arXiv subjects

Christian Lavault

Publications and source records attributed to Christian Lavault.

At least 19 recordsLinked to original sources

arXiv2017

Integral representations and asymptotic behaviour of a Mittag-Leffler type function of two variables

Integral representations play a prominent role in the analysis of entire functions. The representations of generalized Mittag-Leffler type functions and their asymptotics have been (and still are) investigated by plenty of authors in various conditions and cases.The present paper explores the integral representations of a special function extending to two variables the two-parametric Mittag-Leffler type function. Integral representations of this functions within different variation ranges of its arguments for certain values of the parameters are thus obtained. Asymptotic expansion formulas and asymptotic properties of this function are also established for large values of the variables. This yields corresponding theorems providing integral representations as well as expansion formulas.

math.CV↗

arXiv2017

Integral representations and asymptotic behaviours of Mittag-Leffler type functions of two variables

The paper explores various special functions which generalize the two-parametric Mittag-Leffler type function of two variables. Integral representations for these functions in different domains of variation of arguments for certain values of the parameters are obtained. The asymptotic expansions formulas and asymptotic properties of such functions are also established for large values of the variables. This provides statements of theorems for these formulas and their corresponding properties.

math.FA↗

arXiv2017

Fractional calculus and generalized Mittag-Leffler type functions

In this paper, the generalized fractional integral operators of two generalized Mittag-Leffler type functions are investigated. The special cases of interest involve the generalized Fox--Wright function and the generalized M-series and K-function. In the next Section 2 we first recall some generalized fractional integral operators among the most widely used in fractional calculus. Section 3 is devoted to the definitions of M-series and K-function and their relations to special functions. In Sections 4 and 5, effective fractional calculus of the generalized M-series and the K-function is carried out. The last section briefly concludes and opens up new perspectives. The results established herein generalize recent properties of generalized Mittag-Leffler type functions using left-and right-sided generalized fractional differintegral operators. The note results also in important applications in physics and mathematical engineering.

math.CA↗

arXiv2016

Overview of the Heisenberg--Weyl Algebra and Subsets of Riordan Subgroups

In a first part, we are concerned with the relationships between polynomials in the two generators of the algebra of Heisenberg--Weyl, its Bargmann--Fock representation with differential operators and the associated one-parameter group.Upon this basis, the paper is then devoted to the groups of Riordan matrices associated to the related transformations of matrices (i.e. substitutions with prefunctions). Thereby, various properties are studied arising in Riordan arrays, in the Riordan group and, more specifically, in the `striped' Riordan subgroups; further, a striped quasigroup and a semigroup are also examined. A few applications to combinatorial structures are also briefly addressed in the Appendix.

cs.DM↗

arXiv2014

Improvements on the accelerated integer GCD algorithm

The present paper analyses and presents several improvements to the algorithm for finding the $(a,b)$-pairs of integers used in the $k$-ary reduction of the right-shift $k$-ary integer GCD algorithm. While the worst-case complexity of Weber's "Accelerated integer GCD algorithm" is $\cOł(\log_ϕ(k)^2\r)$, we show that the worst-case number of iterations of the while loop is exactly $\tfrac 12 ł\lfloor \log_ϕ(k)\r\rfloor$, where $ϕ:= \tfrac 12 ł(1+\sqrt{5}\r)$.\par We suggest improvements on the average complexity of the latter algorithm and also present two new faster residual algorithms: the sequential and the parallel one. A lower bound on the probability of avoiding the while loop in our parallel residual algorithm is also given.

cs.DC↗

arXiv2013

A distributed prime sieving algorithm based on Scheduling by Multiple Edge Reversal

