on the variance of the internal path length of generalized digital trees the mellin convolution approach abstract this paper studies a generalization of the internal path length of a random digital search tree to bucket trees which are capable of storing up to b records per node we show that under the assumption of the symmetric bernoulli probabilistic model the expected path length of a tree built from n records is asymptotically n log 2 n a 1 log 2 n n and the variance is c e 1 log 2 n n where a and c depend on b only the continuous functions 1 and e 1 are periodic with mean 0 and period 1 the proofs are analytical and make use of generating functions harmonic sums and the mellin integral transform an important and very general tool for the analysis is mellin s convolution integral these results and techniques are motivated by a paper by flajolet and richmond and by kirschenhofer prodinger and szpankowski

effect of noise on the dynamics of a complex map at the period tripling accumulation point as shown recently o b isaeva et al phys rev e 64 055201 2001 the phenomena intrinsic to dynamics of complex analytic maps under appropriate conditions may occur in physical systems we study scaling regularities associated with the effect of additive noise upon the period tripling bifurcation cascade generalizing the renormalization group approach of crutchfield et al phys rev lett 46 933 1981 and shraiman et al phys rev lett 46 935 1981 originally developed for the period doubling transition to chaos in the presence of noise the universal constant determining the rescaling rule for the intensity of the noise in period tripling is found to be gamma 12 206 640 9 em leader numerical evidence of the expected scaling is demonstrated

further results on difference triangle sets further results on the upper bounds for difference triangle sets dts are derived from disjoint difference sets and additive sequences of permutations which greatly improve the known bounds

profiles of tries tries from retrieval are one of the most popular data structures on words they are pertinent to the internal structure of stored words and several splitting procedures used in diverse contexts the profile of a trie is a parameter that represents the number of nodes either internal or external with the same distance from the root it is a function of the number of strings stored in a trie and the distance from the root several if not all trie parameters such as height size depth shortest path and fill up level can be uniformly analyzed through the external and internal profiles although profiles represent one of the most fundamental parameters of tries they have hardly been studied in the past the analysis of profiles is surprisingly arduous but once it is carried out it reveals unusually intriguing and interesting behavior we present a detailed study of the distribution of the profiles in a trie built over random strings generated by a memoryless source we first derive recurrences satisfied by the expected profiles and solve them asymptotically for all possible ranges of the distance from the root it appears that profiles of tries exhibit several fascinating phenomena when moving from the root to the leaves of a trie the growth of the expected profiles varies near the root the external profiles tend to zero at an exponential rate and then the rate gradually rises to being logarithmic the external profiles then abruptly tend to infinity first logarithmically and then polynomially they then tend polynomially to zero again furthermore the expected profiles of asymmetric tries are oscillating in a range where profiles grow polynomially while symmetric tries are nonoscillating in contrast to most shape parameters of random tries studied previously such a periodic behavior for asymmetric tries implies that the depth satisfies a central limit theorem but not a local limit theorem of the usual form also the widest levels in symmetric tries contain a linear number of nodes differing from the order n sqrt log n for asymmetric tries n being the size of the trees finally it is observed that profiles satisfy central limit theorems when the variance goes unbounded while near the height they are distributed according to poisson laws as a consequence of these results we find typical behaviors of the height shortest path fill up level and depth these results are derived here by methods of analytic algorithmics such as generating functions mellin transform poissonization and de poissonization the saddle point method singularity analysis and uniform asymptotic analysis

digital search trees revisited several algorithms have been proposed which build search trees using digital properties of the search keys a general approach to the study of the average case performance of such algorithms is discussed with particular attention to the analysis of the digital search tree structures of coflman and eve specifically the method leads to the solution of a problem left open by knuth finding the average number of nodes in digital search trees with both sons null the paper may be of interest as a survey and tutorial treatment of the analysis of the three primary digital tree search methods digital search trees radix search tries and patricia tries

