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

a bootstrap method for structure detection of narmax models many systems may be described by narmax models using only a few terms however depending on the order of the system the number of candidate terms can become very large selection of a subset of these candidate terms is necessary for an efficient system description this is an unresolved issue in system identification for over parameterized models therefore in this paper we develop a bootstrap structure detection bsd algorithm as a means of determining the structure of highly over parameterized models the performance of this bsd technique was evaluated by using it to estimate the structure of a 1 simple narmax model 2 moderately over parameterized narmax model and 3 highly over parameterized narmax model the results demonstrate that the bsd algorithm is a robust method for detecting the structure of narmax models this method provides accurate estimates of parameter statistics without relying on assumptions made by traditional procedures and yields a parsimonious description of the system

analytic smoothing for equipercentile equating under the common item nonequivalent populations design a cubic spline method for smoothing equipercentile equating relationships under the common item nonequivalent populations design is described statistical techniques based on bootstrap estimation are presented that are designed to aid in choosing an equating method degree of smoothing these include a asymptotic significance tests that compare no equating and linear equating to equipercentile equating b a scheme for estimating total equating error and for dividing total estimated error into systematic and random components the smoothing technique and statistical procedures are explored and illustrated using data from forms of a professional certification test

evidence of codon usage in the nearest neighbor spacing distribution of bases in bacterial genomes a b s t r a c t statistical analysis of whole genomic sequences usually assumes a homogeneous nucleotide density throughout the genome an assumption that has been proved incorrect for several organisms since the nucleotide density is only locally homogeneous to avoid giving a single numerical value to this variable property we propose the use of spectral statistics which characterizes the density of nucleotides as a function of its position in the genome we show that the cumulative density of bases in bacterial genomes can be separated into an average or secular plus a fluctuating part bacterial genomes can be divided into two groups according to the qualitative description of their secular part linear and piecewise linear these two groups of genomes show different properties when their nucleotide spacing distribution is studied in order to analyze genomes having a variable nucleotide density statistically the use of unfolding is necessary i e to get a separation between the secular part and the fluctuations the unfolding allows an adequate comparison with the statistical properties of other genomes with this methodology four genomes were analyzed burkholderia bacillus clostridium and corynebacterium interestingly the nearest neighbor spacing distributions or detrended distance distributions are very similar for species within the same genus but they are very different for species from different genera this difference can be attributed to the difference in the codon usage

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

the size of random fragmentation trees we consider the random fragmentation process introduced by kolmogorov where a particle having some mass is broken into pieces and the mass is distributed among the pieces at random in such a way that the proportions of the mass shared among different daughters are specified by some given probability distribution the dislocation law this is repeated recursively for all pieces more precisely we consider a version where the fragmentation stops when the mass of a fragment is below some given threshold and we study the associated random tree dean and majumdar found a phase transition for this process the number of fragmentations is asymptotically normal for some dislocation laws but not for others depending on the position of roots of a certain characteristic equation this parallels the behavior of discrete analogues with various random trees that have been studied in computer science we give rigorous proofs of this phase transition and add further details the proof uses the contraction method we extend some previous results for recursive sequences of random variables to families of random variables with a continuous parameter we believe that this extension has independent interest

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

on the tensor product of weakly compact operators the identity operator on an infinite dimensional hilbert space shows that neither the projective nor the injective tensor product of two weakly compact operators will be weakly compact in general see holub h1 remark p 10 there are however conditions ensuring this along the studies of diestel and faires df and saksman and tylli st we will add some more proposition 1 due to saksman and tylli deals with the tensor product of a compact and a weakly compact operator answering a question of theirs we consider in proposition 2 the tensor product of two weakly compact operators on spaces with the dunford pettis property having treated the case of absolutely p summing operators in proposition 3 we show in proposition 4 that the injective tensor product of two weakly compact operators on pisier spaces is weakly compact our results concern only the projective and the injective tensor product for the elementary properties of weakly compact operators to be used here we refer to ds vi 4 p 482 5

essential spectra of elementary operators this paper describes the essential spectrum and index function of the operator x axb where a b and x are hilbert space operators analogous results are given for the restriction of this operator to a norm ideal and partial analogues are given for sums of such operators and for the case when the operators act on a banach space