This paper presents a new distributed approach for generating all prime numbers in a given interval of integers. From Eratosthenes, who elaborated the first prime sieve (more than 2000 years ago), to the current generation of parallel computers, which have permitted to reach larger bounds on the interval or to obtain previous results in a shorter time, prime numbers generation still represents an attractive domain of research and plays a central role in cryptography. We propose a fully distributed algorithm for finding all primes in the interval $[2\ldots, n]$, based on the \emph{wheel sieve} and the SMER (\emph{Scheduling by Multiple Edge Reversal}) multigraph dynamics. Given a multigraph $\mathcal{M}$ of arbitrary topology, having $N$ nodes, a SMER-driven system is defined by the number of directed edges (arcs) between any two nodes of $\mathcal{M}$, and by the global period length of all "arc reversals" in $\mathcal{M}$. The new prime number generation method inherits the distributed and parallel nature of SMER and requires at most $n + \lfloor \sqrt{n}\rfloor$ time steps. The message complexity achieves at most $nΔ_N + \lfloor \sqrt{n}\rfloor Δ_N$, where $1\le Δ_N\le N - 1$ is the maximal multidegree of $\mathcal{M}$, and the maximal amount of memory space required per process is $\mathcal{O}(n)$ bits.

cs.DC↗

arXiv2013

A Distributed Algorithm for Constructing a Minimum Diameter Spanning Tree

We present a new algorithm, which solves the problem of distributively finding a minimum diameter spanning tree of any (non-negatively) real-weighted graph $G = (V,E,ω)$. As an intermediate step, we use a new, fast, linear-time all-pairs shortest paths distributed algorithm to find an absolute center of $G$. The resulting distributed algorithm is asynchronous, it works for named asynchronous arbitrary networks and achieves $\mathcal{O}(|V|)$ time complexity and $\mathcal{O}\left(|V|\,|E|\right)$

cs.DC↗

arXiv2013

A Uniform Self-Stabilizing Minimum Diameter Spanning Tree Algorithm

We present a uniform self-stabilizing algorithm, which solves the problem of distributively finding a minimum diameter spanning tree of an arbitrary positively real-weighted graph. Our algorithm consists in two stages of stabilizing protocols. The first stage is a uniform randomized stabilizing {\em unique naming} protocol, and the second stage is a stabilizing {\em MDST} protocol, designed as a {\em fair composition} of Merlin--Segall's stabilizing protocol and a distributed deterministic stabilizing protocol solving the (MDST) problem. The resulting randomized distributed algorithm presented herein is a composition of the two stages; it stabilizes in $O(nΔ+{\cal D}^2 + n \log\log n)$ expected time, and uses $O(n^2\log n + n \log W)$ memory bits (where $n$ is the order of the graph, $Δ$ is the maximum degree of the network, $\cal D$ is the diameter in terms of hops, and $W$ is the largest edge weight). To our knowledge, our protocol is the very first distributed algorithm for the (MDST) problem. Moreover, it is fault-tolerant and works for any anonymous arbitrary network.

cs.DC↗

arXiv2013

A New Modular Division Algorithm and Applications

The present paper proposes a new parallel algorithm for the modular division $u/v\bmod β^s$, where $u,\; v,\; β$ and $s$ are positive integers $(β\ge 2)$. The algorithm combines the classical add-and-shift multiplication scheme with a new propagation carry technique. This "Pen and Paper Inverse" ({\em PPI}) algorithm, is better suited for systolic parallelization in a "least-significant digit first" pipelined manner. Although it is equivalent to Jebelean's modular division algorithm~\cite{jeb2} in terms of performance (time complexity, work, efficiency), the linear parallelization of the {\em PPI} algorithm improves on the latter when the input size is large. The parallelized versions of the {\em PPI} algorithm leads to various applications, such as the exact division and the digit modulus operation (dmod) of two long integers. It is also applied to the determination of the periods of rational numbers as well as their $p$-adic expansion in any radix $β\ge 2$.