a variety of period doubling universality classes in multi parameter analysis of transition to chaos abstract in multi parameter analysis of the onset of chaos non feigenbaum period doubling behavior may occur at some special paths in the parameter space there are two possibilities i the dynamics at the onset of chaos remains essentially one dimensional but the one dimensional map is distorted in such a way that leaves feigenbaum s universality class ii a new mode comes to the threshold of instability and increases the effective dimension of the dynamics we submit a list of one dimensional and two dimensional maps which represent distinct classes of the period doubling universality discuss the properties of the associated types of critical behavior at the border of chaos and demonstrate pictures of the universal parameter space arrangement near the critical points

properties of synchronization in the systems of non identical coupled van der pol and van der pol duffing oscillators broadband synchronization the particular properties of dynamics are discussed for dissipatively coupled van der pol oscillators non identical in values of parameters controlling the hopf bifurcation possibility of a special synchronization regime in an infinitively long band between oscillator death and quasiperiodic areas is shown for such system features of the bifurcation picture are discussed for different values of the control parameters and for the case of additional duffing type nonlinearity analysis of the slow flow equations is presented

multiparameter critical situations universality and scaling in two dimensional period doubling maps we review critical situations linked with period doubling transition to chaos which require using at least two dimensional maps as models representing the universality classes each of them corresponds to a saddle solution of the two dimensional generalization of feigenbaum cvitanovic equation and is characterized by a set of distinct universal constants analogous to feigenbaum s and one type of criticality designated h was discovered by several authors in 80 th in the context of period doubling in conservative dynamics but occurs as well in dissipative dynamics as a phenomenon of codimension 2 second is bicritical behavior which takes place in systems allowing decomposition onto two dissipative period doubling subsystems each of which is brought by parameter tuning onto a threshold of chaos types of criticality designated as fq and c occur in non invertible two dimensional maps we present and discuss a number of realistic systems manifesting those types of critical behavior and point out some relevant conditions of their potential observation in physical systems in particular we indicate a possibility for realization of the h type criticality without vanishing dissipation but with its compensation in a self oscillatory system next we present a number of examples coupled henon like maps coupled driven oscillators coupled chaotic self oscillators which manifest bicritical behavior for fq type we indicate possibility to arrange it in non symmetric systems of coupled period doubling subsystems e g in henon like maps and in chua s circuits for c type we present examples of its appearance in a driven rossler oscillator at the period doubling accumulation on the edge of syncronization tongue and in a model map with the neimark sacker bifurcation

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

asymptotic behavior of the lempel ziv parsing scheme and digital search trees the lempel ziv parsing scheme finds a wide range of applications most notably in data compression and algorithms on words it partitions a sequence of length n into variable phrases such that a new phrase is the shortest substring not seen in the past as a phase the parameter of interest is the number mn of phrases that one can construct from a sequence of length n in this paper for the memoryless source with unequal probabilities of symbols generation we derive the limiting distribution of mn which turns out to be normal this proves a long standing open problem in fact to obtain this result we solved another open problem namely that of establishing the limiting distribution of the internal path length in a digital search tree the latter is a consequence of an asymptotic solution of a multiplicative differential functional equation often arising in the analysis of algorithms on words interestingly enough our findings are proved by a combination of probabilistic techniques such as renewal equation and uniform integrability and analytical techniques such as mellin transform differential functional equations de poissonization and so forth in concluding remarks we indicate a possibility of extending our results to markovian models

bifurcation structure of the c type period doubling transition abstract the period doubling transition to chaos along the edge of an arnold tongue is known to display unusual organization and scaling behavior kuznetsov et al 2005 7 it is also known that forced period doubling systems may be associated with the appearance of so called period doubled tori arneodo et al 1983 15 using the rossler system as an example we present a detailed analysis of the bifurcation structure associated with the forcing of a three dimensional period doubling system we explain how this structure is related to the recently discovered phenomenon of multi layered tori and discuss different bifurcation scenarios that transform a resonance torus into a period doubled ergodic torus similar bifurcation phenomena have recently been observed in a biologically relevant model of kidney blood flow regulation in response to fluctuations in arterial pressure