dynamical model for dna sequences we address the problem of dna sequences developing a dynamical method based on the assumption that the statistical properties of dna paths are determined by the joint action of two processes one deterministic with long range correlations and the other random and ensuremath delta function correlated the generator of the deterministic evolution is a nonlinear map belonging to a class of maps recently tailored to mimic the processes of weak chaos that are responsible for the birth of anomalous diffusion it is assumed that the deterministic process corresponds to unknown biological rules that determine the dna path whereas the noise mimics the influence of an infinite dimensional environment on the biological process under study we prove that the resulting diffusion process if the effect of the random process is neglected is an ensuremath alpha stable l evy process with 12 we also show that if the diffusion process is determined by the joint action of the deterministic and the random process the correlation effects of the deterministic dynamics are cancelled on the short range scale but show up in the long range one we denote our prescription to generate statistical sequences as the copying mistake map cmm we carry out our analysis of several dna sequences and their cmm realizations with a variety of techniques and we especially focus on a method of regression to equilibrium which we call the onsager analysis with these techniques we establish the statistical equivalence of the real dna sequences with their cmm realizations we show that long range correlations are present in exons as well as in introns but are difficult to detect since the exon dynamics is shown to be determined by the entanglement of three distinct and independent cmm s

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

bootstrapping regression models the regression and correlation models are considered it is shown that the bootstrap approximation to the distribution of the least squares estimates is valid and some error bounds are given

some asymptotic theory for the bootstrap efron s bootstrap method of distribution approximation is shown to be asymptotically valid in a large number of situations including t statistics the empirical and quantile processes and von mises functionals some counter examples are also given to show that the approximation does not always succeed

markov chain order estimation with conditional mutual information abstract we introduce the conditional mutual information cmi for the estimation of the markov chain order for a markov chain of k symbols we define cmi of order m i c m as the mutual information of two variables in the chain being m time steps apart conditioning on the intermediate variables of the chain we find approximate analytic significance limits based on the estimation bias of cmi and develop a randomization significance test of i c m where the randomized symbol sequences are formed by random permutation of the components of the original symbol sequence the significance test is applied for increasing m and the markov chain order is estimated by the last order for which the null hypothesis is rejected we present the appropriateness of cmi testing on monte carlo simulations and compare it to the akaike and bayesian information criteria the maximal fluctuation method peres shields estimator and a likelihood ratio test for increasing orders using divergence the order criterion of cmi testing turns out to be superior for orders larger than one but its effectiveness for large orders depends on data availability in view of the results from the simulations we interpret the estimated orders by the cmi testing and the other criteria on genes and intergenic regions of dna chains

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

multifractal analysis of dna sequences using a novel chaos game representation abstract we present a generalization of the standard chaos game representation method introduced by jeffrey to this aim a dna symbolic sequence is mapped onto a singular measure on the attractor of a particular ifs model which is a perfect statistical representation of the sequence a multifractal analysis of the resulting measure is introduced and an interpretation of singularities in terms of mutual information and redundancy statistical dependence among subsequence symbols within the dna sequence is provided the multifractal spectrum is also shown to be more sensitive for detecting dependence structures within the dna sequence than the averaged contribution given by redundancy this method presents several advantages with respect to other representations such as walks or interfaces which may introduce spurious effects in contrast with the results obtained by other standard methods here we note that no general statement can be made on the influence of coding and non coding content on the correlation length of a given sequence

recurrence time statistics in chaotic dynamics i discrete time maps the dynamics of transitions between the cells of a finite phase space partition in a variety of systems giving rise to chaotic behavior is analyzed with special emphasis on the statistics of recurrence times in the case of one dimensional piecewise markow maps the recurrence problem is cast into a renewal process in the presence of intermittency transitions between cells define a non markovian non renewal process reflected in the presence of power law probability distributions and of divergent variances and mean values

fluctuation bounds for entropy production estimators in gibbs measures we give bounds for the fluctuations of estimators of the mean entropy production in gibbsian sources these bounds are valid for every n where n denotes the size length of the sample we consider two estimators which are based on waiting and hitting times

exact goodness of fit tests for markov chains goodness of fit tests are useful in assessing whether a statistical model is consistent with available data however the usual asymptotics often fail either because of the paucity of the data or because a nonstandard test statistic is of interest in this article we describe exact goodness of fit tests for first and higher order markov chains with particular attention given to time reversible ones the tests are obtained by conditioning on the sufficient statistics for the transition probabilities and are implemented by simple monte carlo sampling or by markov chain monte carlo they apply both to single and to multiple sequences and allow a free choice of test statistic three examples are given the first concerns multiple sequences of dry and wet january days for the years 1948 1983 at snoqualmie falls washington state and suggests that standard analysis may be misleading the second one is for a four state dna sequence and lends support to the original conclusion that a second order markov chain provides an adequate fit to the data the last one is six state atomistic data arising in molecular conformational dynamics simulation of solvated alanine dipeptide and points to strong evidence against a first order reversible markov chain at 6 picosecond time steps

