scholarly journals Exponential bounds and tails for additive random recursive sequences

2007 ◽  
Vol Vol. 9 no. 1 (Analysis of Algorithms) ◽  
Author(s):  
Ludger Rüschendorf ◽  
Eva-Maria Schopp
Keyword(s):  
Analysis Of Algorithms ◽  
Sufficient Conditions ◽  
Random Trees ◽  
Divide And Conquer ◽  
Recursive Sequences ◽  
Recursive Structures ◽  
International Audience ◽  
Exponential Bounds ◽  
The Moment

Analysis of Algorithms International audience Exponential bounds and tail estimates are derived for additive random recursive sequences, which typically arise as functionals of recursive structures, of random trees or in recursive algorithms. In particular they arise as parameters of divide and conquer type algorithms. We derive tail bounds from estimates of the Laplace transforms and of the moment sequences. For the proof we use some classical exponential bounds and some variants of the induction method. The paper generalizes results of Rösler (% \citeyearNPRoesler:91, % \citeyearNPRoesler:92) and % \citeNNeininger:05 on subgaussian tails to more general classes of additive random recursive sequences. It also gives sufficient conditions for tail bounds of the form \exp(-a t^p) which are based on a characterization of \citeNKasahara:78.

scholarly journals Polynomial tails of additive-type recursions

2008 ◽  
Vol DMTCS Proceedings vol. AI,... (Proceedings) ◽  
Author(s):  
Eva-Maria Schopp
Keyword(s):  
Tail Probability ◽  
Upper Bounds ◽  
Random Trees ◽  
Divide And Conquer ◽  
Variation Theory ◽  
Recursive Sequences ◽  
Recursive Structures ◽  
International Audience ◽  
Asymptotic Tail Probability ◽  
Polynomial Bounds

International audience Polynomial bounds and tail estimates are derived for additive random recursive sequences, which typically arise as functionals of recursive structures, of random trees, or in recursive algorithms. In particular they arise as parameters of divide and conquer type algorithms. We mainly focuss on polynomial tails that arise due to heavy tail bounds of the toll term and the starting distributions. Besides estimating the tail probability directly we use a modified version of a theorem from regular variation theory. This theorem states that upper bounds on the asymptotic tail probability can be derived from upper bounds of the Laplace―Stieltjes transforms near zero.

scholarly journals On rates of convergence in the probabilistic analysis of algorithms

2021 ◽  
Author(s):  
◽  
Jasmin Straub
Keyword(s):  
Normal Limit ◽  
Analysis Of Algorithms ◽  
Rates Of Convergence ◽  
Random Trees ◽  
Divide And Conquer ◽  
Complexity Measures ◽  
Limit Laws ◽  
Contraction Method ◽  
Normal Limits ◽  
Recursive Structures

Within the last thirty years, the contraction method has become an important tool for the distributional analysis of random recursive structures. While it was mainly developed to show weak convergence, the contraction approach can additionally be used to obtain bounds on the rate of convergence in an appropriate metric. Based on ideas of the contraction method, we develop a general framework to bound rates of convergence for sequences of random variables as they mainly arise in the analysis of random trees and divide-and-conquer algorithms. The rates of convergence are bounded in the Zolotarev distances. In essence, we present three different versions of convergence theorems: a general version, an improved version for normal limit laws (providing significantly better bounds in some examples with normal limits) and a third version with a relaxed independence condition. Moreover, concrete applications are given which include parameters of random trees, quantities of stochastic geometry as well as complexity measures of recursive algorithms under either a random input or some randomization within the algorithm.

scholarly journals A survey of multivariate aspects of the contraction method

2006 ◽  
Vol Vol. 8 ◽  
Author(s):  
Ralph Neininger ◽  
Ludger Rüschendorf
Keyword(s):  
Limit Theorems ◽  
Analysis Of Algorithms ◽  
Branching Processes ◽  
Random Trees ◽  
Open Problems ◽  
Limit Laws ◽  
Contraction Method ◽  
Recursive Sequences ◽  
International Audience ◽  
General Conditions