on the balance property of patricia tries external path length viewpoint abstract in this paper we give exact and asymptotic approximations for the variance of the external path length in a symmetric patricia tree the problem was open up to now we prove that for the binary patricia tree the variance is asymptotically equal to 0 37 n np log 2 n where n is the number of stored records and p x is a periodic function with a very small amplitude this implies that the external path length is asymptotically equal to n log 2 n with probability one i e almost surely these results are next used to show that from the practical average viewpoint the patricia tree does not need to be restructured in order to keep it balanced in general we ask to what extent simpler and more direct algorithms for digital search trees can be expected in practice to match the performance of more complicated worst case asymptotically better ones

the lempel ziv algorithm and message complexity data compression has been suggested as a nonparametric way of discriminating between message sources compressions obtained from a lempel ziv algorithm for relatively short messages such as those encountered in practice are examined the intuitive notion of message complexity has less connection with compression than one might expect from known asymptotic results about infinite messages nevertheless discrimination by compression remains an interesting possibility

average profile of the generalized digital search tree and the generalized lempel ziv algorithm the goal of this research is threefold i to analyze generalized digital search trees ii to derive the average profile i e phrase length of a generalization of the well known parsing algorithm due to lempel and ziv and iii to provide analytic tools to analyze asymptotically certain partial differential functional equations often arising in the analysis of digital trees in the generalized lempel ziv parsing scheme one partitions a sequence of symbols from a finite alphabet into phrases such that the new phrase is the shortest substring seen in the past by at most b 1 phrases b 1 corresponds to the original lempel ziv scheme such a scheme can be analyzed through a generalized digital search tree in which every node is capable of storing up to b strings in this paper we investigate the depth of a randomly selected node in such a tree and the length of a randomly selected phrase in the generalized lempel ziv scheme these findings and some recent results allow us to compute the average redundancy of the generalized lempel ziv code and compare it to the ordinary lempel ziv code leading to an optimal value of b analytic techniques of precise analyses of algorithms are used to establish most of these conclusions

phase transition in a generalized eden growth model on a tree we study analytically the late time statistics of the number of particles in an eden growth model on a tree in this model a cluster grows in continuous time on a binary cayley tree starting from the root by absorbing new particles at the empty perimeter sites at a rate proportional to c l where c is a positive parameter and l is the distance of the perimeter site from the root for c 1 this model corresponds to random binary search trees and for c 2 it corresponds to digital search trees in computer science by introducing a backward fokker planck approach we calculate the mean and the variance of the number of particles at large times and show that the variance undergoes a phase transition at a critical value c sqrt 2 while for c sqrt 2 the variance is proportional to the mean and the distribution is normal for c sqrt 2 the variance is anomalously large and the distribution is non gaussian due to the appearance of extreme fluctuations the model is generalized to one where growth occurs on a tree with m branches and in this more general case we show that the critical point occurs at c sqrt m

laws of large numbers and tail inequalities for random tries and patricia trees we consider random tries and random patricia trees constructed from n independent strings of symbols drawn from any distribution on any discrete space if hn is the height of this tree we show that hn e hn tends to one in probability additional tail inequalities are given for the height depth size internal path length and profile of these trees and ordinary tries that apply without any conditions on the string distributions they need not even be identically distributed

asymptotic properties of data compression and suffix trees recently wyner and ziv see ibid vol 35 p 1250 8 1989 have proved that the typical length of a repeated subword found within the first n positions of a stationary ergodic sequence is 1 h log n in probability where h is the entropy of the alphabet this finding was used to obtain several insights into certain universal data compression schemes most notably the lempel ziv data compression algorithm wyner and ziv have also conjectured that their result can be extended to a stronger almost sure convergence in this paper we settle this conjecture in the negative in the so called right domain asymptotic that is during a dynamic phase of expanding the data base we prove under an additional assumption involving mixing conditions that the length of a typical repeated subword oscillates almost surely a s between 1 h sub 1 log n and 1 h sub 2 log n where d