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

measuring correlations in symbol sequences abstract the paper is devoted to relations between correlation functions and mutual information it is shown that in sequences over an alphabet of symbols statistical dependences are measured by 1 2 independent parameters however not all of them can be determined by autocorrelation functions appropriate sets of correlation functions including crosscorrelations are introduced which allow the detection of all dependences the results are exemplified for binary ternary and quaternary symbol sequences as an application it is discussed that a nonuniform codon usage in protein coding dna sequences introduces periodic correlations even at distances in the order of 1000 base pairs

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

factorial moments analyses show a characteristic length scale in dna sequences a unique feature of most of the dna sequences found through the factorial moments analysis is the existence of a characteristic length scale around which the density distribution is nearly poissonian above this point the dna sequences irrespective of their intron contents show long range correlations with a significant deviation from the gaussian statistics while below this point the dna statistics are essentially gaussian the famous dna walk representation is also shown to be a special case of the present analysis

ordinal association in contingency tables some interpretive aspects abstract two families of models for ordered contingency tables goodman s association models and canonical correlation models are investigated and compared with respect to the interpretation of their parameters we show that the two families of models actually refer to different kinds of ordinal association stochastic order extremity for correlation models and stochastic order entropy for association models this difference is related to the way the two models scale interaction the scale difference is proven to be of substantial consequence especially under strong association

keyword extraction by nonextensivity measure the presence of a long range correlation in the spatial distribution of a relevant word type in spite of random occurrences of an irrelevant word type is an important feature of human written texts we classify the correlation between the occurrences of words by nonextensive statistical mechanics for the word ranking process in particular we look at the nonextensivity parameter as an alternative metric to measure the spatial correlation in the text from which the words may be ranked in terms of this measure finally we compare different methods for keyword extraction

tests of goodness of fit based on the l_2 wasserstein distance we consider the wasserstein distance between a sample distribution and the set of normal distributions as a measure of nonnormality by considering the standardized version of this distance we obtain a version of shapiro wilk s test of normality the asymptotic behavior of the statistic is studied using approximations of the quantile process by brownian bridges this method differs from the ad hoc method of de wet and venter and permits a similar analysis for testing other location scale families

dna viewed as an out of equilibrium structure the complexity of the primary structure of human dna is explored using methods from nonequilibrium statistical mechanics dynamical systems theory and information theory a collection of statistical analyses is performed on the dna data and the results are compared with sequences derived from different stochastic processes the use of 2 tests shows that dna can not be described as a low order markov chain of order up to r 6 although detailed balance seems to hold at the level of a binary alphabet it fails when all four base pairs are considered suggesting spatial asymmetry and irreversibility furthermore the block entropy does not increase linearly with the block size reflecting the long range nature of the correlations in the human genomic sequences to probe locally the spatial structure of the chain we study the exit distances from a specific symbol the distribution of recurrence distances and the hurst exponent all of which show power law tails and long range characteristics these results suggest that human dna can be viewed as a nonequilibrium structure maintained in its state through interactions with a constantly changing environment based solely on the exit distance distribution accounting for the nonequilibrium statistics and using the monte carlo rejection sampling method we construct a model dna sequence this method allows us to keep both long and short range statistical characteristics of the native dna data the model sequence presents the same characteristic exponents as the natural dna but fails to capture spatial correlations and point to point details 2014 american physical society

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

poincare recurrences of dna sequences we analyze the statistical properties of poincare recurrences of homo sapiens mammalian and other dna sequences taken from the ensembl genome data base with up to 15 billion base pairs we show that the probability of poincare recurrences decays in an algebraic way with the poincare exponent 4 even if the oscillatory dependence is well pronounced the correlations between recurrences decay with an exponent 0 6 that leads to an anomalous superdiffusive walk however for homo sapiens sequences with the largest available statistics the diffusion coefficient converges to a finite value on distances larger than one million base pairs we argue that the approach based on poncare recurrences determines new proximity features between different species and sheds a new light on their evolution history

