scholarly journals Average profiles, from tries to suffix-trees

2005 ◽  
Vol DMTCS Proceedings vol. AD,... (Proceedings) ◽  
Author(s):  
Pierre Nicodème
Keyword(s):  
Generating Function ◽  
Mellin Transform ◽  
Poisson Model ◽  
Second Order ◽  
Suffix Trees ◽  
Phase Reversal ◽  
Poisson Law ◽  
International Audience ◽  
Bivariate Generating Function ◽  
Saturation Rate

International audience We build upon previous work of Fayolle (2004) and Park and Szpankowski (2005) to study asymptotically the average internal profile of tries and of suffix-trees. The binary keys and the strings are built from a Bernoulli source $(p,q)$. We consider the average number $p_{k,\mathcal{P}}(\nu)$ of internal nodes at depth $k$ of a trie whose number of input keys follows a Poisson law of parameter $\nu$. The Mellin transform of the corresponding bivariate generating function has a major singularity at the origin, which implies a phase reversal for the saturation rate $p_{k,\mathcal{P}}(\nu)/2^k$ as $k$ reaches the value $2\log(\nu)/(\log(1/p)+\log(1/q))$. We prove that the asymptotic average profiles of random tries and suffix-trees are mostly similar, up to second order terms, a fact that has been experimentally observed in Nicodème (2003); the proof follows from comparisons to the profile of tries in the Poisson model.

scholarly journals Simultaneous generation for zeta values by the Markov-WZ method

2008 ◽  
Vol Vol. 10 no. 3 (Combinatorics) ◽  
Author(s):  
Khodabakhsh Hessami Pilehrood ◽  
Tatiana Hessami Pilehrood
Keyword(s):  
Generating Function ◽  
Simultaneous Generation ◽  
Function Identity ◽  
Zeta Values ◽  
International Audience ◽  
Bivariate Generating Function ◽  
Wz Method

Combinatorics International audience By application of the Markov-WZ method, we prove a more general form of a bivariate generating function identity containing, as particular cases, Koecher's and Almkvist-Granville's Apéry-like formulae for odd zeta values. As a consequence, we get a new identity producing Apéry-like series for all ζ(2n+4m+3),n,m ≥ 0, convergent at the geometric rate with ratio 2−10.

scholarly journals Monadic Second-Order Classes of Forests with a Monadic Second-Order 0-1 Law

2012 ◽  
Vol Vol. 14 no. 1 (Automata, Logic and Semantics) ◽  
Author(s):  
Jason P. Bell ◽  
Stanley N. Burris ◽  
Karen A. Yeats
Keyword(s):  
Explicit Expression ◽  
Generating Function ◽  
Generating Functions ◽  
Second Order ◽  
Radius Of Convergence ◽  
Structure Theory ◽  
Strengthened Version ◽  
International Audience

Automata, Logic and Semantics International audience Let T be a monadic-second order class of finite trees, and let T(x) be its (ordinary) generating function, with radius of convergence rho. If rho >= 1 then T has an explicit specification (without using recursion) in terms of the operations of union, sum, stack, and the multiset operators n and (>= n). Using this, one has an explicit expression for T(x) in terms of the initial functions x and x . (1 - x(n))(-1), the operations of addition and multiplication, and the Polya exponentiation operators E-n, E-(>= n). Let F be a monadic-second order class of finite forests, and let F (x) = Sigma(n) integral(n)x(n) be its (ordinary) generating function. Suppose F is closed under extraction of component trees and sums of forests. Using the above-mentioned structure theory for the class T of trees in F, Compton's theory of 0-1 laws, and a significantly strengthened version of 2003 results of Bell and Burris on generating functions, we show that F has a monadic second-order 0-1 law iff the radius of convergence of F (x) is 1 iff the radius of convergence of T (x) is >= 1.

scholarly journals One-sided Variations on Tries: Path Imbalance, Climbing, and Key Sampling