difference triangle sets from affine planes this article obtains new difference triangle sets using affine planes difference triangle sets have a number of interesting applications in data communications in these applications it is desirable to have difference triangle sets with small scopes

average profile and limiting distribution for a phrase size in the lempel ziv parsing algorithm consider the parsing algorithm developed by lempel and ziv 1978 that partitions a sequence of length n into variable phrases blocks such that a new block is the shortest substring not seen in the past as a phrase in practice the following parameters are of interest number of phrases the size of a phrase the number of phrases of given size and so forth in this paper we focus on the size of a randomly selected phrase and the average number of phrases of a given size the so called average profile of phrase sizes these parameters can be efficiently analyzed through a digital search tree representation for a memoryless source with unequal probabilities of symbols generation the so called asymmetric bernoulli model we prove that the size of a typical phrase is asymptotically normally distributed with mean and variance explicitly computed in terms of digital search trees we prove the normal limiting distribution of the typical depth i e the length of a path from the root to a randomly selected node the latter finding is proved by a technique that belongs to the toolkit of the analytical analysis of algorithms and it seems to be novel in the context of data compression

universal properties of the transition from quasi periodicity to chaos in dissipative systems abstract an exact renormalization group transformation is developed for dissipative systems which describes how the transition to chaos may occur in a continuous and universal manner if the frequency ratio in the quasi periodic regime is held at a fixed irrational value our approach is a natural extension of k a m theory to strong coupling most of our analysis is for analytic circle maps we have found a strong coupling fixed point where invertibility is lost which describes the universal features of the transition to chaos we find numerically that any two such critical maps with the same winding number are c 1 conjugate it follows that the low frequency peaks in an experimental spectrum are universal and we determine how their envelope scales with frequency when the winding number has a periodic continued fraction our renormalization transform has a fixed point and spectra are self similar in addition for a set of non periodic winding numbers with full measure our renormalization transformation yields an ergodic trajectory in a sub space of all critical maps physically one finds singular and universal spectra that do not scale

a generalized suffix tree and its un expected asymptotic behaviors suffix trees find several applications in computer science and telecommunications most notably in algorithms on strings data compressions and codes despite this very little is known about their typical behaviors in a probabilistic framework a family of suffix trees further called b suffix trees built from the first n suffixes of a random word is considered in this family a noncompact suffix tree i e such that every edge is labeled by a single symbol is represented by b 1 and a compact suffix tree i e without unary nodes is asymptotically equivalent to b to infty as n to infty several parameters of b suffix trees are studied namely the depth of a given suffix the depth of insertion the height and the shortest feasible path some new results concerning typical i e almost sure behaviors of these parameters are established these findings are used to obtain several insights into certain algorithms on words molecular biology and universal data compression schemes

on classification with empirically observed statistics and universal data compression classification with empirically observed statistics is studied for finite alphabet sources efficient universal discriminant functions are described and shown to be related to universal data compression it is demonstrated that if one of the probability measure of the two classes is not known it is still possible to define a universal discrimination function which performs as the optimal likelihood ratio discriminant function which can be evaluated only if the probability measures of the two classes are available if both of the probability measures are not available but training vectors from at least one of the two classes are available it is demonstrated that no discriminant function can perform efficiency of the length of the training vectors does not grow at least linearly with the length of the classified vector a universal discriminant function is introduced and shown to perform efficiently when the length of the training vectors grows linearly with the length of the classified sequence in the sense that it yields an error exponent that is arbitrarily close to that of the optimal discriminant function

coupled van der pol duffing oscillators phase dynamics and structure of synchronization tongues abstract synchronization in the system of coupled non identical non isochronous van der pol duffing oscillators with inertial and dissipative coupling is discussed generalized adler s equation is obtained and investigated in the presence of all relevant factors affecting the synchronization non isochronism of the oscillators their non identity coupling of the dissipative and inertial types characteristic symmetries are revealed for the adler s equation responsible for equivalence of some of the factors numerical study of the parameters space of the initial differential equations is carried out using the method of charts of dynamic regimes in the parameter planes results obtained by both these approaches are compared and discussed