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

approaching word length distribution via level spectra abstract treating a text after the removal of paragraphs and punctuations as a spectrum of blanks the distributions of the length of words of ten languages are analyzed using models from the statistical theory of spectra it is found that the ten languages can be classified into two families one with words that follow a wigner like distribution while the words of the other obey a poisson like distribution

probability distribution of intersymbol distances in random symbolic sequences applications to improving detection of keywords in texts and of amino acid clustering in proteins symbolic sequences have been extensively investigated in the past few years within the framework of statistical physics paradigmatic examples of such sequences are written texts and deoxyribonucleic acid dna and protein sequences in these examples the spatial distribution of a given symbol a word a dna motif an amino acid is a key property usually related to the symbol importance in the sequence the more uneven and far from random the symbol distribution the higher the relevance of the symbol to the sequence thus many techniques of analysis measure in some way the deviation of the symbol spatial distribution with respect to the random expectation the problem is then to know the spatial distribution corresponding to randomness which is typically considered to be either the geometric or the exponential distribution however these distributions are only valid for very large symbolic sequences and for many occurrences of the analyzed symbol here we obtain analytically the exact randomly expected spatial distribution valid for any sequence length and any symbol frequency and we study its main properties the knowledge of the distribution allows us to define a measure able to properly quantify the deviation from randomness of the symbol distribution especially for short sequences and low symbol frequency we apply the measure to the problem of keyword detection in written texts and to study amino acid clustering in protein sequences in texts we show how the results improve with respect to previous methods when short texts are analyzed in proteins which are typically short we show how the measure quantifies unambiguously the amino acid clustering and characterize its spatial distribution

the apostol fialkow formula for elementary operators on banach spaces abstract let a a 1 a n and b 1 b n be n tuples of bounded linear operators on a banach space e the corresponding elementary operator e a b is the map s n i 1 a i sb i on l e and e a b denotes the induced operator s n i 1 a i sb i on the calkin algebra c e l e k e here t t k e for t l e we establish that if e has a 1 unconditional basis then dist e a b w c e e a b dist e a b w l e for all elementary operators e a b on l e where w stands for the weakly compact operators there is equality throughout if e l p 1 p canad j math 38 1986 1485 1524 which they proved for e l 2 using the non commutative weyl von neumann theorem due to voiculescu by contrast our arguments are based on subsequence techniques from banach space theory as a byproduct we obtain a positive answer to the generalized fong sourour conjecture for a large class of banach spaces we also explicitly compute the norm of the generalized derivation s as sb on c l 2 this improves a result due to fong and show that the resulting formula fails to hold on c l p

on binary search tree recursions with monomials as toll functions we consider distributional recursions which appear in the study of random binary search trees with monomials as toll functions this extends classical parameters as the internal path length in binary search trees as our main results we derive asymptotic expansions for the moments of the random variables under consideration as well as limit laws and properties of the densities of the limit distributions the analysis is based on the contraction method

a likelihood ratio test against stochastic ordering in several populations abstract the likelihood ratio test is often used to test hypotheses involving a stochastic ordering distribution theory for the likelihood ratio test has been developed only for two stochastically ordered distributions for testing equality of distributions against a stochastic ordering in several populations this paper derives the null asymptotic distribution of the likelihood ratio test statistic which is characterized by minimization problems and has no closed form a monte carlo simulation is conducted to study the limiting distribution because the limiting distribution depends on the specific values of the unknown distributions under the null hypothesis asymptotic and bootstrap approaches are proposed to overcome practical difficulties and implement tests based on the likelihood principle asymptotic validities for these tests are established and simulations are carried out to check their performances for finite sample sizes the tests are applied to an example involving data for survival time f

an analytic approach to the asymptotic variance of trie statistics and related structures we develop analytic tools for the asymptotics of general trie statistics which are particularly advantageous for clarifying the asymptotic variance many concrete examples are discussed for which new fourier expansions are given the tools are also useful for other splitting processes with an underlying binomial distribution we specially highlight philippe flajolet s contribution in the analysis of these random structures

poincare recurrences and multifractal properties of genomic sequences we propose the return times spectra as a tool to analyze the genomic sequences the spectra discriminate between regular chaotic and mixed dynamical systems since generally their decay follows respectively an exponential law a power law or is a linear combination of them the analysis of words of n bases from a dna sequence exhibits the same exponential behavior for the coding and noncoding component the multifractal analysis shows a denite dierence which suggests a less uniform structure for the noncoding part