cs.DC↗

arXiv2013

Worst-Case Analysis of Weber's Algorithm

Recently, Ken Weber introduced an algorithm for finding the $(a,b)$-pairs satisfying $au+bv\equiv 0\pmod{k}$, with $0<|a|,|b|<\sqrt{k}$, where $(u,k)$ and $(v,k)$ are coprime. It is based on Sorenson's and Jebelean's "$k$-ary reduction" algorithms. We provide a formula for $N(k)$, the maximal number of iterations in the loop of Weber's GCD algorithm.

cs.DS↗

arXiv2013

Multiplicate inverse forms of terminating hypergeometric series

The multiplicate form of Gould--Hsu's inverse series relations enables to investigate the dual relations of the Chu-Vandermonde-Gauß's, the Pfaff-Saalschütz's summation theorems and the binomial convolution formula due to Hagen and Rothe. Several identitity and reciprocal relations are thus established for terminating hypergeometric series. By virtue of the duplicate inversions, we establish several dual formulae of Chu-Vandermonde-Gauß's and Pfaff-Saalschütz's summation theorems in Section (3)\cite{ChuVanGauss} and (4)\cite{PfaffSaalsch}, respectively. Finally, the last section is devoted to deriving several identities and reciprocal relations for terminating balanced hypergeometric series from Hagen-Rothe's convolution identity in accordance with the duplicate, triplicate and multiplicate inversions.

math.CO↗

arXiv2011

A note on Prüfer-like coding and counting forests of uniform hypertrees

This note presents an encoding and a decoding algorithms for a forest of (labelled) rooted uniform hypertrees and hypercycles in linear time, by using as few as $n - 2$ integers in the range $[1,n]$. It is a simple extension of the classical Prüfer code for (labelled) rooted trees to an encoding for forests of (labelled) rooted uniform hypertrees and hypercycles, which allows to count them up according to their number of vertices, hyperedges and hypertrees. In passing, we also find Cayley's formula for the number of (labelled) rooted trees as well as its generalisation to the number of hypercycles found by Selivanov in the early 70's.

cs.DM↗

arXiv2010

Reliable Self-Stabilizing Communication for Quasi Rendezvous

The paper presents three self-stabilizing protocols for basic fair and reliable link communication primitives. We assume a link-register communication model under read/write atomicity, where every process can read from but cannot write into its neighbours' registers. The first primitive guarantees that any process writes a new value in its register(s) only after all its neighbours have read the previous value, whatever the initial scheduling of processes' actions. The second primitive implements a "weak rendezvous" communication mechanism by using an alternating bit protocol: whenever a process consecutively writes n values (possibly the same ones) in a register, each neighbour is guaranteed to read each value from the register at least once. On the basis of the previous protocol, the third primitive implements a "quasi rendezvous": in words, this primitive ensures furthermore that there exists exactly one reading between two writing operations All protocols are self-stabilizing and run in asynchronous arbitrary networks. The goal of the paper is in handling each primitive by a separate procedure, which can be used as a "black box" in more involved self-stabilizing protocols.

cs.DC↗

arXiv2008

Convergence of some leader election algorithms

We start with a set of n players. With some probability P(n,k), we kill n-k players; the other ones stay alive, and we repeat with them. What is the distribution of the number X_n of phases (or rounds) before getting only one player? We present a probabilistic analysis of this algorithm under some conditions on the probability distributions P(n,k), including stochastic monotonicity and the assumption that roughly a fixed proportion alpha of the players survive in each round. We prove a kind of convergence in distribution for X_n-log_a n, where the basis a=1/alpha; as in many other similar problems there are oscillations and no true limit distribution, but suitable subsequences converge, and there is an absolutely continuous random variable Z such that the distribution of X_n can be approximated by Z+log_a n rounded to the nearest larger integer. Applications of the general result include the leader election algorithm where players are eliminated by independent coin tosses and a variation of the leader election algorithm proposed by W.R. Franklin. We study the latter algorithm further, including numerical results.

cs.DC↗

arXiv2006

Analysis of an Efficient Distributed Algorithm for Mutual Exclusion (Average-Case Analysis of Path Reversal)