bounds and construction for difference triangle sets the definition of difference triangle sets dtss and is short survey of some of their important applications is given a lower bound on the maximal element in any i j dts is then given and a construction of dtss is described finally tables of the best known lower and upper bounds on optimal dtss are presented

asymptotic behavior of the height in a digital search tree and the longest phrase of the lempel ziv scheme we study the height of a digital search tree dst built from n random strings generated by an unbiased memoryless source i e all symbols are equally likely we shall argue that the height of such a tree is equivalent to the length of the longest phrase in the lempel ziv parsing scheme that partitions a random sequence into n phrases we also analyze the longest phrase in the lempel ziv scheme in which a string of fixed length m is parsed into a random number of phrases in the course of our analysis we shall identify four natural regions of the height distribution and characterize them asymptotically for large n in particular for the region where most of the probability mass is concentrated the asymptotic distribution of the height exhibits an exponential of a gaussian distribution with an oscillating term around the most probable value k 1 lfloor log 2 n sqrt 2 log 2 n log 2 sqrt 2 log 2 n frac 1 log 2 frac 1 2 rfloor 1 more precisely we shall prove that the asymptotic distribution of a dst is concentrated on either the one point k1 or the two points k1 1 and k1 which actually proves slightly modified kesten s conjecture quoted in probab theory related fields 79 1988 pp 509 542 finally we compare our findings for dst with the asymptotic distributions of the height for other digital trees such as tries and patricia tries we derive these results by a combination of analytic methods such as generating functions laplace transform the saddle point method and ideas of applied mathematics such as linearization asymptotic matching and the wkb method our analysis makes certain assumptions about the forms of some of the asymptotic expansions as well as their asymptotic matching we also present detailed numerical verification of our results

a characterization of digital search trees from the successful search viewpoint this paper studies the average complexity of digital search trees from the successful search point of view the average value of the successful search is used to evaluate the search time for a given record the number of comparisons to insert a record etc the average value however is rather a poor measure and the need for higher moments of the successful search is obvious for example the variance provides information on how well a digital tree is balanced the third centralized moment is a measure of the skewness property of the distribution etc in this paper we concentrate on an open problem how to evaluate all moments of the successful search in an asymmetric multiway digital search tree we prove that the mth moment esmn of the successful search sn where n is the number of stored records satisfies esmn lnmn 1 hm1 as n where h1 is the entropy of the alphabet in particular it is shown that the variance of sn is var sn c ln n o 1 for the asymmetric case and var sn o 1 for the symmetric case this paper completes a study undertaken by the author to characterize some digital trees such as regular tries patricia tries and digital search trees from the average viewpoint

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

on the variance of a class of inductive valuations of data structures for digital search let an inductive valuation l on the family of binary tries or patricia tries or digital search trees be defined in the following way l t l tl l tr r t where tl and tr denote the left and right subtrees of t and r depends only on the size the number of records t of t let ln denote l restricted to the trees of size n in theorem 1 we give sufficient conditions on the sequence r t r t for the variance var ln to be of exact order n if the family of tries resp patricia tries resp digital search trees is equipped with the bernoulli model for the symmetric bernoulli model we prove the existence of a continuous periodic function with period 1 such that var ln log2 n n holds

the complete set of hamiltonian intermittency scaling behaviors renormalization group equations describing the phenomenon of intermittency in hamiltonian systems are presented all solutions satisfying certain physical constraints are obtained they are the complete set of simple singularities further considerations lead to precise predictions for scaling behavior at the onset of intermittency

on the average redundancy rate of the lempel ziv code with the k error protocol abstract in this paper we examine the average redundancy rate of a lempel ziv 78 code with the k error protocol storer and reif have studied this modification of the lempel ziv lz scheme and showed that it provides an efficient protection against error propagation while preserving the asymptotic optimality of the code we refine this result by providing an asymptotic expression for the average redundancy rate of this code for memoryless sources using analytic methods we establish our result by exploiting a relationship between a parsing scheme of the lz encoder with the k error protocol and a generalization of the digital search tree dst structure we accompany our analysis with a number of experiments that test the validity of our theoretical result and demonstrate the effects of various additional modifications of the lz algorithm