2007 ◽  
Vol DMTCS Proceedings vol. AH,... (Proceedings) ◽  
Author(s):  
Costas A. Christophi ◽  
Hosam M. Mahmoud
Keyword(s):  
Probability Distribution ◽  
Generating Function ◽  
Path Length ◽  
Mellin Transform ◽  
Moment Generating Function ◽  
Exact Probability ◽  
Digital Trees ◽  
Exact Probability Distribution ◽  
International Audience ◽  
The Mean

International audience One-sided variations on path length in a trie (a sort of digital trees) are investigated: They include imbalance factors, climbing under different strategies, and key sampling. For the imbalance factor accurate asymptotics for the mean are derived for a randomly chosen key in the trie via poissonization and the Mellin transform, and the inverse of the two operations. It is also shown from an analysis of the moving poles of the Mellin transform of the poissonized moment generating function that the imbalance factor (under appropriate centering and scaling) follows a Gaussian limit law. The method extends to several variations of sampling keys from a trie and we sketch results of climbing under different strategies. The exact probability distribution is computed in one case, to demonstrate that such calculations can be done, at least in principle.

scholarly journals Congruence successions in compositions

2014 ◽  
Vol Vol. 16 no. 1 (Combinatorics) ◽  
Author(s):  
Toufik Mansour ◽  
Mark Shattuck ◽  
Mark Wilson
Keyword(s):  
General Formula ◽  
Generating Function ◽  
Asymptotic Estimate ◽  
International Audience ◽  
Limiting Case ◽  
Positive Integers

Combinatorics International audience A composition is a sequence of positive integers, called parts, having a fixed sum. By an m-congruence succession, we will mean a pair of adjacent parts x and y within a composition such that x=y(modm). Here, we consider the problem of counting the compositions of size n according to the number of m-congruence successions, extending recent results concerning successions on subsets and permutations. A general formula is obtained, which reduces in the limiting case to the known generating function formula for the number of Carlitz compositions. Special attention is paid to the case m=2, where further enumerative results may be obtained by means of combinatorial arguments. Finally, an asymptotic estimate is provided for the number of compositions of size n having no m-congruence successions.

scholarly journals Counting occurrences for a finite set of words: an inclusion-exclusion approach

2007 ◽  
Vol DMTCS Proceedings vol. AH,... (Proceedings) ◽  
Author(s):  
Frédérique Bassino ◽  
Julien Clément ◽  
J. Fayolle ◽  
P. Nicodème
Keyword(s):  
Generating Function ◽  
Exclusion Principle ◽  
Complete Proof ◽  
Detailed Proof ◽  
Finite Set ◽  
International Audience ◽  
The One ◽  
Maple Package

International audience In this paper, we give the multivariate generating function counting texts according to their length and to the number of occurrences of words from a finite set. The application of the inclusion-exclusion principle to word counting due to Goulden and Jackson (1979, 1983) is used to derive the result. Unlike some other techniques which suppose that the set of words is reduced (<i>i..e.</i>, where no two words are factor of one another), the finite set can be chosen arbitrarily. Noonan and Zeilberger (1999) already provided a MAPLE package treating the non-reduced case, without giving an expression of the generating function or a detailed proof. We give a complete proof validating the use of the inclusion-exclusion principle and compare the complexity of the method proposed here with the one using automata for solving the problem.

scholarly journals On the Number of 2-Protected Nodes in Tries and Suffix Trees

2012 ◽  
Vol DMTCS Proceedings vol. AQ,... (Proceedings) ◽  
Author(s):  
Jeffrey Gaither ◽  
Yushi Homma ◽  
Mark Sellke ◽  
Mark Daniel Ward
Keyword(s):  
Suffix Trees ◽  
First Order ◽  
International Audience

International audience We use probabilistic and combinatorial tools on strings to discover the average number of 2-protected nodes in tries and in suffix trees. Our analysis covers both the uniform and non-uniform cases. For instance, in a uniform trie with $n$ leaves, the number of 2-protected nodes is approximately 0.803$n$, plus small first-order fluctuations. The 2-protected nodes are an emerging way to distinguish the interior of a tree from the fringe.