The algorithm analysed by Naïmi, Trehe and Arnold was the very first distributed algorithm to solve the mutual exclusion problem in complete networks by using a dynamic logical tree structure as its basic distributed data structure, viz. a path reversal transformation in rooted n-node trees; besides, it was also the first one to achieve a logarithmic average-case message complexity. The present paper proposes a direct and general approach to compute the moments of the cost of path reversal. It basically uses one-one correspondences between combinatorial structures and the associated probability generating functions: the expected cost of path reversal is thus proved to be exactly $H_{n-1}$. Moreover, time and message complexity of the algorithm as well as randomized bounds on its worst-case message complexity in arbitrary networks are also given. The average-case analysis of path reversal and the analysis of this distributed algorithm for mutual exclusion are thus fully completed in the paper. The general techniques used should also prove available and fruitful when adapted to the most efficient recent tree-based distributed algorithms for mutual exclusion which require powerful tools, particularly for average-case analyses.

cs.DC↗

arXiv2006

A distributed approximation algorithm for the minimum degree minimum weight spanning trees

Fischer has shown how to compute a minimum weight spanning tree of degree at most $b Δ^* + \lceil \log\_b n\rceil$ in time $O(n^{4 + 1/\ln b})$ for any constant $b > 1$, where $Δ^*$ is the value of an optimal solution and $n$ is the number of nodes in the network. In this paper, we propose a distributed version of Fischer's algorithm that requires messages and time complexity $O(n^{2 + 1/\ln b})$, and O(n) space per node.

cs.DC↗

arXiv2006

Asymptotic Analysis of a Leader Election Algorithm

Itai and Rodeh showed that, on the average, the communication of a leader election algorithm takes no more than $LN$ bits, where $L \simeq 2.441716$ and $N$ denotes the size of the ring. We give a precise asymptotic analysis of the average number of rounds M(n) required by the algorithm, proving for example that $\dis M(\infty) := \lim\_{n\to \infty} M(n) = 2.441715879...$, where $n$ is the number of starting candidates in the election. Accurate asymptotic expressions of the second moment $M^{(2)}(n)$ of the discrete random variable at hand, its probability distribution, and the generalization to all moments are given. Corresponding asymptotic expansions $(n\to \infty)$ are provided for sufficiently large $j$, where $j$ counts the number of rounds. Our numerical results show that all computations perfectly fit the observed values. Finally, we investigate the generalization to probability $t/n$, where $t$ is a non negative real parameter. The real function $\dis M(\infty,t) := \lim\_{n\to \infty} M(n,t)$ is shown to admit \textit{one unique minimum} $M(\infty,t^{*})$ on the real segment $(0,2)$. Furthermore, the variations of $M(\infty,t)$ on thewhole real line are also studied in detail.

cs.DC↗

arXiv2006

Quasi-Optimal Leader Election Algorithms in Radio Networks with Loglogarithmic Awake Time Slots

A radio network (RN) is a distributed system consisting of $n$ radio stations. We design and analyze two distributed leader election protocols in RN where the number $n$ of radio stations is unknown. The first algorithm runs under the assumption of {\it limited collision detection}, while the second assumes that {\it no collision detection} is available. By ``limited collision detection'', we mean that if exactly one station sends (broadcasts) a message, then all stations (including the transmitter) that are listening at this moment receive the sent message. By contrast, the second no-collision-detection algorithm assumes that a station cannot simultaneously send and listen signals. Moreover, both protocols allow the stations to keep asleep as long as possible, thus minimizing their awake time slots (such algorithms are called {\it energy-efficient}). Both randomized protocols in RN areshown to elect a leader in $O(\log{(n)})$ expected time, with no station being awake for more than $O(\log{\log{(n)}})$ time slots. Therefore, a new class of efficient algorithms is set up that match the $Ω(\log{(n)})$ time lower-bound established by Kushilevitz and Mansour.

cs.DC↗