effect of noise on the critical golden mean quasiperiodic dynamics in the circle map abstract scaling regularities are examined associated with effect of additive noise on a critical circle map at the golden mean rotation number we present an improved numerical estimate for the scaling constant of hamm and graham phys rev a46 1992 6323 responsible for the effect of noise 2 3061852653 decrease of the noise amplitude by this number ensures possibility to distinguish one more level of fractal like structure associated with increase of characteristic time scale by the golden mean factor 5 1 2 numeric results demonstrating evidence of the expected scaling are presented e g portraits of the noisy attractors devil s staircase plots lyapunov charts on the parameter plane in different scales

birth of a new class of period doubling scaling behavior as a result of bifurcation in the renormalization equation it is found that a fixed point of the renormalization group equation corresponding to a system of a unimodal map with extremum of power and a map summarizing values of a function of the dynamical variable of the first subsystem undergoes a bifurcation in the course of increase of it occurs at c 1 984396 and results in a birth of the period 2 stationary solution of the rg equation at 2 this period 2 solution corresponds to the universal period doubling behavior discovered earlier and denoted as the c type criticality kuznetsov and sataev in phys lett a 162 236 242 1992 by combination of analytical methods and numerical computations we obtain and analyze an asymptotic expansion of the period 2 solution in powers of c the developed approach resembles the e expansion in the phase transition theory in which a trivial stationary point of the rg transformation undergoes a bifurcation that gives rise to a new fixed point responsible for the critical behavior with nontrivial critical indices

a note on the height of suffix trees consider a random word in which the individual symbols are drawn from a finite or infinite alphabet with symbol probabilities p i and let h n be the height of the suffix tree constructed from the first n suffixes of this word it is shown that h n is asymptotically close to 2 log n log 1 sum i p i 2 in many respects the difference is o log log n in probability and the ratio tends to one almost surely and in the mean

patricia tries again revisited the patricia trie is a simple modification of a regular trie by eliminating unary branching nodes the patricia achieves better performance than regular tries however the question is how much on the average is the patricia better this paper offers a thorough answer to this question by considering some statistics of the number of nodes examined in a successful search and an unsuccessful search in the patricia tries it is shown that for the patricia containing n records the average of the successful search length s n asymptotically becomes 1 h 1 ln n o 1 and the variance of s n is either var s n c ln n 0 1 for an asymmetric patricia or var s n 0 1 for a symmetric patricia where h 1 is the entropy of the alphabet over which the patricia is built and c is an explicit constant higher moments of s n are also assessed the number of nodes examined in an unsuccessful search u n is studied only for binary symmetric patricia tries we prove that the m th moment of the unsuccessful search length eu m n satisfies lim n eu m n log m 2 n 1 and the variance of u n is var u n 0 87907 these results suggest that patricia tries are very well balanced trees in the sense that a random shape of patriciatries resembles the shape of complete trees that are ultimately balanced trees

using directed hill climbing for the construction of difference triangle sets this correspondence describes the use of the directed hill climbing algorithm for finding improved upper bounds of difference triangle sets

further results on digital search trees abstract in this paper distribution results are proved on the cost of insertion in digital search trees binary tries and patricia tries a method from the calculus of finite differences is used to achieve asymptotic results

disjoint difference sets difference triangle sets and related codes disjoint difference sets dds difference triangle sets dts and related codes are discussed and a recursive construction for dds is given with this construction and the relationship between dds and dts many new upper bounds for dts and some better orthogonal codes are obtained

trajectory scaling function for bifurcations in area preserving maps on the plane abstract the trajectory scaling function for area preserving maps on the plane is found using a calculation of the unstable manifold for the renormalization group operator r t t2 1 with 0 0 internal self similarities of high order cycles and of power spectra are deduced