nearest neighbor spacing distribution of basis in some intron less and intron containing dna sequences abstract we show that the nearest neighbor distribution of distances between basis pairs of some intron less and intron containing coding regions are the same when a procedure called unfolding is applied such a procedure consists in separating the secular variations from the oscillatory terms the form of the distribution obtained is quite similar to that of a random i e poissonian sequence this is done for the humbmyh7cd dromyonma humbmyh7 and dromhc sequences the first two correspond to highly coding regions while the last two correspond to non coding regions we also show that the distributions before the unfolding procedure depend on the secular part but after the unfolding procedure we obtain an striking result all distributions are similar to each other the result becomes independent of the content of introns or the species we have chosen this is in contradiction with the results obtained with the detrended fluctuation analysis in which the correlations yield different results for intron less and intron containing regions

application of information theory to dna sequence analysis a review the analysis of dna sequences through information theory methods is reviewed from the beginning in the 70s the subject is addressed within a broad context describing in some detail the cornerstone contributions in the field the emerging interest concerning long range correlations and the mosaic structure of dna sequences is considered from our own point of view a recent procedure developed by the authors is also outlined

efficient monte carlo procedures for generating points uniformly distributed over bounded regions we consider the monte carlo problem of generating points uniformly distributed within an arbitrary bounded measurable region the class of markovian methods considered generate points asymptotically uniformly distributed within the region computational experience suggests the methods are potentially superior to conventional rejection techniques for large dimensional regions

minima in branching random walks given a branching random walk let m n be the minimum position of any member of the n th generation we calculate mathbfem n to within o 1 and prove exponential tail bounds for mathbfp m n mathbfem n x under quite general conditions on the branching random walk in particular together with work by bramson z wahrsch verw gebiete 45 1978 89 108 our results fully characterize the possible behavior of mathbf em n when the branching random walk has bounded branching and step size

limiting distributions for additive functionals on catalan trees additive tree functionals represent the cost of many divide and conquer algorithms we derive the limiting distribution of the additive functionals induced by toll functions of the form a n when 0 and b log n the so called shape functional on uniformly distributed binary trees sometimes called catalan trees the gaussian law obtained in the latter case complements the central limit theorem for the shape functional under the random permutation model our results give rise to an apparently new family of distributions containing the airy distribution 1 and the normal distribution case b and case a as 0 the main theoretical tools employed are recent results relating asymptotics of the generating functions of sequences to those of their hadamard product and the method of moments

level statistics of words finding keywords in literary texts and symbolic sequences using a generalization of the level statistics analysis of quantum disordered systems we present an approach able to extract automatically keywords in literary texts our approach takes into account not only the frequencies of the words present in the text but also their spatial distribution along the text and is based on the fact that relevant words are significantly clustered i e they self attract each other while irrelevant words are distributed randomly in the text since a reference corpus is not needed our approach is especially suitable for single documents for which no a priori information is available in addition we show that our method works also in generic symbolic sequences continuous texts without spaces thus suggesting its general applicability

statistical properties of dna sequences revisited the role of inverse bilateral symmetry in bacterial chromosomes abstract herein it is shown that in order to study the statistical properties of dna sequences in bacterial chromosomes it suffices to consider only one half of the chromosome because they are similar to its corresponding complementary sequence in the other half this is due to the inverse bilateral symmetry of bacterial chromosomes contrary to the classical result that dna coding regions of bacterial genomes are purely uncorrelated random sequences here it is shown via a renormalization group approach that dna random fluctuations of single bases are modulated by log periodic variations distance series of triplets display long range correlations in each half of the intact chromosome and in protein coding sequences or both long range correlations and log periodic modulations along the whole chromosome hence scaling analyses of distance series of dna sequences have to consider the functional units of bacterial chromosomes

power law exponents characterizing human dna the size distributions of all known coding and noncoding dna sequences are studied in all human chromosomes in a unified approach both introns and intergenic regions are treated as noncoding regions the distributions of noncoding segments p mathit nc s of size s present long tails p mathit nc s ensuremath sim s ensuremath 1 ensuremath ensuremath mu mathit nc with exponents ensuremath mu mathit nc ranging between 0 71 for chromosome 13 and 1 2 for chromosome 19 on the contrary the exponential short range decay terms dominate in the distributions of coding exon segments p c s in all chromosomes aiming to address the emergence of these statistical features minimal stochastic mean field models are proposed based on randomly aggregating dna strings with duplication influx and outflux of genomic segments these minimal models produce both the short range statistics in the coding and the observed power law and fractal statistics in the noncoding dna the minimal models also demonstrate that although the two systems coding and noncoding coexist alternating on the same linear chain they act independently the coding as a closed equilibrium system and the noncoding as an open out of equilibrium one