International audience We survey multivariate limit theorems in the framework of the contraction method for recursive sequences as arising in the analysis of algorithms, random trees or branching processes. We compare and improve various general conditions under which limit laws can be obtained, state related open problems and give applications to the analysis of algorithms and branching recurrences.

scholarly journals Note on the weighted internal path length of b-ary trees

2007 ◽  
Vol Vol. 9 no. 1 (Analysis of Algorithms) ◽  
Author(s):  
Ludger Rüschendorf ◽  
Eva-Maria Schopp
Keyword(s):  
Limit Theorem ◽  
Continuous Time ◽  
Path Length ◽  
Analysis Of Algorithms ◽  
Random Variables ◽  
Time Parameter ◽  
Recursive Sequences ◽  
International Audience

Analysis of Algorithms International audience In a recent paper Broutin and Devroye (2005) have studied the height of a class of edge-weighted random trees.This is a class of trees growing in continuous time which includes many wellknown trees as examples. In this paper we derive a limit theorem for the internal path length for this class of trees.For the proof we extend a limit theorem in Neininger and Rüschendorf (2004) to recursive sequences of random variables with continuous time parameter.

scholarly journals Tail Bounds for the Wiener Index of Random Trees

2007 ◽  
Vol DMTCS Proceedings vol. AH,... (Proceedings) ◽  
Author(s):  
Tämur Ali Khan ◽  
Ralph Neininger
Keyword(s):  
Lower Bounds ◽  
Random Search ◽  
Wiener Index ◽  
Moment Generating Function ◽  
Random Trees ◽  
Upper And Lower Bounds ◽  
Search Trees ◽  
Tail Probabilities ◽  
International Audience ◽  
The Moment

International audience Upper and lower bounds for the tail probabilities of the Wiener index of random binary search trees are given. For upper bounds the moment generating function of the vector of Wiener index and internal path length is estimated. For the lower bounds a tree class with sufficiently large probability and atypically large Wiener index is constructed. The methods are also applicable to related random search trees.

scholarly journals Additive tree functionals with small toll functions and subtrees of random trees

2012 ◽  
Vol DMTCS Proceedings vol. AQ,... (Proceedings) ◽  
Author(s):  
Stephan Wagner
Keyword(s):  
Central Limit Theorem ◽  
Limit Theorem ◽  
Central Limit ◽  
Random Trees ◽  
Divide And Conquer ◽  
Main Motivation ◽  
Additive Tree ◽  
International Audience ◽  
Number Of Leaves ◽  
Labelled Trees

International audience Many parameters of trees are additive in the sense that they can be computed recursively from the sum of the branches plus a certain toll function. For instance, such parameters occur very frequently in the analysis of divide-and-conquer algorithms. Here we are interested in the situation that the toll function is small (the average over all trees of a given size $n$ decreases exponentially with $n$). We prove a general central limit theorem for random labelled trees and apply it to a number of examples. The main motivation is the study of the number of subtrees in a random labelled tree, but it also applies to classical instances such as the number of leaves.

scholarly journals On a New Geometric Constant Related to the Euler-Lagrange Type Identity in Banach Spaces

Mathematics ◽  
2021 ◽  
Vol 9 (2) ◽  
pp. 116
Author(s):  
Qi Liu ◽  
Yongjin Li
Keyword(s):  
Banach Spaces ◽  
Product Space ◽  
Sufficient Conditions ◽  
Normal Structure ◽  
The Other ◽  
Inner Product ◽  
Equivalent Characterization ◽  
Geometric Constant ◽  
Geometric Constants

In this paper, we will introduce a new geometric constant LYJ(λ,μ,X) based on an equivalent characterization of inner product space, which was proposed by Moslehian and Rassias. We first discuss some equivalent forms of the proposed constant. Next, a characterization of uniformly non-square is given. Moreover, some sufficient conditions which imply weak normal structure are presented. Finally, we obtain some relationship between the other well-known geometric constants and LYJ(λ,μ,X). Also, this new coefficient is computed for X being concrete space.