analysis of digital tries with markovian dependency a complete characterization of a digital tree also called a trie is presented from the depth viewpoint in a markovian framework that is under the assumption that symbols in a key are markov dependent the main findings show that asymptotically as the number of keys n tends to infinity the average depth becomes ed sub n approximately 1 h sub 1 log n c and the variance is var d sub n approximately alpha log n c where h sub 1 is the entropy of the markovian dependent alphabet alpha is a parameter of the probabilistic model and c and c are constants the symmetric independent model has alpha 0 hence in this case var d sub n o 1 limiting distribution is also derived for the depth d sub n and in particular it is shown that d sub n tends to the normal distribution in all cases except the symmetric independent model these results extend all previous analyses since most of them have been limited to independent models

two parameter analysis of the scaling behavior at the onset of chaos tricritical and pseudo tricritical points abstract we discuss the so called tricritical points at the border of the period doubling transition to chaos and examine to what extent the associated universality applies to 2d dissipative maps as a concrete example the ikeda map is studied together with its 1d analog for the approximate 1d map the tricritical points appear as the terminal points of feigenbaum s critical curves in the parameter plane for the 2d map the same type of critical behavior does not occur in a rigorous sense it may be observed as a kind of intermediate asymptotics however when one considers a finite number of period doublings we refer to the associated points in the parameter plane as pseudo tricritical for the ikeda map we present estimates of the number of period doublings after which the departure from the tricritical universality becomes essential

some results on tries with adaptive branching we study a modification of digital trees or tries with adaptive multi digit branching such tries can dynamically adjust degrees of their nodes by choosing the number of digits to be processed per lookup while we do not specify any particular method for selecting the degrees of nodes we assume that such selection can be accomplished by examining the number of strings remaining in each sub tree and or estimating parameters of the input distribution we call this class of digital trees adaptive multi digit tries or amd tries and provide a preliminary analysis of their expected behavior in a memoryless model we establish the following results 1 there exist amd tries attaining a constant expected time of a successful search 2 there exist amd tries consuming a linear in the number of strings inserted amount of space 3 both constant search time and linear space usage can be attained if the memoryless source is symmetric we accompany our analysis with a brief survey of several known types of adaptive trie structures and show how our analysis extends and or complements previous results

estimating the information content of symbol sequences and efficient codes several variants of an algorithm for estimating shannon entropies of symbol sequences are presented they are all related to the lempel ziv algorithm 1976 1977 and to recent algorithms for estimating hausdorff dimensions the average storage and running times increase as n and nlog n respectively with the sequence length n these algorithms proceed basically by constructing efficient codes they seem to be the optimal algorithms for sequences with strong long range correlations e g natural languages an application to written english illustrates their use

a universal data compression system a universal data compression algorithm is described which is capable of compressing long strings generated by a finitely generated source with a near optimum per symbol length without prior knowledge of the source this class of sources may be viewed as a generalization of markov sources to random fields moreover the algorithm does not require a working storage much larger than that needed to describe the source generating parameters

on universality for area preserving maps of the plane abstract we present detailed evidence that one parameter families of area preserving maps exhibit cascades of period doubling with universal geometric scaling in the parameter we relate this behaviour to a fixed point equation of the form 1 and det d 1 r 2 r 2 in particular we argue that the scaling transformation r 2 r 2 is conjugate to the transformation 0 x y x y with 2 and in fact 2 we present some numerical evidence that 8 721 1 4 018 1 16 36 where is the asymptotic ratio of the differences of the parameter values corresponding to the successive periods 2k described above

renormalization group analysis of bifurcations in area preserving maps abstract two dimensional area preserving maps can be represented by a generating function the action high orders of successive period doubling bifurcations are studied by writing a renormalization group scheme for this action fixed points and eigenvalues of this scheme are found and interpreted

mellin transforms and asymptotics finite differences and rice s integrals high order differences of simple number sequences may be analysed asymptotically by means of integral representations residue calculus and contour integration this technique akin to mellin transform asymptotics is put in perspective and illustrated by means of several examples related to combinatorics and the analysis of algorithms like digital tries digital search trees quadtrees and distributed leader election

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