibility of our main results

the joint distribution of elastic buckets in multiway search trees random search trees are studied when they grow under a general computer memory management scheme in a general scheme the space is released in buckets of certain predesignated sizes for a search tree with branch factor m the nodes may hold up to m 1 keys suppose the buckets of the memory management scheme that can hold less than m keys have key capacities c 1 ldots c p the search tree must then be implemented with multitype nodes of these capacities after n insertions let x n i be the number of buckets of type i i e of capacity c i 1 leq i leq p the multivariate structure of the tree is investigated for the vector bf x n x n 1 ldots x n p t the asymptotic mean and covariance matrix are determined under practical memory management schemes all variances and covariances experience a phase transition for 3 leq m leq 26 all variances and covariances are asymptotically linear in n for higher branch factors the variances and covariances become a superlinear but subquadratic function of n the joint distribution of bf x n is shown to be multivariate normal in a range of m while the tree is growing conversions between types are necessary a multivariate problem concerning these conversions with an asymptotic multivariate normal distribution is also studied the fixed bucket exact fit and buddy system allocation schemes will serve as illustrating examples

permanence and extinction for a nonlinear diffusive predator prey system abstract in this paper we consider a non autonomous predator prey system in a poor patchy environment under the assumption that the intrinsic growth rates of the species may be negative sufficient conditions are obtained for the permanence and extinction of the system considered

diffusion limited aggregation on a tree we study the following growth model on a regular d ary tree points at distance n adjacent to the existing subtree are added with probabilities proportional to n where 1 is a positive real parameter the heights of these clusters are shown to increase linearly with their total size this complements known results that show the height increases only logarithmically when 1 results are obtained using stochastic monotonicity and regeneration results which may be of independent interest our motivation comes from two other ways in which the model may be viewed as a problem in first passage percolation and as a version of diffusion limited aggregation dla adjusted so that fingering occurs

on growing random binary trees abstract a sequence tn n 1 of nested binary trees generated by an infinite sequence of i i d random variables is studied two absolute constants 1 2 are shown to exist 0 37 lim h n ln n 1 lim h n ln n 2 with probability one here hn and hn are respectively the lengths of the shortest and the longest branches of the tree tn

fluctuations of the front in a one dimensional model of x y 2x we consider a model of the reaction x y to 2x on the integer lattice in which y particles do not move while x particles move as independent continuous time simple symmetric random walks y particles are transformed instantaneously to x particles upon contact we start with a fixed number a ge 1 of y particles at each site to the right of the origin and define a class of configurations of the x particles to the left of the origin having a finite l 1 norm with a specified exponential weight starting from any configuration of x particles to the left of the origin within such a class we prove a central limit theorem for the position of the rightmost visited site of the x particles

domination between trees and application to an explosion problem we define a notion of stochastic domination between trees where one tree dominates another if when the vertices of each are labeled with independent identically distributed random variables one tree is always more likely to contain a path with a specified property sufficient conditions for this kind of domination are 1 more symmetry and 2 earlier branching we apply these conditions to the problem of determining how fast a tree must grow before first passage percolation on the tree exhibits an explosion that is to say infinitely many vertices are reached in finite time for a tree in which each vertex at distance n 1 from the root has f n offspring f nondecreasing an explosion occurs with exponentially distributed passage times if and only if f n 1

phase change of limit laws in the quicksort recurrence under varying toll functions we characterize all limit laws of the quicksort type random variables defined recursively by cal l x n cal l x i n x n 1 i n t n when the toll function tn varies and satisfies general conditions where xn xn in tn are independent in is uniformly distributed over 0 n 1 and cal l x n cal l x n ast when the toll function tn cost needed to partition the original problem into smaller subproblems is small roughly limsup n rightarrow infty log e t n log n le 1 2 xn is asymptotically normally distributed nonnormal limit laws emerge when tn becomes larger we give many new examples ranging from the number of exchanges in quicksort to sorting on a broadcast communication model from an in situ permutation algorithm to tree traversal algorithms etc

periodicity in a nonlinear predator prey system on time scales with state dependent delays abstract with the help of a continuation theorem based on gaines and mawhin s coincidence degree easily verifiable criteria are established for the global existence of positive periodic solutions of a nonlinear state dependent delays predator prey system on time scales

the continuum random tree iii let mathscr r k k geq 1 be random trees with k leaves satisfying a consistency condition removing a random leaf from mathscr r k gives mathscr r k 1 then under an extra condition this family determines a random continuum tree mathscr l which it is convenient to represent as a random subset of l 1 this leads to an abstract notion of convergence in distribution as n rightarrow infty of rescaled random trees mathscr j n on n vertices to a limit continuum random tree mathscr l the notion is based upon the assumption that for fixed k the subtrees of mathscr j n determined by k randomly chosen vertices converge to mathscr r k as our main example under mild conditions on the offspring distribution the family tree of a galton watson branching process conditioned on total population size equal to n can be rescaled to converge to a limit continuum random tree which can be constructed from brownian excursion