singularity analysis hadamard products and tree recurrences we present a toolbox for extracting asymptotic information on the coefficients of combinatorial generating functions this toolbox notably includes a treatment of the effect of hadamard products on singularities in the context of the complex tauberian technique known as singularity analysis as a consequence it becomes possible to unify the analysis of a number of divide and conquer algorithms or equivalently random tree models including several classical methods for sorting searching and dynamically managing equivalence relations

an analysis of random d dimensional quad trees it is shown that the depth of the last node inserted in a random quad tree constructed from independent uniform 0 1 d random vectors is in probability asymptotic to 2 d log n where log denotes the natural logarithm in addition for d 2 exact values are obtained for all the moments of the depth of the last node

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

generating random vectors uniformly distributed inside and on the surface of different regions abstract an acceptance rejection algorithm for generating random vectors uniformly distributed over inside or on the surface of a complex region inserted in a minimal multidimensional rectangle is considered for regions having simple forms simplex hypersphere hyperellipsoid several algorithms are presented as well

entropic fluctuations in dna sequences abstract the local shannon entropy lse in blocks is used as a complexity measure to study the information fluctuations along dna sequences the lse of a dna block maps the local base arrangement information to a single numerical value it is shown that despite this reduction of information lse allows to extract meaningful information related to the detection of repetitive sequences in whole chromosomes and is useful in finding evolutionary differences between organisms more specifically large regions of tandem repeats such as centromeres can be detected based on their low lse fluctuations along the chromosome furthermore an empirical investigation of the appropriate block sizes is provided and the relationship of lse properties with the structure of the underlying repetitive units is revealed by using both computational and mathematical methods sequence similarity between the genomic dna of closely related species also leads to similar lse values at the orthologous regions as an application the lse covariance function is used to measure the evolutionary distance between several primate genomes

structure of subspaces of the compact operators having the dunford pettis property the structure of the subspaces m subset k ell p having the dunford pettis property dpp is studied where k ell p is the space of all compact operators on ell p and 1 p infty the following conditions are shown to be equivalent i m has the dpp ii m is isomorphic to a subspace of c 0 iii the sets sx s in b m subset ell p and s x s in b m subset ell p are relatively compact for all x in ell p and x in ell p the equivalence between i and iii was recently proven in the case of arbitrary hilbert spaces by brown and ulger it is also shown that i and ii are equivalent for subspaces m subset k ell p 1 oplus ldots oplus ell p k this result is optimal in the sense that for 1 p q infty there is a dpp subspace m subset k ell q ell p that fails to be isomorphic to a subspace of c 0

statistics of return times a general framework and new applications in this paper we provide general estimates for the errors between the distribution of the first and more generally the k th return time suitably rescaled and the poisson law for measurable dynamical systems in the case that the system exhibits strong mixing properties these bounds are explicitly expressed in terms of the speed of mixing using these approximations the poisson law is finally proved to hold for a large class of non hyperbolic systems on the interval

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

codon distributions in dna the codons 64 in number are distributed over the coding parts of dna sequences the distribution function is the plot of frequency versus rank of the codons these distributions are characterized by parameters that are almost universal i e gene independent there is but a small part that depends on the gene we present the theory to calculate the universal gene independent part the part that is gene specific however has undetermined overlaps and fluctuations

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

lifting non topological divisors of zero modulo the compact operators this paper provides the first examples of non semifredholm operators s on a banach space such that the left or right multiplication operators r sr or r rs define linear embeddings of the corresponding calkin algebra into itself for instance if s is a bounded linear operator on c 0 1 with closed range such that ker s l1 then there is a constant c 0 with dist sr k c 0 1 c dist r k c 0 1 for all bounded operators r l c 0 1 here k c 0 1 stands for the compact operators on c 0 1 moreover if s l1 l1 has closed range and l1 im s contains no copies of l1 then there is a constant c 0 such that dist rs w l1 c dist r w l1 for all r l l1 here w l1 denotes the weakly compact operators on l1