scholarly journals On cryptographic properties of (n + 1)-bit S-boxes constructed by known n-bit S-boxes

2020 ◽  
Vol 15 (1) ◽  
pp. 258-265
Author(s):  
Yu Zhou ◽  
Daoguang Mu ◽  
Xinfeng Dong
Keyword(s):  
Sufficient Conditions ◽  
Sum Of Squares ◽  
Basic Component ◽  
Necessary And Sufficient Conditions ◽  
Autocorrelation Functions ◽  
Walsh Spectrum ◽  
Cryptographic Algorithms ◽  
Necessary And Sufficient ◽  
Relationship Of

AbstractS-box is the basic component of symmetric cryptographic algorithms, and its cryptographic properties play a key role in security of the algorithms. In this paper we give the distributions of Walsh spectrum and the distributions of autocorrelation functions for (n + 1)-bit S-boxes in [12]. We obtain the nonlinearity of (n + 1)-bit S-boxes, and one necessary and sufficient conditions of (n + 1)-bit S-boxes satisfying m-order resilient. Meanwhile, we also give one characterization of (n + 1)-bit S-boxes satisfying t-order propagation criterion. Finally, we give one relationship of the sum-of-squares indicators between an n-bit S-box S0 and the (n + 1)-bit S-box S (which is constructed by S0).

Central automorphisms of free nilpotent Lie algebras

2017 ◽  
Vol 16 (11) ◽  
pp. 1750205
Author(s):  
Özge Öztekin ◽  
Naime Ekici
Keyword(s):  
Lie Algebras ◽  
Finite Rank ◽  
Characteristic Zero ◽  
Sufficient Conditions ◽  
Nilpotency Class ◽  
Nilpotent Lie Algebra ◽  
Central Automorphism ◽  
Nilpotent Lie Algebras ◽  
Central Automorphisms

Let [Formula: see text] be the free nilpotent Lie algebra of finite rank [Formula: see text] [Formula: see text] and nilpotency class [Formula: see text] over a field of characteristic zero. We give a characterization of central automorphisms of [Formula: see text] and we find sufficient conditions for an automorphism of [Formula: see text] to be a central automorphism.

La structure de classes québécoise au 19e siècle et le problème de l'articulation des modes de production

1981 ◽  
Vol 14 (3) ◽  
pp. 487-518
Author(s):  
Gérald Bernier
Keyword(s):  
Census Data ◽  
New France ◽  
Class Structure ◽  
Capitalist Development ◽  
Single Class ◽  
The Moment ◽  
Empirical Demonstration ◽  
Work Done ◽  
The Impact

The study of social classes in the nineteenth century requires the development of conceptual tools able to explain the impact of the Conquest on the pre-existant social structures in determining transformations of the class structure during the subsequent decades.This article examines the work done on this question by Marxist writers. The author criticizes certain conclusions which have been drawn and which suggest deficiencies at a theoretical level. The objections relate to the marked tendency of these conclusions to perceive the structural effects of the Conquest in terms of the formation of a double-class structure characterized by “ethnic origins.” Specifically, the author challenges the notion of the division itself, as well as the criterion on which the division is based.The author proposes that an analysis centred upon the concepts relating to a problem of the transition and linkage of different modes of production permits a more satisfying interpretation, if accompanied by a certain number of considerations of the “upside” and “downside” of the Conquest. To this end, the argument is based on a characterization of New France in terms of the domination of the relations of production of the feudal type and on an analysis of metropolitan centres with intent to evaluate their level of capitalist development at the moment of their respective colonial penetration in Canada. The results of this approach permit one to posit the existence of a single-class structure, characterized principally by the existence of elements connecting diverse modes and forms of production, whose origin reflects the unequal state of economic development in the two metropolitan centres.The empirical demonstration rests on the census data of 1851–1852 and on the complementary information drawn from the works of historians.

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