scholarly journals Statistics on Lattice Walks and q-Lassalle Numbers

2015 ◽  
Vol DMTCS Proceedings, 27th... (Proceedings) ◽  
Author(s):  
Lenny Tevlin
Keyword(s):  
Positive Integer ◽  
Generating Function ◽  
Catalan Numbers ◽  
Binomial Coefficients ◽  
Integer Coefficients ◽  
Major Index ◽  
Lattice Walks ◽  
International Audience ◽  
The Difference ◽  
Positive Integers

International audience This paper contains two results. First, I propose a $q$-generalization of a certain sequence of positive integers, related to Catalan numbers, introduced by Zeilberger, see Lassalle (2010). These $q$-integers are palindromic polynomials in $q$ with positive integer coefficients. The positivity depends on the positivity of a certain difference of products of $q$-binomial coefficients.To this end, I introduce a new inversion/major statistics on lattice walks. The difference in $q$-binomial coefficients is then seen as a generating function of weighted walks that remain in the upper half-plan. Cet document contient deux résultats. Tout d’abord, je vous propose un $q$-generalization d’une certaine séquence de nombres entiers positifs, liés à nombres de Catalan, introduites par Zeilberger (Lassalle, 2010). Ces $q$-integers sont des polynômes palindromiques à $q$ à coefficients entiers positifs. La positivité dépend de la positivité d’une certaine différence de produits de $q$-coefficients binomial.Pour ce faire, je vous présente une nouvelle inversion/major index sur les chemins du réseau. La différence de $q$-binomial coefficients est alors considérée comme une fonction de génération de trajets pondérés qui restent dans le demi-plan supérieur.

scholarly journals Assessment of low-velocity impact damage in composites by the measure of second-harmonic guided waves with the phase-reversal approach

2019 ◽  
Vol 103 (1) ◽  
pp. 003685041988107 ◽  
Author(s):  
Weibin Li ◽  
Chang Jiang ◽  
Xinlin Qing ◽  
Liangbing Liu ◽  
Mingxi Deng
Keyword(s):  
Guided Waves ◽  
Second Harmonic ◽  
Impact Damage ◽  
Guided Wave ◽  
Second Order ◽  
Low Velocity Impact ◽  
Low Velocity ◽  
Velocity Impact ◽  
Phase Reversal ◽  
The Impact

Structural strength and integrity of composites can be considerably affected by the low-velocity impact damage due to the unique characteristics of composites, such as layering bonded by adhesive and the weakness to impact. For such damage, there is an urgent need to develop advanced nondestructive testing approaches. Despite the fact that the second harmonics could provide information sensitive to the structural health condition, the diminutive amplitude of the measured second-order harmonic guided wave still limits the applications of the second-harmonic generation–based nonlinear guided wave approach. Herein, laminated composites suffered from low-velocity impact are characterized by use of nonlinear guided waves. An enhancement in the signal-to-noise ratio for the measure of second harmonics is achieved by a phase-reversal method. Results obtained indicate a monotonic correlation between the impact-induced damage in composites and the relative acoustic nonlinear indicator of guided waves. The experimental finding in this study shows that the measure of second-order harmonic guided waves with a phase-reversal method can be a promising indicator to impact damage rendering in an improved and reliable manner.

scholarly journals Mellin’s Transform and Application to Some Time Series Models

2014 ◽  
Vol 2014 ◽  
pp. 1-12 ◽  
Author(s):  
S. S. Appadoo ◽  
A. Thavaneswaran ◽  
S. Mandal
Keyword(s):  
Time Series ◽  
Generating Function ◽  
Mellin Transform ◽  
Fuzzy Numbers ◽  
Moment Generating Function ◽  
Time Series Models ◽  
Form Expression ◽  
The Moment ◽  
Close Form ◽  
Skewness And Kurtosis

This paper uses the Mellin transform to establish the means, variances, skewness, and kurtosis of fuzzy numbers and applied them to the random coefficient autoregressive (RCA) time series models. We also give a close form expression to the moment generating function related to fuzzy numbers. It is shown that the results of the proposed time series models are consistent with those of the conventional time series models and that the developed concepts are straightforward and easily implemented.

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